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\begin{document}
\title{The Weyl group of the fine grading of $sl(n,\mathbb{C})$ associated with tensor product of generalized Pauli matrices \footnote{Research supported by NSFC Grant No.10801116 and by
'the Fundamental Research Funds for the Central Universities'}}
\author{Gang HAN \\
{Department of Mathematics, Zhejiang University,}\\{Hangzhou 310027, China}\\
{E-mail: mathhg@hotmail.com}\\Telephone: 086-0571-87953843\\Fax:
086-0571-87953794 }
\date{January 24, 2011}
\maketitle
{\small \noindent \textbf{Abstract}. We consider the fine grading
of $sl(n,\mb C)$ induced by tensor product of generalized Pauli
matrices in the paper. Based on the classification of maximal
diagonalizable subgroups of $PGL(n,\mb C)$ by Havlicek, Patera and
Pelantova, we prove that any finite maximal diagonalizable subgroup
$K$ of $PGL(n,\mb C)$ is a symplectic abelian group and its Weyl
group, which describes the symmetry of the fine grading induced by
the action of $K$, is
just the isometry group of the symplectic abelian group $K$.
For a finite symplectic abelian group, it is also proved that its isometry group is always generated by the transvections
contained in it.
}
\section{Introduction}
\setcounter{equation}{0}\setcounter{theorem}{0}
The study of gradings of Lie algebras and the symmetries of those
gradings is an active research area in recent decades, which are
interesting to both mathematicians and physicians. In physics, Lie
algebras usually play the role as the algebra of infinitesimal
symmetries of a physical system. Knowledge about the gradings of a
Lie algebra will greatly help us to understand better the structure
of the Lie algebra. Study of the symmetries of those gradings offers
a very important tool for describing symmetries in the system of
nonlinear equations connected with contraction of a Lie algebra (see
e.g. \cite{jpt}).
Besides the
famous Cartan decomposition for semisimple Lie algebras, another
well-known example of grading is the grading of $sl(n,\mb C)$ by the
adjoint action of the Pauli group $\Pi_n$ generated by the $n\times
n$ generalized Pauli matrices, which decomposes $sl(n,\mb C)$ into
direct sum of $n^2-1$ one-dimensional subspaces, each of which
consists of semisimple elements.
Let $L$ be a complex simple Lie algebra. Let $\Aut(\L)$ and $\Int(L)$ be respectively the automorphism group and inner automorphism group of $L$, which are
both algebraic groups. A subgroup of $\Aut(\L)$ or $\Int(L)$ is
called \textit{diagonalizable} if it is abelian and consists of
semisimple elements. It is not hard to see that there is a natural
1-1 correspondence between gradings of $L$ and diagonalizable
subgroups of $\Aut(\L)$ (see Section 4). A grading is called
\textit{inner} if the respective diagonalizable subgroup is in
$\Int(\L)$. A grading (resp. inner grading) of $L$ is called fine if it could not be further refined by any other grading (resp. inner grading).
Among the gradings of a Lie algebra, fine (inner) gradings are especially
important. It was shown in \cite{pz} that the fine gradings of
simple Lie algebras correspond to maximal diagonalizable subgroups
(which were called MAD-groups in \cite{pz}) of $\Aut(\L)$. Then fine
inner grading of simple Lie algebras corresponds to maximal
diagonalizable subgroups of $\Int(\L)$. Given a fine (inner) grading
$\Ga$, one can define naturally its Weyl group (see Definition 2.3
of \cite{hg}) to describe its symmetry. Assume $K$ is the maximal
diagonalizable subgroup corresponding to $\Ga$, then one can show
that its Weyl group is isomorphic to the Weyl group of $K$. See Proposition 2.4 and Corollary 2.6 of \cite{hg}.
After many mathematicians and physicians' contribution, the
classification of fine gradings of all the simple Lie algebras are
almost done. For example, it can be found in \cite{e} the
classification of fine gradings of all the classical simple Lie
algebras over an algebraically closed field of characteristic 0.
People have also known a lot about the fine gradings for exceptional
simple Lie algebras, see \cite{k} for a survey of such results. For
the Weyl group of a fine inner grading of a simple Lie algebra $L$,
if the grading is Cartan decomposition (in which case the
corresponding maximal diagonalizable subgroup $K$ of $\Int(L)$ is
just the maximal torus), then it is well-known that the Weyl group
is a finite group generated by reflections; in other cases there is
no general result by far. The next step is to study the case $K$ is
discrete, and people have made some explorations in the case
$L=sl(n,\mb C)$ .
Recall that $PGL(n,\mb C)$ is the inner automorphism group of
$sl(n,\mb C)$. Let us first review the classification of maximal
diagonalizable subgroups of $PGL(n,\mb{C})$, which correspond to
fine inner gradings of $sl(n,\mb C)$. Let $\Pi_n$ be the Pauli group
of $GL(n,\mb{C})$ and $D_n$ be the subgroup of diagonal matrices of
$GL(n,\mb{C})$. Let
$\texttt{P}_n$ and $\texttt{D}_n$ be the respective images of $\Pi_n$ and $D_n$ in $PGL(n,\mb{C})$ under the adjoint
action on $M(n,\mb{C})$. Assume $n=k l_1\cdots l_t$ and each $l_i$
divides $l_{i-1}$. The group $D_k\ot\Pi_{l_1}\ot \cdots
\ot\Pi_{l_t}$ consists of all those elements $A_0\ot A_1\ot\cdots\ot
A_t$ with $A_0\in D_k$ and $A_i\in\Pi_{l_i}$ for $1\le i\le t$. The adjoint action of
$D_k\ot\Pi_{l_1}\ot \cdots \ot\Pi_{l_t}$ on
$$M(k,\mb{C})\ot M(l_1,\mb{C})\ot\cdots\ot M(l_t,\mb{C})\cong M(n,\mb{C})$$ induces the embedding
$$ \texttt{D}_k\times \texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_t}\hookrightarrow PGL(n,\mb{C}). $$ If we identify
$ \texttt{D}_k\times \texttt{P}_{l_1}\times \cdots \times
\texttt{P}_{l_t}$ with its image, then it was shown by Havlicek,
Patera and Pelantova in Theorem 3.2 of \cite{hpp} that any maximal
diagonalizable subgroup $K$ of $PGL(n,\mb{C})$ is conjugate
to one and only one of the $ \texttt{D}_k\times \texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_t}$.
Let $K$ be a discrete maximal diagonalizable subgroup of $PGL(n,\mb
C)$. Then $K\cong\texttt{P}_{l_1}\times\texttt{P}_{l_2}\times \cdots
\times \texttt{P}_{l_t}$ where $n=l_1 l_2\cdots l_t$ and each $l_i$
divides $l_{i-1}$. Then the fine grading induced by $K$ also
decomposes $sl(n,\mb C)$ into $n^2-1$ one-dimensional subspaces,
each of which consists of semisimple elements. We will show in
Section 5 that there
is a nonsingular anti-symmetric pairing $<,>$ on $K$,
such that
$(K,<>)$ is a nonsingular symplectic abelian group (see Definition \ref{d8}). Moreover the pairing $<,>$ is invariant under the Weyl group of
$K$. It is shown in Proposition \ref{d6} that there is a one-to-one correspondence between conjugacy classes
of finite maximal diagonalizable subgroups of $\PGL(n,\mathbb{C})$
and finite symplectic abelian groups of order $n^2$. The following
theorem about the structure of the isometry group of a finite
nonsingular symplectic abelian group is Theorem \ref{c3}. For the
definition of a transvection on a symplectic abelian group, see
Definition \ref{d4}.
\begin{theorem}\lb{cc3}
Let $(H,<,>)$ be a finite nonsingular symplectic abelian group. Then
its isometry group is generated by the set of transvections in it.
\end{theorem}
If $K=P_n$, then in the important paper \cite{jpt} the authors showed that the respective Weyl group is $\SL(2,\mb Z_n)$. If $m=p^2$ with $p$ a
prime and $K=\texttt{P}_p\times \texttt{P}_p$, then in \cite{pst}
the authors proved that the respective Weyl group is
$\Sp(4,\mathbb{Z}_p)$. Next in \cite{hg} we dealt with the case
$n=m^k$ ($m$ may not be a prime) and $K=\texttt{P}_m^{k}$ is the
$k$-fold direct product of $\texttt{P}_m$, and proved that the Weyl
group is isomorphic to $\Sp(2k,\mathbb{Z}_n)$ and is generated by
transvections. Then, in this paper we deal with the general case
that $K$ is an arbitrary discrete maximal diagonalizable subgroup,
and prove the following result in Theorem \ref{d7} generalizing the
previous result.
\begin{theorem}\lb{dd9}
Let $K$ be a finite maximal diagonalizable subgroup of
$G=\PGL(n,\mathbb{C})$ and $W_G(K)$ be its Weyl group. Then $W_G(K)$
equals the isometry group of $(K,<,>)$, and is generated by the set
of transvections in it.
\end{theorem}
The paper is organized as follows. The definition and classification of
finite nonsingular symplectic abelian groups will be reviewed in
Section 2. Then in Section 3 we will define transvections on a
finite symplectic abelian group and prove Theorem \ref{cc3}. Next in
section 4 we will review the definitions of the grading of a simple
Lie algebra and prove some important properties. Then in Section 5
we will define the anti-symmetric pairing on any finite maximal
diagonalizable subgroup of $\PGL(n,\mathbb{C})$ and prove the $1-1$
correspondence between conjugacy classes of finite maximal
diagonalizable subgroups of $\PGL(n,\mb C)$ and nonsingular
symplectic abelian groups of order $n^2$. In the last section,
Theorem \ref{dd9} will be proved.\bigskip
Finally we introduce some notations in the paper.
For a finite set $S$, we will use $|S|$ to denote its cardinality.
For any $n\in \mb Z_+$, let $\mb Z_n=\mb Z/n\mb
Z=\{\bar{0},\bar{1},\cdots,\overline{n-1}\}$. For simplicity
we will just use $i$ to denote $\bar{i}$ for $i=0,1,\cdots,n-1$.
Let $\omega_n=e^{2\pi i/n}$ and
$C_n=\{\omega_n^i|i=0,1,\cdots,n-1\}$ be the cyclic group of order
$n$ generated by $\omega_n$. Sometimes we will identify $\mb Z_n$
with $C_n$ by mapping $i$ to $\omega_n^i$.
\bigskip
\centerline{\textit{Acknowledgments}}
The research is supported by NSFC Grant No.10801116 and by 'the
Fundamental Research Funds for the Central Universities'. It is
finished during the author's visit at the department of mathematics
in MIT in 2011. He acknowledges the hospitality of MIT, and would
like to take this opportunity to heartily thank David Vogan for
drawing his attention to this problem and for Vogan's great
generosity in sharing his immense knowledge with him during the
research. Proposition \ref{d0} is due to him.
\section{Classification of finite nonsingular symplectic abelian groups
}
\setcounter{equation}{0}\setcounter{theorem}{0}
We will follow the definition of symplectic abelian groups in
\cite{ka}, which is defined with respect to any field. But for our
purpose we will always assume the field to be $\mb C$,
and we will write our abelian groups additively in Section 2 and 3.
Let $H$ be an abelian group. Recall that an abelian group is
automatically a $\mb Z$-module.
\begin{defi}\lb{d8}
A map $$<,>:H\times H\rt \mb C^{\times}$$ is called a pairing of $H$ into $\mb C^{\times}$ if
$<,>$ is $\mb Z$-bilinear. The pairing is called anti-symmetric if
for all $a,b\in H$, $$<a,b>=<b,a>^{-1}.$$ An anti-symmetric pairing
$<,>$ is called nonsingular if $<a,b>=1$ for any $b\in H$ implying
$a=0$.
\end{defi}
\begin{defi}
Assume $<,>$ is an anti-symmetric pairing of $H$ into $\mb C^{\times}$. Then $(H,<,>)$
is called a \textit{symplectic abelian group}. A symplectic abelian group $(H,<,>)$ is said to be
\textit{nonsingular} if $<,>$ is nonsingular.
\end{defi}
Now assume that $(H,<,>)$ is a nonsingular symplectic abelian group.
A subgroup $H_0$ of $H$ is called a \textit{nonsingular symplectic
abelian subgroup} if the restriction $<,>|H_0$ is nonsingular.
Two subgroups $H_1$ and $H_2$ of $H$ are said to be
\textit{orthogonal}, written $H_1\perp H_2$, if $<a,b>=1$ for any
$a\in H_1, b\in H_2$.
Two symplectic abelian
groups are said to be \textit{isometric} if there is a group isomorphism
between them preserving the respective pairings.
If
$H_1,H_2,\cdots,H_n$ is a family of nonsingular symplectic
abelian subgroups of $H$ such that
$$H=H_1\oplus H_2\oplus\cdots\oplus H_n$$ and $$H_i\perp H_j,\ i\neq j,$$
then we will say that $H$ is the orthogonal direct sum of symplectic
abelian subgroups $H_1,H_2,\cdots,H_n$.
Assume $n\in \mb Z^{+}$ and $n>1$. If a pair of elements $u,v\in H$
of order $n$ satisfying $<u,v>=\omega_n$, then we call $(u,v)$ a
\textit{hyperbolic pair} of order $n$ in $H$. Let \be \mb H_n=\mb
Z_n\times \mb Z_n\lb{d3}\ee and $<(i,j),(k,l)>=\omega_n^{il-jk}$ be
the pairing on $\mb H_n$, which is clearly nonsingular and
anti-symmetric. Then $(\mb H_n,<,>)$ (or just $\mb H_n$) is a
nonsingular symplectic abelian group, called the \textit{hyperbolic
group} of rank $n$. Note that in \cite{ka} the rank $n$
for a hyperbolic group is assumed to be a power of a prime, but we
will not have this restriction in this paper. Let $u_1=(1,0)\in \mb
H_n,v_1=(0,1)\in\mb H_n$. Then the hyperbolic pair $(u_1,v_1)\in\mb
H_n^2$ is called the \textit{standard hyperbolic pair} of $\mb H_n$.
\begin{lem}\lb{e}
Let $H$ be a finite symplectic abelian group. If $a,b\in H$ are both
of order $n$ and $<a,b>=\omega_n$, then $a$ and $b$ generate a
subgroup $K$ isometric to $\mb H_n$.
\end{lem}
\bp Assume for some $i,j\in \mb Z$, $ia+jb=0$. Then
$<a,ia+jb>=\omega_n^j=1$ thus $n|j$. Similarly $n|i$. So
$K=\{ia+jb|i,j\in \mb Z_n\}$. Then $$K\rt \mb H_n,\ ia+jb\mapsto
(i,j)$$ is clearly an isometry of symplectic abelian groups. \ep
\begin{lem}\lb{c}
If $m$ and $n$ are relatively prime, then $\mb H_{mn}\cong \mb H_m
\op \mb H_n$ as symplectic abelian groups.
\end{lem}
\bp Let $(u_1,v_1)\in \mb H_m^2$ (resp. $(u_2,v_2)\in \mb H_n^2$) be
the standard hyperbolic pair for $\mb H_m$ (resp. $\mb H_n$). Let
$$a=u_1+u_2\in \mb H_m \op \mb H_n,\ b=v_1+v_2\in \mb H_m
\op \mb H_n.$$ The order of $a$ and $b$ are both $mn$ as $m$ and $n$
are relatively prime.
One has $<a,b>=<u_1,v_1><u_2,v_2>=\omega_m
\omega_n=\omega_{mn}^{m+n}$. As $mn$ and $m+n$ are also relatively
prime, $<a,ib>=\omega_{mn}$ for some integer $i$. Clearly
$\ord(ib)$, the order of $ib$, is still $mn$. Then by Lemma \ref{e},
the subgroup of $\mb H_m \op \mb H_n$ generated by $a$ and $ib$ is
isometric to $\mb H_{mn}$. As $|\mb H_{mn}|=|\mb H_m \op \mb H_n|$,
$$\mb H_m \op \mb H_n\rt \mb H_{mn},\ ja+k(ib)\mapsto (j,k)$$ is an
isometry of symplectic abelian groups. \ep
\begin{lem}\lb{x}
Let $(H,<,>)$ be a symplectic abelian group. Let $a,b\in H$ with
$ord(a)=i,\ ord(b)=j$. Assume $<a,b>=x$.
(1) If $l$ is the minimal positive integer such that $x^l=1$, then
$l|i$ and $l|j$.
(2) If $i,j$ are relatively prime then $x=1$.
\end{lem}
\bp (1) As $x^i=<ia,b>=<0,b>=1$, one has $l|i$. Similarly $l|j$.
(2) Apply (1). \ep
For any prime $p$ dividing $|H|$, we will always denote the $p$-Sylow subgroup of $H$ by $H(p)$.
\begin{theorem}\lb{d1}[Lemma 1.6 and Theorem 1.8 of \cite{ka}]
Let $(H,<,>)$ be a finite nonsingular symplectic abelian group. Then
(1) $H$ is an orthogonal direct sum of all its Sylow subgroups.
(2) Assume that $H(p)$ is a $p$-Sylow subgroup of $H$. Then $$H(p)\cong \mb H_{p^{r_1}}\op \mb H_{p^{r_2}}\op\cdots \op \mb
H_{p^{r_s}}$$ for some positive integers $r_1,r_2,\cdots,r_s$ with
$r_i\geq r_{i+1}$.
(3) $H$ is an orthogonal direct sum of hyperbolic subgroups $\mb
H_n$, where each $n$ is a power of some prime.
\end{theorem}
\bp (1) It is proved in Lemma 1.6 of \cite{ka}. In fact it is an
easy consequence of Lemma \ref{x} (2).
(2) This is proved in Theorem 1.8 of
\cite{ka} implicitly.
(3) It is a consequence of (1) and (2).
It is also proved in Theorem 1.8 of
\cite{ka}.
\ep
\begin{coro}\lb{d9}
Let $(H,<,>)$ be a nonidentity finite nonsingular symplectic abelian
group. Then there exists positive integers $l_1,l_2,\cdots,l_k$ with
each $l_i|l_{i-1}$ such that \be H\cong \mb H_{l_1}\op \mb
H_{l_2}\op \cdots \op \mb H_{l_k}\lb{b}.\ee Such positive integers
are uniquely determined by $H$.
\end{coro}
\bp The existence of such positive integers and the isometry
(\ref{b}) follow directly from (1) and (2) of Theorem \ref{d1}.
According to the structure theorem of finite abelian groups, such
positive integers are uniquely determined by $H$. \ep
Let $(H,<,>)$ be a nonsingular symplectic abelian group and $H_0$ be a finite nonsingular symplectic subgroup. For any $a\in H$ of
order $n$ and any $j\in \mb
Z$, define
\be \omega_n^j\cdot a=_{def}ja.
\lb{t1}\ee
Assume that $H_0\cong \mb H_{l_1}\op \mb H_{l_2}\op \cdots \op
\mb H_{l_k}$ with $l_i|l_{i-1}$ for all $i$. For $i=1,\cdots,k$ let
$(a_i,b_i)$ be a hyperbolic pair of order $l_i$ for $\mb H_{l_i}$.
Let \be\pi:H\rt H_0,\ c\mapsto \sum_{i=1}^k(<c,b_i>\cdot
a_i-<c,a_i>\cdot b_i).\lb{p}\ee Let $H_0^{\perp}=\{a\in H|<a,b>=0,
\forall\ b\in H_0\}$.
\begin{prop}
(1)$H_0^{\perp}$ is a symplectic subgroup of $H$ and $H=H_0\op
H_0^{\perp}$ is a direct sum of nonsingular symplectic abelian
subgroups.
(2)The map $\pi$ is independent of the hyperbolic pairs $(a_i,b_i)$
chosen.
\end{prop}
\bp
(1) First it is clear that $H_0^{\perp}$ is a subgroup of $H$.
$H_0\cap H_0^{\perp}=0$ as \\ $<,>|H_0$ is nonsingular. Let $c\in
H$. Assume $<c,a_i>=\omega_{l_i}^{t_i}$ for $i=1,\cdots,k$, then
\bee\begin{split}<\pi(c),a_i>&=<-<c,a_i>b_i,a_i>\\&=<-t_i
b_i,a_i>\\&=<b_i,a_i>^{-t_i}=\omega_{l_i}^{t_i}.\end{split}\eee So
$$<c,a_i>=<\pi(c),a_i>$$ for
$i=1,\cdots,k$. Similarly $$<c,b_i>=<\pi(c),b_i>$$ for
$i=1,\cdots,k$. Thus one has $c-\pi(c)\in H_0^{\perp}$ and \be
c=\pi(c)+(c-\pi(c))\in H_0+H_0^{\perp}.\lb{q}\ee So $ H=H_0\op
H_0^{\perp}.$ It is clear that $<,>|H_0^{\perp}$ must also be
nonsingular as $<,>$ is nonsingular. Thus $H=H_0\op H_0^{\perp}$ is a direct sum of
nonsingular symplectic abelian subgroups.
(2) Let $\pi^{'}:H\rt
H_0$ be defined as in (\ref{p}) with respect to another choice of
hyperbolic pairs $(a_i^{'},b_i^{'})$ for each $\mb H_{l_i}$. Then
for any $c\in H$ one also has $$c=\pi^{'}(c)+(c-\pi^{'}(c))\in
H_0\op H_0^{\perp}.$$ Comparing to (\ref{q}), as $H=H_0\op
H_0^{\perp}$ is a direct sum, one must have \\ $\pi(c)=\pi^{'}(c)$
for any $c\in H$. \ep We call the map $\pi:H\rt H_0$ defined in
(\ref{p}) the \textit{projection} of $H$ onto $H_0$.
\section{Transvections and isometry groups of finite nonsingular symplectic abelian groups
}
\setcounter{equation}{0}\setcounter{theorem}{0}
Let $(H,<,>)$ be a finite nonsingular symplectic abelian group.
\begin{defi}
Let $\Sp(H)$ be the set of isometries of $H$ onto itself. Then $\Sp(H)$ is clearly a group, called the isometry group
of $H$.
\end{defi}
If $H=H_1\op H_2$ is a direct sum of nonsingular symplectic abelian
subgroups then it is clear that
$$\Sp(H_1)\times \Sp(H_2)\rt \Sp(H),\
(\phi,\nu)(a,b)=(\phi(a),\nu(b))$$ embeds $\Sp(H_1)\times \Sp(H_2)$
as a subgroup of $\Sp(H)$. For any symplectic subgroup $H_0$ of $H$,
as $H=H_0\op H_0^{\perp}$, we will always regard $\Sp(H_0)$ as a
subgroup of $\Sp(H)$ by the embedding
$$\Sp(H_0)\hookrightarrow \Sp(H_0)\times \Sp(H_0^{\perp})\st
\Sp(H),\ \phi\mapsto (\phi,1),$$ where 1 denotes the identity map on
$H_0^{\perp}$.
\begin{prop}
Let $H$ be a finite nonsingular symplectic abelian group. Then $\Sp(H)$ acts transitively on the set of hyperbolic pairs in $H$ with the same order.
\end{prop}
\bp Assume $(a_1,b_1)$ and $(a_2,b_2)$ are two hyperbolic pairs in
$H$ with order $n$. Let $H_i=Span (a_i,b_i)$ for $i=1,2$. Then $H_i$
is a symplectic subgroup of $H$ isometric to $\mb H_n$. Let
$\phi_1:H_1\rt H_2$ be the isometry such that $\phi_1(a_1)=a_2,\
\phi_1(b_1)=b_2.$
One has $H=H_1\op H_1^{\perp}$ and $H=H_2\op H_2^{\perp}$. By
Theorem \ref{d1} (3), $H_1^{\perp}\cong H_2^{\perp}$. Fix an
isometry $\phi_2:H_1^{\perp}\rt H_2^{\perp}$. Then
$$\phi=(\phi_1,\phi_2):H_1\op H_1^{\perp}\rt H_2\op H_2^{\perp},\
(c,d)\mapsto (\phi_1(c),\phi_2(d))$$ is in $\Sp(H)$ and maps
$(a_1,b_1)$ to $(a_2,b_2)$. Thus $\Sp(H)$ acts transitively on the
set of hyperbolic pairs in $H$ with the same order. \ep
Let $$\wh H=_{def}Hom(H,\mb C^{\times})$$ be the abelian group of characters of $H$.
For any $a\in H$, define
$$\ga_a:H\rt \mb C^{\times},\ b\mapsto <a,b>.$$
\begin{lem} The map \be \varphi:H\rt \wh H,\ a\mapsto \ga_a\lb{g}\ee is an isomorphism of abelian groups.\end{lem}
\bp
As $<,>$ is $\mb Z$-bilinear, $\va$ is $\mb Z$-linear. As $<,>$ is
nonsingular, $\va$ is one-to-one, thus is onto as $|H|=|\wh H|$.
\ep
\begin{defi}\lb{a5}
For any $\ga\in \wh H$, let $$\ga^*=_{def}\varphi^{-1}(\ga).$$
\end{defi}
Then for any $a\in H$, $\ga(a)=<\ga^*,a>$.
As (\ref{g}) is an isomorphism, $\ga$ and $\ga^*$ have the same
order, and for $\ga_1,\ga_2\in \wh H$,
\be (\ga_1+\ga_2)^*=\ga_1^*+\ga_2^*.\lb{b3}\ee
Recall that for a vector space $V$ over $\mb C$ (or other field), a linear map
$\phi\in \GL(V)$ is called a transvection if
$$\phi(v)=v+\lambda(v)u$$ for some $\lambda\in V^*, u\in V$ satisfying
$\lambda(u)=0$. If $V$ is a symplectic vector space with $<,>$ the
anti-symmetric pairing on it, then any transvection of $V$
preserving the form $<,>$ must be of the form
\be\phi(v)=v-k<u,v>u\lb{d5}\ee for some $u\in V,\ k\in \mb C$. One
knows that $\SL(V)$ and $\Sp(V)$ are both generated by their
transvections. One can define transvections for a symplectic abelian
group analogously.
For any $b\in H$ with $b\neq 0$, assume $\ord(b)=m$. For any $a\in
H$, $<b,a>=\ga_b(a)$ takes value in the cyclic group
$C_m=\{\omega_m^i|i=0,1,\cdots,m-1\}$. Recall the convention in
(\ref{t1}).
\begin{defi}\lb{d4} For any $b\in H$ with $b\neq 0$ and $k\in\mb Z$,
define a homomorphism \be s_{b,k}:H\rt H,\ a\mapsto a-k(<b,a>\cdot
b),\lb{g3}\ee and call it a transvection on $H$. Using the
identification $\varphi$ of $H$ and $\wh H$, for any $\ga\in \wh H$
define $ s_{\ga,k}= s_{\ga^*,k}$. Denote $s_{b,1}$ (resp.
$s_{\ga,1}$) by $s_{b}$ (resp. $s_{\ga}$) for simplicity.
\end{defi}
Then one
has \be s_\ga(a)=a-\ga(a)\ga^*.\lb{b9}\ee
\begin{lem} (1) For any $b\in H$ with $b\neq 0$ and any $k,j\in\mb Z$, one has
\be s_{b,0}=1,\lb{g0}\ee \be s_{b,k}
s_{b,j}= s_{b,k+j}, \lb{g1}\ee and
\be s_{b,k}^{-1}= s_{b,-k}\lb{g2}\ee
(2) If $\ord(b)=m$, them $\{ s_{b,k}|k\in\mb Z\}$ is a cyclic
group of order $m$ generated by $ s_{b,1}$.
\end{lem}
\bp (1) (\ref{g0}) follows from the definition (\ref{g3}). For any
$a\in H$, \bee\begin{split} s_{b,k}
s_{b,j}(a)&= s_{b,k}(a-j(<b,a>\cdot b))\\&=(a-j(<b,a>\cdot
b))-k(<b,a-j(<b,a>\cdot b)>\cdot b)\\&=a-(k+j)(<b,a>\cdot b)\\&=
s_{b,k+j}(a),\end{split}\eee So (\ref{g1}) holds. Then (\ref{g2})
follows from (\ref{g0}) and (\ref{g1}).
(2) It follows from (1). \ep
\begin{lem}
One has $ s_{b,k}\in \Sp(H)$, where $b\in H$ with $b\neq 0$ and $k\in\mb
Z$.
\end{lem}
\bp By (2) of last lemma one only need to show $ s_b\in \Sp(H)$. By
(\ref{g3}) and (\ref{g2}) it is clear that $ s_b$ is a $\mb
Z$-linear isomorphism of $H$, so we only need to prove that $ s_b$
preserves $<,>$. Assume $a,c\in H$ and $<b,a>=\omega_m^i$,
$<b,c>=\omega_m^j$. Then \bee
\begin{split}
< s_b(a), s_b(c)>&=<a-<b,a>b,c-<b,c>b>\\
&=<a-ib,c-jb>\\&=<a,c> <b,c>^{-i}<a,b>^{-j}\\
&=<a,c> \omega_m^{j(-i)}\omega_m^{(-i)(-j)}\\
&=<a,c>.
\end{split}
\eee
\ep
Let \be Q(H)=_{def}< s_{b,k}|0\neq b\in H,k\in \mb Z>\ee be the
subgroup of $\Sp(H)$ generated by all the transvections. It is clear
that $Q(H)$ is generated by those $ s_b$ with $b\in H$ and $b\neq
0$. For any element $b\neq 0$ in a nonsingular symplectic subgroup
$H_0$ of $H$, $ s_b\in Q(H_0)$ can be regarded as in $Q(H)$ since
$b\in H$. Thus $Q(H_0)$ can be naturally regarded as a subgroup of
$Q(H)$.
Let $\GL(2,\mathbb{Z}_n)$ be the group of $2\times 2$ invertible
matrices in $M(2,\mathbb{Z}_n)$. Let $$\SL(2,\mb Z_n)=\{A\in
\GL(2,\mathbb{Z}_n)|\det(A)=1\in\mb Z_n\}.$$ Let $$J=\left(
\begin{array}{cc}
0 & 1\\
-1 & 0 \\
\end{array}
\right)\in M(2,\mb Z_n)$$
and $$\Sp(2,\mb Z_n)=\{A\in \GL(2,\mathbb{Z}_n)|A^t J A=J\}.$$ It is
easily verified that $\SL(2,\mb Z_n)=\Sp(2,\mb Z_n)$.
\begin{lem}\lb{h}
One has $\Sp(\mb H_n)\cong \Sp(2,\mb Z_n)=\SL(2,\mb Z_n)$ and $\Sp(\mb
H_n)=Q(\mb H_n)$. In particular, $\Sp(\mb H_n)$ is generated by $
s_{u_1}$ and $ s_{v_1}$, where $(u_1,v_1)$ is the standard
hyperbolic pair of $\mb H_n$.
\end{lem}
\bp Note that $(u_1,v_1)$ is an (ordered) $\mb Z_n$-basis for $\mb
H_n$. For any $\varphi\in \Sp(\mb H_n)$, one has
$$\va(u_1)=a_{11}u_1 +a_{21}v_1,\ \va(v_1)=a_{12}u_1+a_{22}v_1$$ where $a_{ij}\in\mb Z_n$. Then with respect to the $\mb Z_n$-basis $(u_1,v_1)$, the matrix of $\va$ is defined to be \be C=\left(
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}\\
\end{array}
\right),\lb{}\ee which is in $\GL(2,\mathbb{Z}_n)$.
This defines a map $\Sp(\mb H_n)\rt \GL(2,\mb Z_n)$, which is an injective homomorphism. For any $a\in\mb H_n$, $a=i u_1+j v_1$
with $i,j\in\mb Z_n$. Then the coordinate $\wt a$ of $a$ is denoted $\wt a=(i,j)^t$, the transpose of $(i,j)$. It is clear that \be \wt{\va(a)}=C \wt a.\lb{n}\ee
If we identify $$C_n=\{\omega_n^i|i=0,1,\cdots,n-1\}\rt\mb Z_n,\
\omega_n^i\rt i,$$ then the matrix of the pairing $<,>$ in the $\mb
Z_n$-basis $(u_1,v_1)$ is $J$.
For any $a,b\in\mb H_n$, one has \be <a,b>=\wt a^t J \wt b\lb{m}.\ee
As $\va\in \Sp(\mb H_n)$, by (\ref{n}) and (\ref{m}) one has
$$C^t J C=J,$$ thus $C\in \Sp(2,\mb Z_n)$. So $\Sp(\mb H_n)\st \Sp(2,\mb Z_n)=\SL(2,\mb Z_n)$.
As $$ s_{u_1}(u_1)=u_1,\ s_{u_1}(v_1)=v_1-<u_1,v_1>u_1=v_1-u_1$$ so
the matrix of $ s_{u_1}$ is $$A=\left(
\begin{array}{cc}
1 & -1\\
0 & 1 \\
\end{array}
\right) .$$
As $$ s_{v_1}(u_1)=u_1-<v_1,u_1>v_1=u_1+v_1,\ s_{v_1}(v_1)=v_1$$ so
the matrix of $ s_{v_1}$ is $$B=\left(
\begin{array}{cc}
1 & 0\\
1 & 1 \\
\end{array}
\right) .$$
It is clear that $A,B$ generate $\SL(2,\mb Z_n)$, thus $\Sp(\mb
H_n)=Q(\mb H_n)= \SL(2,\mb Z_n)$. \ep
\begin{coro}\lb{l}
For any $a\in \mb H_n$, if $\ord(a)=n$ then $a$ is conjugate to $u_1$ under $\Sp(\mb H_n)$; otherwise $a$ is conjugate to $l u_1$ for
some $l\in\mb Z_n$.
In particular all the elements in $\mb H_n$ with order $n$ are
conjugate under $\Sp(\mb H_n)$.
\end{coro}
\bp Assume $a=(i,j)$. As $\mb Z_n$ is a principal ideal ring, the
ideal $I(i,j)$ in $\mb Z_n$ generated by $i,j$ must be generated by
some $l\in\mb Z_n$. So $I(i,j)=I(l)$.
If $\ord(a)=n$, $I(l)=\mb Z_n$. So there exists $k,m\in \mb Z_n$
such that $ik+jm=1$. Let $A=\left(
\begin{array}{cc}
k& m\\
-j & i \\
\end{array}
\right)\in \SL(2,\mb Z_n)$. Then $A(i,j)^t=(1,0)^t$, so $a$ is conjugate to $u_1$ by $A$.
If $\ord(a)<n$, then $(i,j)=l(i^{'},j^{'})$ for some
$a^{'}=(i^{'},j^{'})\in \mb H_n$ as $l$ is the greatest common
devisor of $i,j$. Then $\ord(a^{'})=n$ and there exists some $A\in
\SL(2,\mb Z_n)$ such that $A(a^{'})^t=(1,0)^t$ and $A a^t=(l,0)^t$.
Thus $a$ is conjugate to $l u_1$ by $A$. \ep
\begin{lem}\lb{o}
Let $H=\mb H_n\op \mb H_n$ and $\phi:H\rt H,\ (a,b)\mapsto (b,a)$. Then $\phi\in Q(H)$.
\end{lem}
\bp
Let $(u_1,v_1)$ (resp. $(u_2,v_2)$) be the standard hyperbolic pair
in $\mb H_n\op 0$ (resp. $0\op \mb H_n$). Then $\phi$ maps
$u_1$ to $u_2$ and $v_1$ to $v_2$. Let $x=v_1+v_2$, then $(u_1,x)$ and
$(u_2,x)$ are both hyperbolic pairs of order $n$. Let $H_1=Span
(u_1,x)$, then $H=H_1\op H_1^{\perp}$. By Corollary \ref{l}, there
exists $$\tau=(\tau^{'},1)\in \Sp(H_1)\times \Sp(H_1^{\perp})\st
\Sp(H)$$ such that $\tau(x)=u_1$. Similarly there exists $\va\st
\Sp(H)$ such that $\va(u_2)=x$. Then $\tau\va(u_2)=u_1$. Let
$v_2^{'}=\tau\va(v_2)$.
Assume $<v_2^{'},v_1>=\omega_n^i$. Then
$$ s_{u_1}^{i-1}(u_1)=u_1,$$
$$ s_{u_1}^{i-1}(v_2^{'})=v_2^{'}-(i-1)(<u_1,v_2^{'}>\cdot u_1)=v_2^{'}-(i-1)u_1.$$ Let $v_2^{''}=v_2^{'}-(i-1)u_1$. Then $<v_2^{''},v_1>=\omega_n$.
Let $q=v_2^{''}-v_1$. Note $\ord(q)=n$, then $$ s_q(u_1)=u_1-<q,u_1>\cdot q=u_1,$$
$$ s_q(v_2^{''})=v_2^{''}-<q,v_2^{''}>\cdot q=v_2^{''}-\omega_n\cdot q=v_2^{''}-q=v_1.$$ So the map $\nu= s_q s_{u_1}^{i-1}
\tau\va\in Q(H)$ maps $(u_2,v_2)$ to $(u_1,v_1)$. Then $\nu\phi$
fixes $(u_1,v_1)$ and maps its orthocomplement $0\op \mb H_n$
isometrically onto itself. Then there exists
$\theta=(1,\theta^{'})\in \Sp(\mb H_n)\times \Sp(\mb H_n)$ such that
$\theta\nu\phi=1$. So $\phi\in Q(H)$ as $\theta$ and $\nu$ are both
generated by transvections.
\ep
\begin{lem}
Assume that $H=H(p)$ for some prime $p$. Then by Theorem \ref{d1} (2), $H=\mb H_{p^{r_1}}\op \mb H_{p^{r_2}}\op\cdots \op \mb H_{p^{r_s}}$ for some positive integers
$r_1,r_2,\cdots,r_s$ with $r_i\geq r_{i+1}$. For any $a=(a_1,\cdots,a_s)\in H$ with order $p^{r_1}$, there exists $\phi\in Q(H)$ such that
$\phi(a)=b=(b_1,\cdots,b_s)$ with $\ord({b_1})=p^{r_1}$.
\end{lem}
\bp
In this case one has $\ord(a)=Max_{i=1}^s\{\ord(a_i)\}$. If $\ord(a_1)=p^{r_1}$ then we take $\phi=1$. If $\ord(a_i)=p^{r_1}$ for some $i>1$, then $r_i=r_1$.
Then as the subgroups $\mb H_{p^{r_1}}$ and $\mb H_{p^{r_i}}$ of $H$ are isometric, by Lemma \ref{o} there exists some
$\phi\in Q(\mb H_{p^{r_1}} \op \mb H_{p^{r_i}})\st Q(H)$
such that $$\phi(a)=\phi(a_1,\cdots,a_i,\cdots,a_s)=(a_i,\cdots,a_1,\cdots,a_s)=b. $$ Then $b_1=a_i$ and $\ord({b_1})=p^{r_1}$.
\ep
\begin{lem}
Assume $H=\mb H_{l_1}\op \mb H_{l_2}\op \cdots \op \mb H_{l_k}$ with $l_i|l_{i-1}$ for all $i$. Then for any $a=(a_1,\cdots,a_k)\in H$ with order $l_1$,
there exists $\phi\in Q(H)$ such that $\phi(a)=b=(b_1,\cdots,b_k)$ with $\ord(b_1)=l_1$.
\end{lem}
\bp Let $p_1,\cdots,p_s$ be the set of primes dividing $|H|$. Then
$H=\op_i H(p_i)$ and $H(p_i)=\mb H_{l_1}(p_i)\op \mb H_{l_2}(p_i)\op
\cdots \op \mb H_{l_k}(p_i)$. Let $\pi_i:H\rt H(p_i)$ be the
projection. Then $$\pi_i(a)=(a_{1i},a_{2i},\cdots,a_{ki})\in \mb
H_{l_1}(p_i)\op \mb H_{l_2}(p_i)\op \cdots \op \mb H_{l_k}(p_i).$$
One has $a=\sum_i \pi_i(a)$. By last lemma there exists $\phi_i\in
Q(H(p_i))\st Q(H)$ such that
$\phi_i(\pi_i(a))=(b_{1i},b_{2i},\cdots,b_{ki})$ with
$\ord(b_{1i})=\ord(\pi_i(a))$. Let $\phi=\Pi_i \phi_i$. Then
$\phi(a)=b=(b_1,b_2,\cdots,b_k)\in \mb H_{l_1}\op \mb H_{l_2}\op
\cdots \op \mb H_{l_k}$, where $b_1=\sum_i b_{1i}$ and
$\ord(b_{1})=\ord(b)=\ord(a)=l_1$.
\ep
\begin{lem}
Assume $H=\mb H_{l_1}\op \mb H_{l_2}\op \cdots \op \mb H_{l_k}$ with $l_i|l_{i-1}$ for all $i$. Then for any $a=(a_1,a_2,\cdots,a_k)\in H$ with order $l_1$,
there exists $\phi\in Q(H)$ such that $\phi(a)=b=(u_1,0,\cdots,0)$, where $u_1=(1,0)\in \mb H_{l_1}$.
\end{lem}
\bp By last lemma we can assume that $\ord({a_1})=l_1$. Then use
induction on the number $t$ of nonzero elements in
$\{a_1,\cdots,a_k\}$. The case $t=1$ follows from Corollary \ref{l}.
Assume there are $t=l\geq 2$ nonzero elements in
$\{a_1,a_2,\cdots,a_k\}$ and the result holds for $l-1$. Without
loss of generality we can assume $a_2\neq 0$. As $\ord({a_1})=l_1$,
there exists some $b_1\in \mb H_{l_1}\st H$ such that
$<b_1,a_1>=\omega_{l_1}$. Let $b=(b_1,a_2,0\cdots,0)$. Then
$\ord({b})=l_1$ and $<b,a>=\omega_{l_1}$, so \be \begin{split}
s_b(a)&=a-<b,a>\cdot b\\&=a-\omega_{l_1}\cdot b=a-b
\\&=(a_1-b_1,0,a_3,\cdots,a_k).
\end{split}\ee So $ s_b(a)=(a_1-b_1,0,a_3,\cdots,a_k)$ differs with $a$
only in the first and second position. As $ s_b(a)$ has order $l_1$
and has $l-1$ nonzero elements, by induction there exists
$\phi_1\in Q(H)$ such that $\phi_1( s_b(a))=(u_1,0,\cdots,0)$. Then
$\phi=\phi_1 s_b\in Q(H)$ has the desired property and the result
holds for $t=l$.
\ep
\begin{coro}\lb{j}
$Q(H)$ acts transitively on the set of elements in $H$ with maximal order.
\end{coro}
\begin{lem}
Assume $H=\mb H_{l_1}\op \mb H_{l_2}\op \cdots \op \mb H_{l_k}$ with $l_i|l_{i-1}$ for all $i$ and $G=\Sp(H)$. For $i=1,\cdots,k$ let $(u_i,v_i)$ be the standard hyperbolic pair in $\mb H_{l_i}$. Then (1) $G_{u_1}=Q(H)_{u_1}$. (2) $G=Q(H)$.
\end{lem}
\bp We will prove them by induction on $k$.
The case $k=1$ follows from Lemma \ref{h} as we proved there
$\Sp(H)=Q(H)$ if $H=\mb H_{l_1}$. Let $k>1$. Assume (1) and (2) holds for
$k-1$.
Assume $\si\in G_{u_1}$. As
$<u_1,v_1>=<\si(u_1),\si(v_1)>=<u_1,\si(v_1)>$,
$<u_1,\si(v_1)-v_1>=0$ so
$$\si(v_1)=v_1+j_1 u_1+\sum_{i=2}^k (p_i u_i+ q_i v_i)$$ for some
$j_1,p_i,q_i\in \mb Z$. By Corollary \ref{l} there exists $\phi_i\in
Q(\mb H_{l_i})\st Q(H)$ such that $\phi_i(p_i u_i+ q_i v_i)=j_i u_i$
for $i\geq 2$. Let $\phi=\Pi_{i=2}^k\ \phi_i$. Then $\phi\in
Q(H)_{u_1}$ and $$\phi \si(v_1)=v_1+j_1 u_1+\sum_{i=2}^k j_i u_i.$$
For any $i$ with $2\le i\le k$, $ s_{u_1+u_i}(u_t)=u_t$ for
$t=1,\cdots,k$. As $\ord(u_1+u_i)=l_1$, \be
\begin{split} s_{u_1+u_i}(v_1)&=v_1-<u_1+u_i,v_1>\cdot (u_1+u_i)\\&=v_1-\omega_{l_1}\cdot (u_1+u_i)\\&=v_1-(u_1+u_i).\end{split}\ee
Let $\tau=\Pi_{i=2}^k\ s_{u_1+u_i}^{j_i}$. Then $\tau\in
Q(H)_{u_1}$ and
$$ \tau(\phi\si(v_1)
)=v_1+(j_1-\sum_{i=2}^k j_i)u_1.$$
So $\tau\phi\si$ preserves $\mb H_{l_1}$ and also $\mb
H_{l_1}^{\perp}=\mb H_{l_2}\op \cdots \op \mb H_{l_k}$ thus
$$\tau\phi\si\in \Sp(\mb H_{l_1})\times \Sp(\mb H_{l_2}\op \cdots
\op \mb H_{l_k}).$$ By induction $$\Sp(\mb H_{l_1})\times \Sp(\mb
H_{l_2}\op \cdots \op \mb H_{l_k})=Q(\mb H_{l_1})\times Q(\mb
H_{l_2}\op \cdots \op \mb H_{l_k}),$$ so $$\tau\phi\si\in Q(\mb
H_{l_1})\times Q(\mb H_{l_2}\op \cdots \op \mb H_{l_k})\st Q(H).$$
As $\tau,\phi\in Q(H)$, one also has $\si\in Q(H)$. So $\si\in
Q(H)_{u_1}$ then $G_{u_1}\st Q(H)_{u_1}$. As $G\supset Q(H)$, one
must have $G_{u_1}= Q(H)_{u_1}$. So (1) holds for $k$.
By Corollary \ref{j}, $Q(H)$ acts transitively on the set of
elements in $H$ with maximal order, so does $G$. As
$$G_{u_1}=Q(H)_{u_1}\ \an \ |G/G_{u_1}|=|Q(H)/Q(H)_{u_1}|,$$ so
$|G|=|Q(H)|$ thus $G=Q(H)$. So (2) also holds for $k$. \ep
Now we have proved the following theorem.
\begin{theorem}\lb{c3}
Let $H$ be a finite nonsingular symplectic abelian group. Then
$\Sp(H)$ is generated by the set of transvections on $H$.
\end{theorem}
\begin{rem}
If $H=\mb H_n^k$, the $k$-fold direct sum of $\mb H_n$, then by
choosing some suitable $\mb Z_n$-basis of $\mb H_n$, it is easy to
see that
$$\Sp(\mb H_n^k)\cong\Sp(2k,\mb Z_n),$$ where
$$\Sp(2k,\mathbb{Z}_n)=\{A\in GL(2k,\mathbb{Z}_n)|A^t J_{2k}
A=J_{2k}\}$$ with $$J_{2k}=\left(
\begin{array}{ccccc}
0 & 1 & & &\\
-1 & 0 & & &\\
& & \ddots & &\\
& & & 0 & 1 \\
& & &-1 & 0 \\
\end{array}
\right)\in M(2k,\mathbb{Z}_n).$$ Thus this theorem implies
that in particular $\Sp(2k,\mb Z_n)$ is generated by the
transvections.
\end{rem}
\section{Fine gradings of Lie algebras
and their Weyl groups }
\setcounter{equation}{0}\setcounter{theorem}{0}
4.1. The details of this subsection can be found in Section 4 of \cite{hg}. We include it for completeness.
We always assume that $L$ is a complex simple Lie algebra. Let
$\Aut(L)$ be its automorphism group and $\Int(L)$ the identity
component of $\Aut(L)$, called the inner automorphism group
of $L$. It is clear that $\Aut(L)$ and $\Int(L)$ are both algebraic groups.
Let $\Lambda$ be a additive abelian group. A
$\Lambda$-\textit{grading} $\Gamma$ on $L$ is the decomposition of
$L$ into direct sum of subspaces $$\Gamma: L=\oplus_{\ga\in \Lambda}
\ L_\ga$$ such that
$$[L_\ga,L_\dt]\st L_{\ga+\dt},\ \forall\\ \ga,\dt\in \Lambda.$$ Let
$\ddt=\{\ga\in \Lambda|L_\ga\neq 0\}$. We will always
assume that $\Lambda$ is generated by $\ddt$, otherwise it could be
replaced by its subgroup generated by $\ddt$. So $\Lambda$ is always finitely generated.
Given a $\Lambda$-grading $\Gamma$ on $L$, let $$K=\widehat
\Lambda=_{def} \Hom(\Lambda, \mathbb{C}^{\times})$$ be the abelian
group of characters of $\Lambda$. Then $K$ acts on $L$ by
$$\si\cdot X=\si(\ga) X,\ \forall\\ X\in L_\ga,\ \forall\ \ga\in \Lambda,\
\forall\\si\in K.$$ This defines an injective homomorphism $K\rt
\Aut(L)$. So $K$ can be viewed as a subgroup of $\Aut(L)$. Recall
that an algebraic group is called \textit{diagonalizable} if it is
abelian and consists of semisimple elements. It is easy to see that
$K$ is a diagonalizable algebraic subgroup of $\Aut(L)$.
Conversely, given a diagonalizable algebraic subgroup $K$ of
$\Aut(L)$, let $$\Lambda=\widehat K=_{def} \Hom(K,
\mathbb{C}^{\times})$$ be the (additive) abelian group of
homomorphisms from $K$ to $\mb C^{\times}$ as algebraic groups. Then
one has a $\Lambda$-grading on $L$: $$\Ga:\ \ L=\oplus_{\gamma\in
\Lambda}\ L_\ga,$$ where $L_\ga=\{X\in L|\si \cdot X=\ga(\si) X,\
\forall\ g\in K\}.$ Let $$\ddt=\ddt(L,K)=_{def} \{\ga\in
\Lambda|L_{\ga}\neq 0\}.$$ We call $\ddt$ the set of \textit{roots}
of $K$ in $L$.
Thus there is a natural one-to-one correspondence between gradings of
$L$ by finitely generated abelian groups and diagonalizable
algebraic subgroups of $\Aut(L)$. A grading is called \textit{inner} if the respective diagonalizable subgroup is in
$\Int(\L)$. A grading (resp. inner grading) of $L$ is called fine if it could not be further refined by any other grading (resp. inner grading).
It is clear that the bigger the
diagonalizable algebraic subgroup $K$ is, the finer the
corresponding grading is. Thus a grading (resp. inner grading) $\Ga$ on $L$ is
fine if and only if the corresponding diagonalizable subgroup $K$
is a maximal diagonalizable subgroup of $\Aut(L)$ (resp. $\Int(L)$).
Let $G$ be either $\Aut(L)$ or $\Int(L)$. Let $K$ be a {maximal} diagonalizable subgroup of $G$,
$\Ga$ the grading on $L$ induced by the action of $K$. One could
define the Weyl group $W_G(\Ga)$ of the grading $\Ga$ with respect
to $G$, see Definition 2.3 of \cite{hg}, to describe the symmetry of
the grading $\Ga$.
One has the following result in \cite{hg}.
\begin{prop}[Corollary 2.6 of \cite{hg}]
Let $L$ be a simple Lie algebra and $G=\Int(L)$. Let $K$ be a maximal diagonalizable subgroup of $G$ and $\Ga$ be the corresponding grading
on $L$ induced by the action of $K$. Let $W_G(K)=N_G(K)/K$ be the Weyl group
of $K$ with respect to $G$. Then one has $W_G(\Ga)=W_G(K)$. \end{prop}
\bigskip
4.2. From now on we will always assume $K\st G=\Int(L)$ to be a {finite
maximal diagonalizable subgroup} and $\ddt=\ddt(L,K)$. Let $B$ be the Killing form on
$L$. Recall that a linear subspace $S$ of $L$ is called a toral
subalgebra if $[S,S]=0$ and the endomorphism $\ad_X$ is semisimple
for each $X\in S$.
As $L$ is simple, the adjoint map
$$\ad:L\rt \ad(L),\ X\mapsto \ad_X$$ is a $G$-equivariant isomorphism.
\begin{defi} For any
$\ga\in \ddt$, let
$$L_{[\ga]}=_{def}\oplus_{k\in \mathbb{Z}}\ L_{k\ga}.$$ \end{defi}
\begin{prop}\lb{d0}
(1) One has $L_0=0$, i.e., $0\notin \ddt$.
(2) Assume $\ga,\dt\in\ddt$ and $\ga+\dt\neq 0$. Then $B|L_\ga\times
L_\dt=0$. For any $X\in L_\ga$ with $X\neq 0$, there exists $Y\in
L_{-\ga}$ such that $B(X,Y)\neq 0$.
(3) For any $\ga\in \ddt$, $L^{\Ker \ga}=L_{[\ga]}$ and is a toral
subalgebra of $L$. One has $Lie\ Z(Ker \ga)_0=ad(L_{[\ga]})$ and
$Z(Ker \ga)_0$ is an algebraic torus (isomorphic to some
$(\mathbb{C}^\times)^i$).
\end{prop}
\bp (1) As $K$ is a maximal diagonalizable subgroup, $Z_G(K)=K$ by Lemma 2.2 of \cite{hg}. As
$K$ is finite, $$\ad(L^K)=\ad(L)^K=\Lie Z_G(K)=\Lie K=0.$$ So
$L_0=L^K=0$.
(2) Assume $\ga+\dt\neq 0$. For any $X\in L_\ga$ and $Y\in L_\dt$,
$\ad_X \ad_Y$ maps each $L_\zeta$ into $L_{\zeta+\ga+\dt}$ thus
$B(X,Y)=0$. Then $B|L_\ga\times L_\dt=0$. Because $B$ is nonsingular
on $L$, the second statement then follows from the first one.
(3) We first prove $L^{\Ker \ga}=L_{[\ga]}$. Choose some $\si\in K$
satisfying $\ga(\si)=\omega_m$, where $m$ is the order of $\ga$.
Then $\ga(\si)$ is a generator of the cyclic group $\ga(K)\cong
K/\Ker \ga$. $L^{\Ker \ga}$ is the direct sum of those
$L_\bt$ with $\bt$
being identity on $\Ker \ga$. Then $\bt(\si)=\omega_m^k$ for some
integer $k$ as $\si^m\in \Ker \ga $. Then
$\bt(\si)=\ga(\si)^{k}=(k\ga)(\si)$. As $\Ker \ga$ and $\si$
generate $K$, $\bt=k\ga$. Thus $L^{\Ker \ga}=\oplus_{k}
L_{k\ga}=L_{[\ga]}$.
One has $[L_0,L_{[\ga]}]=0$ as $L_0=0$. Then by Proposition 3.6 of
\cite{h}, $L_{[\ga]}$ is a toral subalgebra of $L$. As
$\Lie\Int(L)=\ad(L)$, it is clear that
$$\Lie\ Z(\Ker \ga)_0=\Lie\ Z(\Ker \ga)=\ad(L)^{\Ker \ga}=\ad(L^{\Ker
\ga})=\ad(L_{[\ga]}).$$Thus $Z(Ker \ga)_0$ is an algebraic torus.
\ep
\begin{rem}If $L$ is a semisimple Lie algebra then all the
results in this section still hold.
\end{rem}
\section{Finite maximal diagonalizable subgroups of $\PGL(n,\mb C)$ and anti-symmetric pairings on them}
\setcounter{equation}{0}\setcounter{theorem}{0}
Let $n\in \mathbb{Z}_+$. Recall $\omega_n=e^{2\pi i/n}$. Let
$$Q_n=diag(1,\omega_n,\omega_n^2,\cdots,\omega_n^{n-1})$$ and $$P_n=\left(
\begin{array}{ccccc}
0 & 1 & 0 &\cdots & 0 \\
0 & 0 & 1 &\cdots & 0 \\
\vdots& \vdots& \vdots & \ddots & \vdots \\
0 & 0 & 0 &\cdots & 1 \\
1 & 0 & 0 &\cdots & 0\\
\end{array}
\right).$$ Let $\Pi_n=\{\omega_n^j P_n^k Q_n^l\ |j,k,l=0,1,\cdots,n-1\}$. Note $P_n Q_n=\omega_n Q_n P_n$. This is a subgroup of $\GL(n,\mathbb{C})$,
called the \textit{Pauli group} of rank $n$.
Let $D_n$ be the subgroup of diagonal matrices
of $\GL(n,\mb{C})$. Let $\texttt{P}_n$ and $\texttt{D}_n$
be the respective images of $\Pi_n$ and $D_n$ under the adjoint action on $M(n,\mb C)$. One knows that
$$\texttt{P}_n=\{Ad_{P_n}^iAd_{Q_n}^j|i,j=0,\cdots,n-1\}\cong \mb
Z_n\times \mb Z_n,$$ and that
$\texttt{D}_n$ and $\texttt{P}_n$ are both maximal
diagonalizable subgroups of $\PGL(n,\mathbb{C})$.
Let $L=sl(n,\mathbb{C})$ and $G=\Int(L)\cong \PGL(n,\mathbb{C})$.
One has the standard isomorphism
$$M=M(t,\mathbb{C})\ot M(l_1,\mathbb{C})\ot \cdots \ot M(l_k,\mathbb{C})\rt M(n,\mathbb{C}),$$ where $n=t l_1\cdots l_k$ and $l_i|l_{i-1}$ for all $i$. It induces injective homomorphisms \be
D_t\ot\Pi_{l_1}\ot\cdots\ot \Pi_{l_k}\st S=\GL(t,\mathbb{C})\ot
\GL(l_1,\mb C)\ot \cdots \ot \GL(l_k,\mathbb{C})\rt
\GL(n,\mathbb{C}).\lb{a8}\ee
Let $A=A_0\ot A_1\ot\cdots \ot A_k\in S$. Then for $X=X_0\ot X_1\ot\cdots\ot X_k\in M$,
$$\Ad_A(X)=\Ad_{A_0}(X_0)\ot\Ad_{A_1}(X_1)\ot\cdots\ot \Ad_{A_k}(X_k).$$
Thus the adjoint action induces homomorphisms $$\GL(n,\mathbb{C})\rt \PGL(n,\mathbb{C}),$$
\bee \GL(t,\mathbb{C})\ot
\GL(l_1,\mb C)\ot \cdots \ot \GL(l_k,\mathbb{C})\rt \PGL(t,\mathbb{C})\times \PGL(l_1,\mb C)\times \cdots \times
\PGL(l_k,\mathbb{C}),\eee
\be\lb{a4} A=A_0\ot A_1\ot\cdots \ot A_k\mapsto Ad_A=(\Ad_{A_0},\Ad_{A_1},\cdots,\Ad_{A_k})\ee and by restriction
$$D_t\ot\Pi_{l_1}\ot\cdots\ot \Pi_{l_k}\rt \texttt{D}_t\times \texttt{P}_{l1}\times\cdots\times\texttt{P}_{l_k}.$$
Thus by (\ref{a8}) one has injective homomorphisms
\be \CD \texttt{D}_t\times
\texttt{P}_{l1}\times\cdots\times\texttt{P}_{l_k}\st
\PGL(t,\mathbb{C})\times \PGL(l_1,\mb C)\times \cdots \times
\PGL(l_k,\mathbb{C}) @>\phi>> \PGL(n,\mathbb{C}).
\endCD\lb{a9}
\ee
We will identify $\texttt{D}_t\times \texttt{P}_{l_1}\times \cdots
\times \texttt{P}_{l_k}$ with its image in $\PGL(n,\mb{C}).$
\begin{theorem}[Theorem 3.2 of \cite{hpp}] Any maximal diagonalizable subgroup of $\PGL(n,\mb{C})$ is conjugate to one and only one of the $\texttt{D}_t
\times \texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_k}$ with $n=t l_1\cdots l_k$ and each $l_i$ dividing $l_{i-1}$.
\end{theorem}
\begin{coro} \lb{b2} Any finite maximal diagonalizable subgroup of $\PGL(n,\mb{C})$ is conjugate to one and only one of the $\texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_k}$
with $n=l_1\cdots l_k$ and each $l_i$ dividing $l_{i-1}$.
\end{coro}
In the case $K=\texttt{P}_{l_1}\times \cdots \times
\texttt{P}_{l_k}$ is a finite maximal diagonalizable subgroup of
$\PGL(n,\mb{C})$, (\ref{a9}) becomes
\be \CD
\texttt{P}_{l1}\times\cdots\times\texttt{P}_{l_k}\st
\PGL(l_1,\mb C)\times \cdots \times
\PGL(l_k,\mathbb{C}) @>\phi>> \PGL(n,\mathbb{C})
\endCD\lb{b0}
\ee
\bigskip
Let $K\st \PGL(n,\mb{C})$ be a finite maximal diagonalizable
subgroup. Let
$$p:\GL(n,\mathbb{C})\rt \PGL(n,\mathbb{C})$$ be the projection.
\begin{defi} \lb{d} For
any $\si\in K$ fix some $\widetilde{\si}\in p^{-1}(\si)$. For any
$\si,\tau\in K$, $\widetilde{\si} \widetilde{\tau}
\widetilde{\si}^{-1} \widetilde{\tau}^{-1}=lI_n$ as
$p(\widetilde{\si} \widetilde{\tau} \widetilde{\si}^{-1}
\widetilde{\tau}^{-1})=1$. Clearly $l$ is independent of the
preimages $\widetilde{\si},\widetilde{\tau}$ chosen. Define
$<\si,\tau>= l$.
\end{defi}
\begin{lem}[Lemma 3.4 of \cite{hg}] The map $<,>:K\times K\rt \mathbb{C}^{\times}$ is an anti-symmetric pairing on $K$, which is invariant under $N_G(K)$.
\end{lem}
\begin{prop}\lb{b1}
Let $G=\PGL(n,\mb{C})$ and $K$ be a maximal diagonalizable subgroup
of $G$. If $K=\texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_k}$
with $n=l_1\cdots l_k$ and each $l_i$ dividing $l_{i-1}$, then $<,>$
is nonsingular on $K$. Thus $(K,<,>)$ is a nonsingular symplectic
abelian group isometric to $\mb H_{l_1}\op \cdots \op \mb H_{l_k}$.
\end{prop}
\bp For $i=1,\cdots,k$, let
$$\si_i=(1,\cdots,1,Ad_{P_{l_i}},1,\cdots,1)\in K,
\ \tau_i=(1,\cdots,1,Ad_{Q_{l_i}},1,\cdots,1)\in K$$ where
$Ad_{P_{l_i}}$ and $Ad_{Q_{l_i}}$ are in the $i$-th position. Then
$\{\si_i,\tau_i|i=1,\cdots,k\}$ is a set of generators of $K$ and
any element in $K$ can be written uniquely as
$\si_1^{i_1}\tau_1^{j_1}\cdots \si_k^{i_k}\tau_k^{j_k}$. By simple
computation one has for $i \neq j$,
$$<\si_i,\tau_i>=\omega_{l_i},\ <\si_i,\tau_j>=1,$$ $$<\si_i,\si_j>=1,\ <\tau_i,\tau_j>=1.$$
Thus $(\si_i,\tau_i)$ is a hyperbolic pair of order $l_i$ and spans
the symplectic subgroup $\texttt{P}_{l_i}$ isometric to $\mb
H_{l_i}$. It is clear that such subgroups $\texttt{P}_{l_i}$ are
mutually orthogonal to each other. The map
\be\lb{c4}\texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_k}\rt
\mb H_{l_1}\op \cdots \op \mb H_{l_k},\
\si_1^{i_1}\tau_1^{j_1}\cdots \si_k^{i_k}\tau_k^{j_k}\mapsto
((i_1,j_1),\cdots,(i_k,j_k))\ee is clearly an isometry of
nonsingular symplectic abelian groups. \ep
As a corollary of Corollary \ref{b2}, Proposition \ref{b1} and Corollary
\ref{d9} one has the following result.
\begin{prop}\lb{d6}
There is a one-to-one correspondence between conjugacy class of
finite maximal diagonalizable subgroups of $\PGL(n,\mathbb{C})$ and
nonsingular symplectic abelian groups of order $n^2$ .
\end{prop}
\section{Weyl groups of finite maximal diagonalizable subgroups of $\PGL(n,\mb C)$ }
\setcounter{equation}{0}\setcounter{theorem}{0}
Recall that $L=sl(n,\mb C)$ and $K$ is a finite maximal
diagonalizable subgroup of $G=\PGL(n,\mb{C})$. First we will
describe the grading of $sl(n,\mb C)$ and $gl(n,\mb C)$ induced by
the action of $K$.
At first let $K=\texttt{P}_n$. The character group $\wh{K}$, written
additively, is generated by $\bt_n$ and $\al_n$, which are dual to
$\Ad_{P_n},\Ad_{Q_n}$:
$$\bt_n(\Ad_{P_n})=\omega_n,\ \bt_n(\Ad_{Q_n})=1,$$
$$\al_n(\Ad_{P_n})=1,\ \al_n(\Ad_{Q_n})=\omega_n.$$
Thus $\wh{K}=\{i\bt_n+j\al_n|(i,j)\in \mathbb{Z}_n\times
\mathbb{Z}_n\}\cong \mathbb{Z}_n^2$.
One has
$$\bt_n(\Ad_{P_n}^{i}\Ad_{Q_n}^{j})=\omega_n^i=<\Ad_{Q_n}^{-1},\Ad_{P_n}^{i}\Ad_{Q_n}^{j}>$$
and
$$ \al_n(\Ad_{P_n}^{i}\Ad_{Q_n}^{j})=\omega_n^j=<\Ad_{P_n},\Ad_{P_n}^{i}\Ad_{Q_n}^{j}>,$$
so $$\bt_n^*=\Ad_{Q_n}^{-1},\ \al_n^*=\Ad_{P_n}.$$ As by (\ref{b3}) one has $(\ga+\dt)^*=\ga^*\dt^*$,
\be \lb{a1}(i\bt_n+j\al_n)^*=\Ad_{P_n}^{j}\Ad_{Q_n}^{-i}.\ee As
$$\Ad_{Q_n}(Q_n^i P_n^j)=\omega_n^{-j}Q_n^i P_n^j=(i\bt_n-j\al_n)(\Ad_{Q_n})Q_n^i P_n^j,$$
$$\Ad_{P_n}(Q_n^i P_n^j)=\omega_n^{i}Q_n^i P_n^j=(i\bt_n-j\al_n)(\Ad_{P_n})Q_n^i P_n^j,$$ one has
\be \lb{z} Q_n^i P_n^j\in L_{i\bt_n-j\al_n}.\ee In particular
$$P_n\in L_{-\al_n},\ Q_n\in L_{\bt_n}.$$ Note that $tr (Q_n^i P_n^j)=0$ unless
$(i,j)=(0,0)$. Let \be X_{i\bt_n-j\al_n}=Q_n^i P_n^{j}.\lb{a2}\ee
Then one has the following gradings
$$gl(n,\mathbb{C})=\oplus_{(i,j)}\ \mathbb{C} Q_n^i P_n^j=\op_{\ga\in \wh K}\ \mb C X_{\ga},$$
$$sl(n,\mathbb{C})=\oplus_{(i,j)\neq (0,0)}\ \mathbb{C} Q_n^i P_n^j=\op_{\ga\neq 0}\ \mb C X_{\ga}.$$ So
$\ddt(gl(n,\mathbb{C}),K)=\wh{K}$ and
$\ddt(sl(n,\mathbb{C}),K)=\wh{K}\setminus \{0\}$. Note that each
root space is one-dimensional, and for any $\ga\in\wh K$, by
(\ref{a1}) and (\ref{a2}) one has \be
\ga^*=(Ad_{X_\ga})^{-1}.\lb{a3}\ee The following result is
originally Theorem 10 of \cite{jpt}, for its proof we refer the
readers to Proposition 4.4 of \cite{hg}.
\begin{theorem}
Let $G=\PGL(n,\mb C)$ and $K=\texttt{P}_n$. One has $W_G(K)\cong
\SL(2,\mb Z_n)$ and is generated by $s_{\al_n}$ and $s_{\bt_n}$.
\end{theorem}
Next let $K=\texttt{P}_{l_1}\times \cdots \times \texttt{P}_{l_k}$
with $n=l_1\cdots l_k$ and each $l_i$ dividing $l_{i-1}$. As
$M(n,\mb C)=M(l_1,\mb C)\ot\cdots\ot M(l_k,\mb C)$, and $M(l_i,\mb
C)=\op_{\ga\in \wh{\texttt{P}_{l_1}}}\mb C X_\ga$, one has
$$ M(n,\mb C)=\op_{(\ga_1,\cdots,\ga_k)}\ \mb C X_{\ga_1}\ot\cdots\ot X_{\ga_k}.$$
Note that $\wh K=\wh{\texttt{P}_{l_1}}\times \cdots \times
\wh{\texttt{P}_{l_k}}$. Let $\ga=(\ga_1,\cdots,\ga_k)\in \wh K$ with
$\ga_i\in \wh{\texttt{P}_{l_i}}$. For any
$\si=(\si_1,\cdots,\si_k)\in K$, $\ga(\si)=\ga_1(\si_1)\cdots
\ga_k(\si_k)$ and one has \bee
\begin{split} \si\cdot X_{\ga_1}\ot\cdots\ot
X_{\ga_k}&=\si_1\cdot X_{\ga_1}\ot\cdots\ot \si_k\cdot
X_{\ga_k}\\&=\ga_1(\si_1)X_{\ga_1}\ot\cdots\ot
\ga_k(\si_k)X_{\ga_k}\\&=\ga(\si) X_{\ga_1}\ot\cdots\ot
X_{\ga_k}.\end{split}\eee
So $X_{\ga_1}\ot\cdots\ot
X_{\ga_k}\in L_\ga$. Note that $tr (X_{\ga_1}\ot\cdots\ot
X_{\ga_k})=\Pi_i tr(X_{\ga_i})$, which is nonzero if and only if
$\ga_1=\cdots=\ga_k=0$. Let
$$Y_\ga=X_{\ga_1}\ot\cdots\ot X_{\ga_k}.$$ Then
$$gl(n,\mathbb{C})=\op_{\ga\in \wh K}\ \mb C Y_\ga$$ and
$$sl(n,\mathbb{C})=\op_{\ga\neq 0}\ \mb C Y_\ga.$$
So $\ddt(gl(n,\mb C),K)=\wh K$ and $\ddt(sl(n,\mb C),K)=\wh
K\setminus \{0\}$. Note that each root space is also one-dimensional
and consists of semisimple elements.
\begin{lem}For any $\ga\in \wh K$, $\ga^*=(\Ad_{Y_\ga})^{-1}$. \end{lem}
\bp For any $Ad_X\in K$, as $Y_\ga$ is invertible, $$Y_\ga^{-1} X
Y_\ga X^{-1}=Y_\ga^{-1} (\ga(Ad_X)Y_\ga)=\ga(Ad_X) I,$$ so
$<(\Ad_{Y_\ga})^{-1},Ad_X>=\ga(Ad_X)$. Thus
$\ga^*=(\Ad_{Y_\ga})^{-1}$.
\ep
Recall in (\ref{b0}) one has the
embedding \bee \CD
\texttt{P}_{l1}\times\cdots\times\texttt{P}_{l_k}\st \PGL(l_1,\mb
C)\times \cdots \times \PGL(l_k,\mathbb{C}) @>\phi>>
\PGL(n,\mathbb{C}).
\endCD
\eee
Let $N(\texttt{P}_{l_i})$ be
the normalizer of $\texttt{P}_{l_i}$ in $\PGL(l_i,\mathbb{C})$, then
clearly $\phi$ restricts to \bee \CD
N(\texttt{P}_{l_1})\times\cdots\times N(\texttt{P}_{l_k}) @>\phi>>
\PGL(n,\mathbb{C}).
\endCD
\eee The left hand side is in $N_G(K)$. As
$$N(\texttt{P}_{l_i})/\texttt{P}_{l_i}\cong \SL(2,\mathbb{Z}_{l_i}),$$ one has
\be\lb{s} \SL(2,\mathbb{Z}_{l_1})\times\cdots\times
\SL(2,\mathbb{Z}_{l_k})\st W_G(K).\ee
Let $\ga\in \ddt(sl(n,\mb C),K)$ and $G=\PGL(n,\mb C)$. Assume the order of $\ga$ is $m$ and choose $\si\in K$ satisfying $\ga(\si)=\omega_m$. As $\si\in Z(Ker \ga)$,
$\Ad_\si$ maps $Z(Ker \ga)_0$ into $Z(Ker \ga)_0$. Let $f_\si:G\rt
G,\ \eta\mapsto\si\eta\si^{-1}\eta^{-1}$. Then $f_\si(Z(Ker
\ga)_0)\st Z(Ker \ga)_0$. Denote
$Z(\Ker \ga)_0$ by $Z_0$.
\begin{lem}\lb{a6}
(1) The map $f_\si:Z_0\rt Z_0,\ \eta\mapsto\si\eta\si^{-1}\eta^{-1}$ is a
continuous epimorphism.
(2) Assume $\ga^*\in Z_0$. Then there exists $\zeta\in Z_0$ with
$f_\si(\zeta)=\ga^*$. One has $\zeta\in N_G(K)$ and
$Ad_\zeta:K\rt K,\ \tau\mapsto \zeta\tau\zeta^{-1}$ is just the transvection $$ s_\ga:K\rt K,\ \tau\mapsto \tau(\ga^*)^{-\ga(\tau)}$$ as in
(\ref{b9}). (Note that as a subgroup of G,
$K$ is a multiplicative abelian group.) Thus $ s_\ga\in W_G(K)$.
\end{lem}
\bp As it was shown in Proposition \ref{d0} (3) that $L_{[\ga]}$ is
a toral subalgebra of $L$, the lemma follows from Lemma 3.7 of
\cite{hg}. \ep
\begin{lem}\lb{c1} For each $\ga\in\ddt(sl(n,C),K)$, $ s_\ga\in W_G(K)$. \end{lem}
\bp Assume
$\ga=(a_1\bt_{l_1}+b_1\al_{l_1},\cdots,a_k\bt_{l_k}+b_k\al_{l_k})$,
then $$Y_\ga =Q_{l_1}^{a_1}P_{l_1}^{-b_1}\ot\cdots\ot
Q_{l_k}^{a_k}P_{l_k}^{-b_k}.$$
By Corollary \ref{l} and (\ref{s}) there exists some
$Y_\dt=Q_{l_1}^{c_1}\ot\cdots\ot Q_{l_k}^{c_k}$ such that
$Ad_{Y_\ga}$ is conjugate to $Ad_{Y_\dt}$ under $N_G(K)$. Assume $
Y_\dt=Q_{l_1}^{c_1}\ot\cdots\ot Q_{l_k}^{c_k}$ as an element of
$\GL(n,\mathbb{C})$ has order $m$. Then $L_{i\dt}=\mb C Y_{\dt}^i$
for $i=1,\cdots,m-1$ and $L_{[\dt]}=\oplus_{i=1}^{m-1} \mb C
Y_\dt^i$ is an abelian Lie algebra consisting of semisimple
elements. We will show $\Ad_{Y_\dt}\in Z(\Ker \dt)_0$, then
$\Ad_{Y_\ga}\in Z(\Ker \ga)_0$ as $Ad_{Y_\ga}$ and $Ad_{Y_\dt}$ are
conjugate. Thus $\ga^*=(\Ad_{Y_\ga})^{-1}\in Z(\Ker \ga)_0$ and $
s_\ga\in W_G(K)$ by Lemma \ref{a6}.
The set $D_i$ of
eigenvalues of $Q_{l_i}^{c_i}$ is a cyclic group for each $i$. Let
$D$ be the set of eigenvalues of $Y_\dt$. For any $a,b\in D$,
$a=a_1\cdots a_k,\ b=b_1\cdots b_k$ with $a_i,b_i\in D_i$. Then
$ab^{-1}=(a_1 b_1^{-1})\cdots (a_k b_k^{-1})\in D$. So $D$ is also a
group. As the order of $Y_\dt$ is $m$, $D$ is a subgroup of the
cyclic group $C_m$. Then $D=C_m$ as the order of $Y_\dt$ is $m$.
Let $\omega=\omega_m$ then in some suitable basis of
$\mathbb{C}^{n}$,
$$Y_\dt=diag(1,\cdots,1,\omega,\cdots,\omega,\cdots\cdots,\omega^{m-1},\cdots,\omega^{m-1}),$$
where for $j=0,1,\cdots,m-1,$ there are $t_j$ copies of $\omega^j$
on the diagonal with each $t_j>0$. Let $s=\frac{2\pi i}{m}$ and
$$A=diag(0,\cdots,0,s,\cdots,s,\cdots\cdots,(m-1)s,\cdots,(m-1)s),$$ where for $j=0,1,\cdots,m-1$ there are $t_j$ copies of $js$ on the
diagonal. Then $exp(A)=Y_\dt$.
Let $D=(d_{i,j})_{m\times m}$ where $d_{i,j}=w^{ij}$ for
$i,j=0,1,\cdots,m-1$. As $D$ is invertible, there are unique complex
numbers $c_0,c_1,\cdots,c_{m-1}$ satisfying
$$D\cdot (c_0,c_1,\cdots,c_{m-1})^{t}=(0,s,\cdots,(m-1)s)^{t}.$$
Then
$\sum_{i=0}^{m-1}c_i Y_\dt^i=A$ and $exp(\sum_{i=0}^{m-1}c_i
Y_\dt^i)=Y_\dt.$ Then as $$\sum_{i=1}^{m-1}c_i\ {Y_\dt^i}\in
L_{[\dt]}\ \an \ \Lie Z(\Ker \dt)_0=\ad L_{[\dt]}$$ one has
$$\Ad_{Y_\dt}=exp(\ad(\sum_{i=1}^{m-1}c_i\ {Y_\dt^i}))\in Z(\Ker
\dt)_0.$$ \ep
Let $K$ be a finite maximal diagonalizable subgroup of
$G=\PGL(n,\mathbb{C})$. Recall that $K$ has a $W_G(K)$-invariant
anti-symmetric pairing $<,>$ and $(K,<,>)$ is a nonsingular
symplectic abelian group by Proposition \ref{b1}. Thus
$W_G(K)\st\Sp(K)$, where $\Sp(K)$ is the isometry group of
$(K,<,>)$.
\begin{theorem}\lb{d7}
The Weyl group $W_G(K)$ equals $\Sp(K)$, and is generated by
the set of transvections $ s_\ga$ with $\ga\in\ddt(sl(n,\mb C),K)$.
\end{theorem}
\bp By Lemma \ref{c1} one has $s_\ga\in W_G(K)$ for each
$\ga\in\ddt(sl(n,C),K)$. As $\ddt(sl(n,C),K)=\wh K\setminus \{0\}$,
$W_G(K)$ contains all $ s_\si$ with $\si$ a nonidentity element in
$K$. By Theorem \ref{c3} all such $ s_\si$ generate $\Sp(K)$, thus
$W_G(K)=\Sp(K)$.
\ep
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According to the Toronto Star’s Damien Cox, the Leafs are expected to sign 24 year old forward prospect Leo Komarov to an entry-level contract at some point in the near future. The Maple Leafs hold Komarov’s NHL rights as a 6th round selection, 180th overall back in 2006. Komarov was born in Estonia, but grew up playing hockey in Finland where currently holds a duel-citizenship along with Russia. He has spent the last two years playing in the KHL, earning himself an all-star appearance this past season. In 52 games played for OHK Dynamo Moskva, he recorded 14 goals and 12 assists, in addition to 70 penalty minutes. Komarov also impressed during a brief playoff run, scoring 4 times and adding 2 helpers in just 6 games.
Drawing comparisons to the likes of Jarkko Ruutu or Steve Ott, Komarov is described as a classic hockey pest who plays with a great deal of tenacity, heart and energy. He is a hitting machine who covers a lot ice very quickly and finishes all his checks with force. Standing at roughly 5’11 200 lbs, Komarov is not a big guy but he is an excellent skater whose uptempo style of play can be a strong asset on the forecheck. Komarov is a particularly effective player in open ice, taking away time and space with good positional instincts combined with remarkable speed and an active stick. He also engages well along the boards in the offensive zone and shows a willingness to drive to the net with the puck.
Komarov has played a little bit of centre, but is likely a better fit playing the wing. While the young Estonian doesn’t quite have the hands of a top six forward, he possesses a decent shot and can be counted on to chip in offensively every now and then. Komarov’s greatest strength is the ability to absolutely infuriate his opponents by playing hard and tight, chirping all game long, and occasionally pushing the limits of dirty play. During his time in the Finnish SM-Liiga, he was named the league’s “Most Hated Player” three times. However, Komarov antics can also frustrate his own coaches and teammates by losing his temper and taking unnecessary or ill-advised penalties.
At 24 years of age, with an NHL ready body and two years of experience playing against men in the KHL, it won’t take long for Komarov to see some NHL action on the team’s 4th line. He may challenge for a checking role in September, but if sent to the Marlies, he should be ready for an injury call-up to the big club as early as midseason next year. In time, Komarov could develop into an effective bottom six agitator.
—
It’s also interesting to note that this is yet another John Ferguson Jr. draft pick that appears poised to one day wear a Toronto Maple Leafs’ sweater. Even just from the 2006 and 2007 drafts, Fergson’s regime has produced NHL’ers Nikolai Kulemin, James Reimer, Carl Gunnarsson, Jiri Tlusty and Viktor Stalberg, with prospects Matt Frattin, Juraj Mikus, Korbinian Holzer, and Leo Komarov not far behind. Say what you want about him, but the man can really draft.
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The funny poems will surely make you chuckle & help convey the right message also. Many of these poems are of the short nature. This is generally my favorite because they are easier to comprehend and are quick to read and memorize. They should be suitable for children, so if you like these versions better, feel free to use them. Get access to all Funny poems, Funny love poems, Short funny poem.sheets,:
'Haha! Cat, you pay the fees
For acting so bad with your peers! '
'Help, please'
'Call me bad boy, I love to tease
I'm getting outta here, say cheers! '
Then a bird came flying in the street
With his wings, following a beat
Tictactictactictac
Tictactictactictac< br>He saw something red
So he stopped above the cat's head:
'Oh, poor white cat
Are you trapped in this hat? '
'Damn I am! '
'Let me help you, sad little cat
Here, you're out, care for a chat? '
Rumors I heard
Say the cat ate the bird.
Funny - Fat Cat
Rita El Khoury
Funny is laughter
Funny is fun
Funny is healthy
Funny is nice
Funny is amusing
Funny is comical
A pelt of pranks
Above the normal ranks
Is funny when it is on
Yet quite sad when gone
Funny
Sylvia Chidi
t's funny how hello is always finished with goodbye
it's funny how good memories can make you cry
it's funny how forever never seems to last
it's funny how much you would lose if you forgot about your past
it's funny how ?friends? can just leave when you are down
it's funny how when you need someone they never are around
it's funny how people change and think they are so much better
it cant be packed in one letter
it's funny how one night can contain so much regret
because the funniest part of all, is none of thats funny to me
Its Funny
jutyum brendaro
I'M Funny
BRad Ical
Believe me, I run from dying
I know, it's so dumb, but funny trying
WAR, around me everywhere
SOLDIERS, killing everyday, I swear.
Truly, I have nothing to fear
Because when I look in the mirror
I see myself with no hair
A jock, playing, fakinit, like its there
I know... you hate that, my dear
Made your point of view Claire
Telling me, if I show up, you will run out there
Hiding from me, on your feet, downstairs
Anyway, sweetie, anyway, truly I don't care
Whatever you say honey, you love me
With your smile you illuminate my world
You can't imagine my happy feelings,
whenever I tell you, that, without you,
my life has no meaning at all
I love myself, maybe, but I've loved you before
Knowing in your heart Im not outdoors
Because you love me too and I am sure
I Hope, my life is long, because we are made for it
You are what a man wants from this life and more
I love you, its true and beautiful
So let us live our magic story
All thank to My dear friend Cruz
Funny Wear, Steel Soldiers
Molay Toufik
Welcome to the Colosseum
Where pain is pleasure and it is our pleasure to be of service
Lay back on the rack
The waxing poetics are hot and the iced humor is very dry
The fence sitter splinters are sharp but the elitist tongues are sharper
There are many painful pleasures which abound
We guarantee your total satisfaction
Or we'll give you fifty lashes of your back for free
For no one ever enters with a smile
Who doesn't leave here with a grimace
(2007)
A Funny Thing Happened On The Way To The Forum
Ted Sheridan
My daddy's so funny; always making me laugh
But, he's the best daddy anybody could have
Every time he comes around he has something funny to say
On days when I feel down daddy can always makes my day
My daddy's so funny even when he's not trying to be
He has a way with words and gestures that always tickles me
He makes funny expressions, and he's so witty too
He's so funny he'll make you forget you were feeling blue
My daddy's so funny, but he is also sweet
He will do anything to help someone else; even get them off the street
You can find him givin' change, or just standing up talkin'
To a person that most people would look at and just keep on walkin'
My daddy's so funny, but he is such a good provider
He would do anything to take care of his family, even use the car like a taxi driver
I'll never forget the time he got up at 5: 00 in the mornin'
He walked me all the way to my train stop, even though it was stormin'
My daddy's so funny, but helped raise us girls so well
When my mama told us ?no, ? we couldn't run to him and tell
He taught us to be dainty and to act like ladies
He was always involved in our lives from the time that we were babies
My daddy's so funny, but he takes good care of my mother
No matter how tough times would get, he always stood by her
He has been with her for more than 50 years
With her he has shared much marital bliss and many tears
My daddy's so funny, but there is something you should know
He might laugh and joke a lot, but he takes life seriously though
His personality and the things he does are not for him to be seen
He just is who he is, and really loves people; from that we can all glean
My daddy's so funny, when you ask him what he has to give
He'll tell you in a minute, I got mo' cold cash than you got days to live!
My daddy is so funny, when I just think about him I laugh
I thank God for my daddy, one of the greatest gifts any girl could have.
My Daddy's So Funny
Deidre Blair
all the money
to the last penny
was found to be proxy
in that country
all essentials
should be freely available
the governance ordered
which brought in this disorder
who does not have a need
for the essentials
right from the cop to the thief
had a need for it that day
opened stores were looted
closed one's were broke open
hand in glove were they
the cop and the thief
a commotion in town
every law broken
the temples barren
even the priest lost his calm
a situation which would not calm down
the governing body had almost broke down
a let loose situation for the hooligans
WHO DOES NOT HAVE A NEED FOR ESSENTIALS
GOD INTERVENED
A FUNNY SITUATION DEMANDED...
A SUPER POWER'S INTERVENTION
AN EARTH QUAKE ROCKED THE TOWN
ALL LOOTERS WERE DRIVEN OUT OF STORES
TO SEE MAN MADE ESSENTIALS CRUMBLE
LIFE THE MOST ESSENTIAL
# A Funny Situation Demanded.....
Samanyan Lakshminarayanan
You broke my heart
So many times
Was he sweeter
Is he better
Was the grass greener
On the other side
Wipe your eyes
Please don't cry
You meet a monster
Clever like Dracula
Please don't bother me
I am eating lobster
With my brother
I hate your sorry lines
And the stupid lies
Get another guy
My love bye, bye
Shut the door
I need time to recover
Sorry it's over
. It's Over... (Funny)
Eyan Desir
Funny to be a Century
And see the People going by
I should die of the Oddity
But then?I'm not so staid as He
Does he keep His Secrets safely very
Were He to tell extremely sorry
This Bashful Globe of Ours would be
So dainty of Publicity
Funny to Be A Century
Emily Dickinson
hei, hei, hei, hei,
heih, heih, heih, heih,
who says one, is only one
One only one and remain only one,
One and one not make eleven,
join to make it, only remain one,
one will always remain only one,
try, try and try, one will remain one
One mixing with one
make it within one,
one minus one will not remain one;
life is zero without only one,
Even God is one, add no more one,
Add more and more, may not look even,
Christ is one, Allah is one,
They represent one. Mighty only one,
Heaven is one, hell also one,
universe is one and earth also one,
Good is one and bad also one,
kind may be different but meaning only one
What is continent and what is region
Who is African and who is Asian
What is Chilean and what is Nigerian
Someone may come and say I am Serbian,
Not all religions but only human one,
Love them all and hate no one,
reject no one and extend embrace,
No love and hate but smiling only face
I continue to bow,
Lye always low,
Love to man and woman,
Oh, God, show me only one
Love and hate can't ever one,
sorrow and joy also not one,
poles remain apart and not called one,
Only HE may appear omni and look only one
Funny 1
Hasmukh Amathalal.
Funny
Sandra Osborne never are around
it's funny how people change and think they're so much better
it's funny how many lies are packed into one love letter
it's funny how one night can contain so much regret
it's funny how you can forgive but not forget
it's funny how ironic life turns out to be
but the funniest part of all, is none of thats funny to me
Funny... But Not
arianna loshnowsky
Funny you should say,
That the world is spining,
Funny you should say,
That Manchester is winning.
Funny you should say,
That time flies fast,
Funny you should say,
That's all in the past.
Funny you should say,
That death is dawning near,
Funny you should say,
He turned a deaf ear.
Funny you should say,
That life is full of honey,
But for that you need money,
Hmmm... funny.
Funny
La Que Escribe Poesia
It is not only funny
But very astonishing and clumsy
We all know about the holiness of boobs
Holy mention too can be found in books
Women are much more praiseworthy
Some may wear the ugliness gifted from almighty
The deformity and unnatural growth is not in our hands
The body as such will be submerged and melted at the end
All nourishment and food is catered from that part
Our life gets shape and growth from the very start
Let us adore and praise the ladies for commendable job
We should not denigrate their honor and rob
It may be shown as world record
Many may take it as mere reference for looking forward
Any good looking natural piece deserves to be praised
But wrong way of projection should also be outrightly condemned
It must be taken as deformity in body
It is not done for records by somebody
It is really annoying for its wrong way of presentation
Otherwise it is not even worth for its mention
The opinion and perception may vary
The impression and meaning may also differently carry
Yet the basic concept of giving honor to ladies need not be compromised
Ladies deserve good comments and treatment as consciously promised
Only Funny
Hasmukh Amathalal
Something's wrong with me
Or them
I'm talking the men and women
Who deliver the weather
All of 'em like heat
While I'm a blizzard through and through.
You Think That's Funny
Charles Chaim Wax.
Funny Drinks
DAVID GERARDINO
It's not funny when
you laugh and call
me names
I don't smile when
you push me down
and laugh
I don't find it very funny
when you dis me and
say it was just a joke
I'm not the one who laughs
when you call my name but
turn away
Your jokes are not funny
and never will be so stop
saying them
Not Funny
Mariana Zita
There once was an old Irish Setter,
So ugly that no one would pet her,
Then one day she was struck
By a huge garbage truck,
I swear the dog looks so much better.
Morbidly Funny Doggy Rhyme
Robert Mestre.
A Story Of Funny Students
Rajaram Ramachandran
Its Funny...
MeganOlivia Maxwell
he's been waiting for this his whole life...
he's wanted her for a long time...
he's finally going to lose his virginity...
he's dreamed about this for years...
he's been in a daze scince she told him it was tim...
he's bought all the condoms he could buy...
he's waiting impatiently for the time to pass...
he's going to become a man...
(she shows up)
he's almost riping her clothes off...
she's almost riping his clothes off...
they've been goig out for 5 years...
he's been waiting even longer...
he's got it all in a plan...
then she whispers 'i'm a man'
This Is So Funny I Laughed While Writing It Lol
Melissa Broomhead
For two lovebirds, I met a Stratford on Avon
With the first initials D & T
They know who they are.
Love is a funny thing,
one minute you are up,
then the next you are down.
Then your head is spinning around,
your seeing stars,
dancing on the moon.
Love is a funny thing,
and that is for sure.
Love Is A Funny Thing
David Harris Moniter was a teacher
A shortcut was a dangerous way
And Shout-Outs done by preachers would ever!
The Age Before The Computer (Really Funny)
Devang Gandhi.
I have eaten
the plums
that were in
the icebox
and which
you were probably
saving
for breakfast
Forgive me
they were delicious
so sweet
and so cold.
| 97,789
|
\begin{document}
\maketitle
\noindent
\begin{abstract}
\noindent
Hannes Leitgeb formulated eight norms for theories of truth in his paper: `What Theories of Truth Should be Like (but Cannot be)'.
We shall present in this paper a mathematical theory of truth for suitably constructed languages which contain the first-order language of set theory, and prove that it satisfies all those norms.
\vskip12pt
\noindent {{\bf MSC:} 00A30, 03B10, 03C62, 03D60, 03F50, 06A07, 47H04, 47H10}
\vskip12pt
\noindent{\bf Keywords:} Set theory, Zermelo-Fraenkel, model, first--order, theory, language, sublanguage, formal, sentence, truth, provability, fixed point, chain, transfinite sequence.
\end{abstract}
\baselineskip 16pt
\section{Introduction}\label{S1}
Hannes Leitgeb formulated in \cite{[16]} the following norms for theories of truth:
\begin{itemize}
\item [(a)] Truth should be expressed by a predicate (and a theory of syntax should be available).
\item[(b)] If a theory of truth is added to mathematical or empirical theories, it
should be possible to prove the latter true.
\item[(c)] The truth predicate should not be subject to any type restrictions.
\item[(d)] $T$-biconditionals should be derivable unrestrictedly.
\item[(e)] Truth should be compositional.
\item[(f)] The theory should allow for standard interpretations.
\item[(g)] The outer logic and the inner logic should coincide.
\item[(h)] The outer logic should be classical.
\end{itemize}
We present a mathematical theory of truth (shortly MTT) that satisfies norms (a)--(h). It is formulated for suitably constructed languages which contain the first-order language $L=\{\in\}$ of set theory. They are sublanguages of $\mathcal L=\{\in,T\}$, where $T$ is a monadic predicate. While not entailing all formalities of first-order languages, they are closed under logical connectives and quantifiers of first--order predicate logic.
Zermelo-Fraenkel (ZF) set theory, sets of its minimal model $M$, concepts definable in $L$ and classical two-valued logic are main tools. MTT proves ZF set theory modelled by $M$ true. As stated in \cite{Ho}, ``First-order
Zermelo-Fraenkel set theory is widely regarded as the standard of
mathematical correctness, in the sense that a proof is correct if and only if it
can be formalised as a formal proof in set theory."
As shown in \cite{[BG]}, ``with some elementary assumptions about truth and logic, a logical disaster ensues."
It is caused by Liar paradox.
Paradoxes led Zermelo to axiomatize set theory. By \cite{[16]} Tarski ``excluded all Liar-like sentences from being
well-formed" from his theory of truth (cf. \cite{[20]}).
Construction of sublanguages enables MTT to satisfy also the following norms:
\begin{itemize}
\item[(i)] The mathematical theory of truth should be free from paradoxes.
\item[(j)] Truth should be defined for languages within the very languages themselves.
\end{itemize}
MTT is immune to `Tarski's Commandment' (cf. \cite{[MG]}), to Tarski's Undefinability Theorem (cf. \cite{[20]}), to Tarskian hierarchies (cf. \cite{[H]}), and to `Revenge of the Liar' (cf. \cite{Fe}).
Model $M$ can be replaced by any countable model of ZF or
other set theory. Examples of such theories are presented, e.g., in \cite{[Ku]}.
The language $L$ can be replaced by the language of arithmetic, and $M$ by a standard model of arithmetic. In \cite{[9]} MTT is extended to a more general class of languages.
PKF theory (cf. \cite{[Ha],[8]}) axiomatizes Kripke's theory of truth (cf. \cite{[15]}). Both inner and outer logic of PKF are partial, whence norms (h) and (j) are not valid.
Kripke's theory don't satisfy norms (d) and (g) because of partially interpreted truth predicate and three-valued inner logic.
Theory KF, which axiomatizes Kripke's closed-off models (cf. \cite{[F],Fe,[15]}), don't satisfy norms (d), (g) and (i).
Norm (g) is rejected in \cite{Fe}.
Preferring norm (g) over any mixture of two different logics, and (h) over partial logic, we use classical logic and construct sublanguages.
Modifying Wittgenstein: Whereof one must not speak, thereof one can be silent.
\section{Preliminaries}\label{S3}
As for the used terminology, cf. e.g., \cite{[11],[Ku]}. Denote by $M$ the minimal, countable, transitive and well-founded model of ZF set theory constructed in \cite{[4]}. It follows from \cite[Lemma II.8.22]{[Ku]} that for each sentence $A$ of the language $L=\{\in\}$ either $A$ or its negation $\neg A$ is
true in $M$ in the sense defined in \cite{[Ku]} (cf. \cite[II.7 and p. 237]{[Ku]}).
$M$ contains by \cite{[4],[13]} natural numbers.
Assume that their names, called numerals, are defined in $L$, e.g., as in \cite{[Fi]}.
First we shall construct a family of sublanguages for the language $\mathcal L=\{\in,T\}$, where $T$ is a monadic predicate.
Denote by $D$ the set of
G\"odel numbers of sentences of $\mathcal L$ in its fixed G\"odel numbering. Let \#$A$
denote the G\"odel number of a sentence $A$ of $\mathcal L$, and $\left\lceil A\right\rceil$ the numeral of \#$A$.
Given a subset $U$ of $D$,
define subsets $G(U)$ and $F(U)$ of $D$ by following rules (iff abbreviates if and only if):
\begin{enumerate}
\item[(r1)] If $A$ is in $L$, then \#$A$ is in $G(U)$ iff $A$ is true in $M$, and in $F(U)$ iff $\neg A$ is true in $M$.
\item[(r2)] If $A$ is in $\mathcal L$, then \#$T(\left\lceil A\right\rceil)$ is in $G(U)$ iff \#$A$ is in $U$, and in $F(U)$ iff
\#[$\neg A$] is in $U$.
\end{enumerate}
Sentences determined by rules (r1) and (r2), i.e., all sentences $A$ of $L$ and those sentences $T(\left\lceil A\right\rceil)$ of $\mathcal L$ for which \#$A$ or \#[$\neg A$] is in $U$, are called {\em basic sentences}.
Next five rules deal with logical connectives. Let $A$ and $B$ be sentences of $\mathcal L$.
\begin{enumerate}
\item[(r3)] Negation rule: \#[$\neg A$] is in $G(U)$ iff \#$A$ is in $F(U)$, and in $F(U)$ iff \#$A$ is in $G(U)$.
\item[(r4)] Disjunction rule: \#[$A\vee B$]
is in $G(U)$ iff \#$A$ or \#$B$ is in $G(U)$, and in $F(U)$ iff \#$A$ and \#$B$
are in $F(U)$.
\item[(r5)] Conjunction rule: \#[$A\wedge B$]
is in $G(U)$ iff \#[$\neg A\vee \neg B$] is in $F(U)$ iff (by (r3) and (r4)) both \#$A$ and \#$B$ are in $G(U)$. Similarly, \#[$A\wedge B$] is in $F(U)$ iff \#[$\neg A\vee \neg B$] is in $G(U)$ iff \#$A$ or \#$B$ is in $F(U)$.
\item[(r6)] Implication rule: \#[$A\rightarrow B$] is in $G(U)$ iff \#[$\neg A\vee B$] is in $G(U)$ iff (by (r3) and (r4)) \#$A$ is in $F(U)$ or \#$B$ is in $G(U)$. \#[$A\rightarrow B$] is in $F(U)$ iff \#[$\neg A\vee B$] is in $F(U)$ iff \#$A$ is in $G(U)$ and \#$B$ is in $F(U)$.
\item[(r7)] Biconditionality rule:
\#[$A \leftrightarrow B$] is in $G(U)$ iff \#$A$ and \#$B$ are both in $G(U)$ or both in $F(U)$, and in $F(U)$ iff \#$A$ is in $G(U)$ and
\#$B$ is in $F(U)$, or \#$A$ is in $F(U)$ and \#$B$ is in $G(U)$.
\end{enumerate}
When $A(x)$ is a formula in $L$, then quantifications $\exists xA(x)$ and $\forall xA(x)$ are sentences of $L$. Hence rule (r1) is applicable for them.
So it suffices to set rules for $\exists xT(x)$ and $\forall xT(x)$. Assume that the set $X$ of numerals of
G\"odel numbers of sentences of $\mathcal L$ is the intended domain of discourse for $T$. We set the following rules:
\begin{enumerate}
\item[(r8)] \#[$\exists xT(x)$] is in $G(U)$ iff \#$T(\mathbf n)$ is in $G(U)$ for some $\mathbf n\in X$, and \#[$\exists xT(x)$] is in $F(U)$ iff \#$T(\mathbf n)$ is in $F(U)$ for every $\mathbf n\in X$.
\item[(r9)] \#[$\forall xT(x)$] is in $G(U)$ iff \#$T(\mathbf n)$ is in $G(U)$ for every $\mathbf n\in X$, and \#[$\forall xT(x)$] is in $F(U)$ iff \#$T(\mathbf n)$ is in $F(U)$ at least for one $\mathbf n\in X$.
\end{enumerate}
Rules (r1)--(r9) and induction on the complexity of formulas
determine uniquely subsets $G(U)$ and $F(U)$ of $D$ whenever $U$ is a subset of $D$.
Sublanguages $\mathcal L_U$ of $\mathcal L$ formed by those sentences $A$ of $\mathcal L$ for which \#$A$ is in $G(U)$ or in $F(U)$ contain all sentences of $L$ and are closed under logical connectives and quantifiers.
We say that a subset $U$ of $D$ is {\em consistent} iff both \#$A$ and \#[$\neg A$] are not in $U$ for any sentence $A$ of $\mathcal L$. For instance, the empty set $\emptyset$ is consistent.
The following two lemmas have counterparts in \cite{[10]}.
\begin{lemma}\label{L01} Let $U$ be a consistent subset of $D$. Then $G(U)\cap F(U)=\emptyset$,
i.e., every sentence of $\mathcal L_U$ belongs either to $G(U)$ or to $F(U)$.
\end{lemma}
\begin{proof} If $A$ is in $L$, then by rule (r1) and \cite[Lemma II.8.22]{[Ku]}, \#$A$ is not in $G(U)\cap F(U)$.
By rule (r2) \#$T(\left\lceil A\right\rceil)$ is in $G(U)$ iff \#$A$ is in $U$, and in $F(U)$ iff
\#[$\neg A$] is in $U$. Thus \#$T(\left\lceil A\right\rceil)$
cannot be both in $G(U)$ and in $F(U)$ because $U$ is consistent.
Make an induction hypothesis:
\begin{enumerate}
\item[(h0)] $A$ and $B$ are such sentences of $\mathcal L$ that neither \#$A$ nor \#$B$ is in $G(U)\cap F(U)$.
\end{enumerate}
As shown above, (h0) holds if $A$ and $B$ are basic sentences.
If \#[$\neg A$] is in $G(U)\cap F(U)$, then \#$A$ is in $F(U)\cap G(U)$. Hence, if (h0) holds, then \#[$\neg A$] is not in $G(U)\cap F(U)$.
If \#[$A\vee B$] is in $G(U)\cap F(U)$, then \#$A$ or \#$B$ is in $G(U)$, and both \#$A$ and \#$B$ are in $F(U)$ by (r4), so that \#$A$ or
\#$B$ is in $G(U)\cap F(U)$. Hence, if (h0) holds, then \#[$A\vee B$] is not in $G(U)\cap F(U)$.
\#[$A\wedge B$] cannot be in $G(U)\cap F(U)$, for otherwise both \#$A$ and \#$B$ are in $G(U)$, and at least one of \#$A$ and \#$B$ is in $F(U)$, contradicting with (h0).
If \#[$\neg A$] is in $G(U)\cap F(U)$, then \#$A$ is in $F(U)\cap G(U)$, and (h0) is not valid. Thus, under the hypothesis (h0) neither
\#[$\neg A$] nor \#$B$ is in $G(U)\cap F(U)$. This result and the above result for disjunction imply that \#[$\neg A\vee B$], or equivalently,
\#[$A\rightarrow B$], is not in $G(U)\cap F(U)$. Similarly, \#[$A\leftrightarrow B$] is not in $G(U)\cap F(U)$, for otherwise, \#$A$ or
\#$B$ would be in $G(U)\cap F(U)$ by rule (r7), contradicting with (h0).
It remains to show that \#[$\exists xT(x)$] and \#[$\forall xT(x)$] are not in $G(U)\cap F(U)$.
If $U$ is empty, then $T(\mathbf n)$ is by rule (r2) neither in $G(U)$ nor in $F(U)$ for any $\mathbf n\in X$. Thus \#[$\exists xT(x)$] is by rule (r8) neither in $G(U)$ nor in $F(U)$, and hence not in $G(U)\cap F(U)$.
If $U$ is not empty, then \#$A$ is in $U$ for some $A$ in $\mathcal L$. Since $U$ is consistent, then \#$[\neg A]$ is not in $U$. By rule (r2), \#$T(\left\lceil A\right\rceil)$ is not in $F(U)$. Thus \#[$\exists xT(x)$] is by rule (r8) not in $F(U)$, and hence not in $G(U)\cap F(U)$.
Because $U$ is consistent, it is a proper subset of $D$.
Thus there is $n\in D$ such that $n\not\in U$. But $n=\#A$ for some sentence $A$ of $\mathcal L$, whence \#$T(\mathbf n)=\#T(\left\lceil A\right\rceil)$
is not in $G(U)$ by rule (r2). Consequently, \#[$\forall xT(x)$] is by rule (r9) not in $G(U)$, and hence not in $G(U)\cap F(U)$.
The above results and induction on the complexity of formulas imply that \#$A$ is not in $G(U)\cap F(U)$ for any sentence $A$ of $\mathcal L$.
\end{proof}
\begin{lemma}\label{L201} If $U$ is a consistent subset of $D$, then both $G(U)$ and $F(U$ are consistent.
\end{lemma}
\begin{proof} If $G(U)$ is not consistent, then there is such a sentence $A$ of $\mathcal L$, that \#$A$ and \#[$\neg A$] are in $G(U)$. Because \#[$\neg A$] is in $G(U)$, then \#$A$ is also in $F(U)$ by rule (r3), and hence in $G(U)\cap F(U)$. But then, by Lemma \ref{L01}, $U$ is not consistent.
Consequently, if $U$ is consistent, then $G(U)$ is consistent. The proof that $F(U)$ is consistent if $U$ is, is similar.
\end{proof}
\section{A mathematical theory of truth and its properties}\label{S4}
Recall that
$D$ denotes the set of G\"odel numbers of sentences of the language $\mathcal L=\{\in,T\}$. Given a subset $U$ of $D$, let $G(U)$ and $F(U)$ be the subsets of $D$ constructed in Section \ref{S3}.
\begin{definition}\label{D41} {\it Assume that $U$ is a consistent subset of $D$, and that $U=G(U)$.
Denote by $\mathcal L_U$ the language containing those sentences $A$ of $\mathcal L=\{\in,T\}$ for which \#$A$ is in $G(U)$ or in $F(U)$.
A sentence $A$ of $\mathcal L_U$ is interpreted as true iff \#$A$ is in $G(U)$, and as false iff \#$A$ is in $F(U)$.
$T$ is called a truth predicate for $\mathcal L_U$}. \end{definition}
The existence of consistent subsets $U$ of $D$ satisfying $U=G(U)$, including the smallest one, is proved in Section \ref{S5}.
\smallskip
The main goal of this paper is to prove the following Theorem for the above defined mathematical theory of truth (shortly MTT).
\begin{theorem}\label{T31} Assume that $U$ is a consistent subset of $D$, and that $U=G(U)$. Then MTT satisfies all the norms (a)--(j) presented in the Introduction.
\end{theorem}
\begin{proof} (a) The syntax of MTT is comprised by logical symbols of first-order predicate logic, nonlogical symbols $\in$ and $T$, natural numbers as constants, numerals as terms, and variables ranging in $M$. $T$ is the truth predicate for $\mathcal L_U$, by Definition \ref{D41}.
(b) Assume that $A$ is a sentence of $L$. By \cite[Lemma II.8.22]{[Ku]} either $A$ is true in $M$, or $\neg A$ is true in $M$ ($A$ is false in $M$).
-- $A$ is true in $M$ iff \#$A$ is in $G(U)$, by rule (r1), iff \#$A$ is in $U$, because $U=G(U)$, iff \#$T(\left\lceil A\right\rceil)$ is in $G(U)$ by rule (r2), iff $T(\left\lceil A\right\rceil)$ is true, by Definition \ref{D41}.
-- $\neg A$ is true in $M$ iff \#[$\neg A$] is in $G(U)$, by rule (r1), iff \#[$\neg A$] is in $U$, because $U=G(U)$, iff \#$\left\lceil T(A)\right\rceil$ is in $F(U)$, by rule (r2), iff $T(\left\lceil A\right\rceil)$ is false, by Definition \ref{D41}.
Consequently, a sentence $A$ of $L$ is true in $M$ iff $T(\left\lceil A\right\rceil)$ is true, and false in $M$ iff $T(\left\lceil A\right\rceil)$ is false.
Thus MTT proves the $ZF$ set theory modelled by $M$ true.
(c) $T$ is by Definition \ref{D41} a truth predicate for $\mathcal L_U$, and is not subject to any restrictions within $\mathcal L_U$.
(d) Let $A$ be a sentence of $\mathcal L_U$. Since $U$ is consistent, then \#$A$ is by Lemma \ref{L01} either in $G(U)$ or in $F(U)$.
Applying rules (r2) and (r3), and the assumption $U=G(U)$, we obtain
\newline
-- \#$A$ is in $G(U)$ iff \#$A$ is in $U$ iff
\#$T(\left\lceil A\right\rceil)$ is in $G(U)$;
\newline
-- \#$A$ is in $F(U)$ iff \#[$\neg A$] is in $G(U)$ iff \#[$\neg A$] is in $U$ iff \#$T(\left\lceil A\right\rceil)$ is in $F(U)$.
\newline
Consequently, \#$A$ and \#$T(\left\lceil A\right\rceil)$ are both either in $G(U)$ or in $F(U)$. Thus \#[$A\leftrightarrow T(\left\lceil A\right\rceil)$] is by rule (r7) in $G(U)$, so that $A\leftrightarrow T(\left\lceil A\right\rceil)$ is true by Definition \ref{D41}. This holds for every sentence $A$ of $\mathcal L_U$, whence $T$-biconditionals $A\leftrightarrow T(\left\lceil A\right\rceil)$ are derivable unrestrictedly in $\mathcal L_U$.
(e) It follows from Definition \ref{D41} that `\#$A$ is in $G(U)$' can be replaced by `$A$ is true' and `\#$A$ is in $F(U)$' by
`$A$ is false'.
Thus rules (r3)--(r9) imply that the truth in MTT is compositional.
(f) MTT allows for standard interpretations. $M$ is by \cite{[13]} a standard model of set theory.
(g) The outer logic and the inner logic coincide in MTT because of (d), as stated \cite{[16]}.
(h) By the proof of (e) the logic of $\mathcal L_U$, i.e., the inner logic of MTT is classical. This and (g) imply that also the outer logic of MTT is classical.
(i) Since $U$ is consistent, it follows from Lemmas \ref{L01} and \ref{L201} that every sentence of $\mathcal L_U$ is either true or false in the sense of Definition \ref{D41}. Thus MTT is free from paradoxes. Only mathematical methods are used in it's formulation, whence it is a mathematical theory.
(j) By Definition \ref{D41} truth for the language is defined within the language $\mathcal L_U$ itself.
\end{proof}
\begin{remark}\label{R41} If $\theta$ is a monadic predicate of $L$, there is such a sentence $A$ in $L$ that biconditionality $A\leftrightarrow\neg\theta(\left\lceil A\right\rceil)$ is provable from axioms of ZF set theory (cf., e.g., \cite[Lemma IV.5.31]{[Ku]}).
Thus $T$-biconditionality rule (d) does not hold in $L$ when $T=\theta$. In particular, $L$ does not contain its truth predicate.
This holds also for the language $\mathcal L$.
To satisfy all norms (a)--(j) it is then essential that those languages $\mathcal L_U$ for which MTT is formulated differ both from $L$ and from $\mathcal L$.
This holds in our approach, since $G(U)\cup F(U)$ is a proper and nonempty subset of $D$ when $U$ is a consistent fixed point of $G$.
The language $\mathcal L_U$ corresponding to the smallest consistent fixed point $U$ of $G$ relates to the language of grounded sentences defined in \cite{[10],[15]}.
See also \cite{[6]}, where considerations are restricted to signed statements of $\mathcal L$.
MTT is only a {\em theory} of truth depending, e.g., on the semantic notion 'true in $M$'. Moreover, the construction of $M$ starts from the set $\omega\cup\{\omega\}$, the existence of which is a matter of faith.
\end{remark}
\section{Appendix}\label{S5}
Let $\mathcal P$ denote the family of all consistent subsets of the set $D$ of
G\"odel numbers of sentences of $\mathcal L$. Fixed points of the set mapping $G:=U\mapsto G(U)$ from $\mathcal P$ to $\mathcal P$, i.e., those $U\in \mathcal P$ for which $U=G(U)$, have a central role in the formulation of MTT. Before the proof of our main fixed point theorem we prove some
auxiliary results.
\begin{lemma}\label{L202} Let $U$ and $V$ be sets of $\mathcal P$, and assume that $U \subseteq V$. If $A$ is a sentence of $\mathcal L$, then
\#$A$ is in $G(V)$ whenever it is in $G(U)$,
and \#$A$ is in $F(V)$ whenever it is in $F(U)$.
\end{lemma}
\begin{proof} Assume that $U\subseteq V$. Consider first basic sentences. Let $A$ be a sentence of $L$. By rule (r1) \#$A$ is in $G(U)$ and also in $G(V)$ iff $A$ is true in $M$, and both in $F(U)$ and in $F(V)$ iff $\neg A$ is true in $M$.
Let $A$ be a sentence of $\mathcal L$. If \#$T(\left\lceil A\right\rceil)$ is in $G(U)$, then \#$A$ is in $U$ by rule (r2). Because $U\subseteq V$, then \#$A$ belongs to $V$, whence \#$T(\left\lceil A\right\rceil)$ is in $G(V)$ by rule (r2).
If \#$T(\left\lceil A\right\rceil)$ is in $F(U)$, then \#[$\neg A$] is in $U$ by rule (r2). Since $U\subseteq V$, then \#[$\neg A$] belongs to $V$, so that \#$T(\left\lceil A\right\rceil)$ is in $F(V)$ by rule (r2).
Thus all basic sentences satisfy the lemma.
Assume that $A$ is a sentence of $\mathcal L$. If \#[$\neg A$] is in $G(U)$ but not in $G(V)$, then \#$A$ is in $F(U)$ but not in $F(V)$ by rule (r3). If \#[$\neg A$] is in $F(U)$ but not in $F(V)$, then \#$A$ is in $G(U)$ but not in $G(V)$ by rule (r3). Hence, if $A$ satisfies the lemma, then also $\neg A$ satisfies it.
Make an induction hypothesis:
\begin{enumerate}
\item[(h1)] $A$ and $B$ are such sentences of $\mathcal L$ that \#$A$ is in $G(V)$ if it is in $G(U)$, \#$A$ is in $F(V)$ if it is in $F(U)$,
\#$B$ is in $G(V)$ if it is in $G(U)$, and \#$B$ is in $F(V)$ if it is in $F(U)$.
\end{enumerate}
If \#[$A\vee B$] is in $G(U)$, then \#$A$ or \#$B$ is in $G(U)$ by rule (r4). By (h1) \#$A$ or \#$B$ is in $G(V)$, so that \#[$A\vee B$] is in $G(V)$. If \#[$A\vee B$] is in $F(U)$, then \#$A$ and \#$B$ are in $F(U)$ by rule (r4), and hence also in $F(V)$, by (h1), so that \#[$A\vee B$] is in $F(V)$.
Thus $A\vee B$ satisfies the lemma if (h1) holds.
If \#[$A\wedge B$] is in $G(U)$, then both \#$A$ and \#$B$ are in $G(U)$ by rule (r5), and hence also in $G(V)$, by (h1). Thus \#[$A\wedge B$] is in $G(V)$. If \#[$A\wedge B$] is in $F(U)$, then \#$A$ or \#$B$ is in $F(U)$ by rule (r5), and hence also in $F(V)$, by (h1), whence \#[$A\wedge B$] is in $F(V)$.
Thus $A\wedge B$ satisfies the lemma if (h1) holds.
If \#[$A\rightarrow B$] is in $G(U)$, then \#[$\neg A$] or \#$B$ is in $G(U)$, i.e., \#$A$ is in $F(U)$ or \#$B$ is in $G(U)$. Then, by (h1), \#$A$ is in $F(V)$ or \#$B$ is in $G(V)$, i.e., \#[$\neg A$] or \#$B$ is in $G(V)$. Thus \#[$A\rightarrow B$] is in $G(V)$. If \#[$A\rightarrow B$] is in $F(U)$, then \#$A$ is in $G(U)$ and \#$B$ is in $F(U)$. This implies by (h1) that \#$A$ is in $G(V)$ and \#$B$ is in $F(V)$, so that \#[$A\rightarrow B$] is in $F(V)$.
Thus $A\rightarrow B$ satisfies the lemma if (h1) holds.
If \#[$A\leftrightarrow B$] is in $G(U)$, then \#$A$ and \#$B$ are both in $G(U)$ or both in $F(U]$, and hence both in $G(V)$ or both in $F(V)$ if (h1) holds, so that \#[$A\leftrightarrow B$] is in $G(V)$.
If \#[$A\leftrightarrow B$] is in $F(U)$, then \#$A$ is in $G(U)$ and \#$B$ is in $F(U)$ or vice versa. Thus \#$A$ is in $G(V)$ and
\#$B$ is in $F(V)$ or vice versa if (h1) holds, so that \#[$A\leftrightarrow B$] is in $F(V)$. Thus $A\leftrightarrow B$ satisfies the lemma if (h1) holds.
Because (h1) holds for basic sentences, the above results and induction on the complexity of expressions imply that logical connectives of sentences of $\mathcal L$ satisfy the lemma.
It follows from the proof of Lemma \ref{L01} that \#[$\exists xT(x)]$ is neither in $F(U)$ nor in $F(V)$.
If \#[$\exists xT(x)$] is in $G(U)$, then $T(\mathbf n)$ is in $G(U)$ for some $\mathbf n\in X$. Since $\mathbf n=\left\lceil A\right\rceil$ for some sentence $A$ of $\mathcal L$, then \#$T(\left\lceil A\right\rceil)$ is in $G(U)$. Thus \#$A$ is by rule (r2) in $U$, and hence also in $V$. Then, by rule (r2), \#$T(\left\lceil A\right\rceil)=\#T(\mathbf n)$ is in $G(V)$, so that \#[$\exists xT(x)]$ is in $G(V)$ by rule (r8). Consequently,
\#[$\exists xT(x)$] is in $G(V)$ whenever it is in $G(U)$.
The proof of Lemma \ref{L01} implies that \#[$\forall xT(x)$] is neither in $G(U)$ nor in $G(V)$. If \#[$\forall xT(x)$] is in $F(U)$, then $T(\mathbf n)$ is in $F(U)$ for some $\mathbf n\in X$. But $\mathbf n=\left\lceil A\right\rceil$ for some sentence $A$ of $\mathcal L$, so that \#$T(\left\lceil A\right\rceil)$ is in $F(U)$. So \#[$\neg A$] is by rule (r2) in $U$, and hence also in $V$. Thus, by rule (r2), \#$T(\left\lceil A\right\rceil)=\#T(\mathbf n)$ is in $F(V)$, so that \#[$\exists xT(x)]$ is in $F(V)$ by rule (r8). Consequently, \#[$\forall xT(x)$]
is in $F(V)$ whenever it is in $F(U)$.
The above results imply the conclusion of the lemma.
\end{proof}
According to Lemma \ref{L201} the mapping $G:=U\mapsto G(U)$ maps $\mathcal P$ into $\mathcal P$.
Assuming that $\mathcal P$ is ordered by inclusion, the above lemma implies the following result.
\begin{lemma}\label{L203} $G$ is order preserving in $\mathcal P$, i.e.,
$G(U)\subseteq G(V)$ whenever $U$ and $V$ are sets of $\mathcal P$ and $U\subseteq V$.
\end{lemma}
As above lemmas, also the next lemma has a counterpart in \cite{[10]}.
\begin{lemma}\label{L204}
(a) If $\mathcal W$ is a chain in $\mathcal P$, then the union
$\cup \mathcal W=\cup\{U\mid U\in \mathcal W\}$ is consistent.
(b) The intersection $\cap \mathcal W=\cap\{U\mid U\in \mathcal W\}$ of every nonempty subfamily $\mathcal W$ of $\mathcal P$ is a consistent subset of $D$.
\end{lemma}
\begin{proof} (a) Assume on the contrary that $\cup \mathcal W$ is not consistent. Then there is a such a sentence $A$ of $\mathcal L$
that both \#$A$ and \#[$\neg A$] are in $\cup \mathcal W$. Thus
$\mathcal W$ has a member, say $U$, which contains \#$A$,
and a member, say $V$, which contains \#[$\neg A$]. If $\mathcal W$ is a chain, then $U\subseteq V$ or $V\subseteq U$. In former case $V$ and in latter case
$U$ contains both \#$A$ and \#[$\neg A$]. But this is impossible because $\mathcal W$ is a subfamily of $\mathcal P$. This proves (a).
(b) The intersection $\cap \mathcal W$ is a subset of $D$, and is contained in every member of $\mathcal W$. Hence $\cap \mathcal W$
is consistent, for otherwise there is such a sentence $A$ in $\mathcal L$ that both \#$A$ and \#[$\neg A$] are in $\cap \mathcal W$. Then every member of $\mathcal W$ would also contain both \#$A$ and \#[$\neg A$]. But this is impossible because every member of
$\mathcal W$ is consistent. This proves (b).
\end{proof}
In the formulation and the proof of our main fixed point theorem we use transfinite sequences of $\mathcal P$ indexed by von Neumann ordinals. Such a sequence $(U_\lambda)_{\lambda\in\alpha}$ of $\mathcal P$ is said to be strictly increasing if
$U_\mu\subset U_\nu$ whenever $\mu\in\nu\in\alpha$, and strictly decreasing if
$U_\nu\subset U_\mu$ whenever $\mu\in\nu\in\alpha$.
A set $V$ of $\mathcal P$ is called {\em sound} iff $V\subseteq G(V)$.
The following fixed point theorem is an application of Lemmas \ref{L201}, \ref{L203} and \ref{L204}.
\begin{theorem}\label{T1}
If $V\in \mathcal P$ is sound, then there exists the smallest of those fixed points of $G$ which contain $V$. This fixed point is the last member of the union of those transfinite sequences
$(U_\lambda)_{\lambda\in\alpha}$ of $\mathcal P$ which satisfy
\begin{itemize}
\item[(C)] $(U_\lambda)_{\lambda\in\alpha}$ is strictly increasing,
$U_0=V$, and if $0\in\mu\in \alpha$, then
$U_\mu = \underset{\lambda\in\mu}{\bigcup}G(U_\lambda)$.
\end{itemize}
\end{theorem}
\begin{proof} Let $V\in\mathcal P$ be sound.
Transfinite sequences of $\mathcal P$ satisfying (C) are called $G$-sequences.
We shall first show that $G$-sequences are nested:
\begin{enumerate}
\item[(1)] {\it Assume that $(U_\lambda)_{\lambda\in\alpha}$ and $(V_\lambda)_{\lambda\in\beta}$ are $G$-sequences, and that $\{U_\lambda\}_{\lambda\in\alpha} \not\subseteq \{V_\lambda\}_{\lambda\in\beta}$.
Then $(V_\lambda)_{\lambda\in\beta} = (U_\lambda)_{\lambda\in\beta}$}.
\end{enumerate}
By the assumption of (1) $\mu = \min \{\lambda\in\alpha\mid U_\lambda\not\in\{V_\lambda\}_{\lambda\in\beta}\}$ exists, and
$\{U_\lambda\}_{\lambda\in\mu} \subseteq \{V_\lambda\}_{\lambda\in\beta}$.
Properties (C) imply by transfinite induction that $U_\lambda=V_\lambda$ for each $\lambda\in\mu$. To prove that $\mu=\beta$,
make a counter-hypothesis: $\mu\in\beta$. Since $\mu\in\alpha$ and $U_\lambda=V_\lambda$ for each $\lambda\in\mu$, it follows from properties (C) that $U_\mu = \underset{\lambda\in\mu}{\bigcup}G(U_\lambda) = \underset{\lambda\in\mu}{\bigcup}G(V_\lambda)=V_\mu$, which is
impossible, since $V_\mu\in \{V_\lambda\}_{\lambda\in\beta}$, but $U_\mu\not\in \{V_\lambda\}_{\lambda\in\beta}$. Consequently, $\mu=\beta$ and $U_\lambda=V_\lambda$ for each $\lambda\in\beta$, whence
$(V_\lambda)_{\lambda\in\beta} =(U_\lambda)_{\lambda\in\beta}$.
By definition, every $G$-sequence $(U_\lambda)_{\lambda\in\alpha}$ is a function $\lambda\mapsto U_\lambda$ from $\alpha$ into $\mathcal P$. Property (1) implies that these functions are compatible. Thus their union is by \cite[Theorem 2.3.12]{[12]} a function with values in $\mathcal P$, the domain being the union of all index sets of $G$-sequences. Because these index sets are ordinals, then their union is also an ordinal by \cite[I.8.10]{[Ku]}. Denote it by $\gamma$. The union function can be represented as a sequence $(U_\lambda)_{\lambda\in\gamma}$ of $\mathcal P$. It is strictly increasing as a union of strictly increasing nested sequences.
To show that $\gamma$ is a successor, assume on the contrary that $\gamma$ is a limit ordinal. Given $\nu\in\gamma$, then $\mu=\nu\cup\{\nu\}$ and $\alpha=\mu\cup\{\mu\}$ are in $\gamma$, and $(U_\lambda)_{\lambda\in\alpha}$ is a $G$-sequence. Denote $U_\gamma = \underset{\lambda\in\gamma}{\bigcup}G(U_\lambda)$.
$G$ is order preserving by Lemma \ref{L203}, and $(U_\lambda)_{\lambda\in\gamma}$ is a strictly increasing sequence of $\mathcal P$. Thus $\{G(U_\lambda)\}_{\lambda\in\gamma}$ is a chain in $\mathcal P$, whence $U_\gamma$ is consistent by Lemma \ref{L204}(a). Moreover,
$U_\nu\subset U_\mu=\underset{\lambda\in\mu}{\bigcup}G(U_\lambda)\subseteq U_\gamma$. This holds for each $\nu\in\gamma$, whence $(U_\lambda)_{\lambda\in\gamma\cup\{\gamma\}}$ is a $G$-sequence. This is impossible, since $(U_\lambda)_{\lambda\in\gamma}$ is the union of all $G$-sequences.
Consequently, $\gamma$ is a successor, say $\gamma=\alpha\cup\{\alpha\}$. Thus $U_\alpha$ is the last member of $(U_\lambda)_{\lambda\in\gamma}$,
$U_\alpha=\max\{U_\lambda\}_{\lambda\in\gamma}$, and $G(U_\alpha)=\max\{G(U_\lambda)\}_{\lambda\in\gamma}$. Moreover, $(U_\lambda)_{\lambda\in\gamma}$ is a $G$-sequence, for otherwise $(U_\lambda)_{\lambda\in\alpha}$ would be the union of all $G$-sequences. In particular,
$U_\alpha=\underset{\lambda\in\alpha}{\bigcup}G(U_\lambda)\subseteq \underset{\lambda\in\gamma}{\bigcup}G(U_\lambda)=G(U_\alpha)$,
so that $U_\alpha\subseteq G(U_\alpha)$. Equality holds, since otherwise the longest $G$-sequence $(U_\lambda)_{\lambda\in\gamma}$ could be extended by $U_\gamma= \underset{\lambda\in\gamma}{\bigcup}G(U_\lambda)$.
Thus $U_\alpha$ is a fixed point of $G$ in $\mathcal P$.
Assume that $W\in\mathcal P$ is a fixed point of $G$, and that $V\subseteq W$. Then $U_0=V\subseteq W$. If $0\in\mu\in\gamma$, and $U_\lambda\subseteq W$ for each $\lambda\in\mu$, then $G(U_\lambda)\subseteq G(W)$ for each $\lambda\in\mu$, whence $U_\mu=\underset{\lambda\in\mu}{\bigcup}G(U_\lambda)\subseteq G(W)=W$. Thus, by transfinite induction, $U_\mu\subseteq W$ for each $\mu\in\gamma$. Thus $U_\alpha\subseteq W$ so that $U_\alpha$ is the smallest fixed point of $G$ that contains $V$.
\end{proof}
As a consequence of Theorem \ref{T1} we obtain.
\begin{corollary}\label{C1} $G$ has the smallest fixed point.
\end{corollary}
\begin{proof} The empty set $\emptyset$ is both sound and consistent. Thus the smallest fixed point of $G$ that contains $\emptyset$ exists by Theorem \ref{T1}. It is the smallest fixed point of $G$, since every fixed point of $G$ contains $\emptyset$.
\end{proof}
Next we prove that every consistent subset of $D$ has the greatest sound and consistent subset.
\begin{theorem}\label{T2} The equation $V=W\cap G(V)$ has for each consistent subset $W$ of $D$ the greatest solution $V$ in $\mathcal P$. It is the greatest sound set of $\mathcal P$ that is contained in $W$. $V$ is the last member of
the union of those transfinite sequences
$(V_\lambda)_{\lambda\in\alpha}$ of $\mathcal P$ which satisfy
\begin{itemize}
\item[(D)] $(V_\lambda)_{\lambda\in\alpha}$ is strictly decreasing,
$V_0=W$, and if $0\in\mu\in \alpha$, then
$V_\mu = W\cap(\underset{\lambda\in\mu}{\bigcap}G(V_\lambda))$.
\end{itemize}
\end{theorem}
\begin{proof} Assume that $W$ is a consistent subset of $D$. Like the proof that the union of all $G$-sequences is a $G$-sequence one can prove that the union of all transfinite sequences which have properties (D) given in Theorem \ref{T2} has property (D). Let $(V_\lambda)_{\lambda\in\gamma}$
be that sequence.
Denote $V=W\cap(\underset{\lambda\in\gamma}{\bigcap}G(V_\lambda))$. Because $W$ and the sets $V_\lambda$, $\lambda\in\gamma$, are consistent, it follows from Lemma \ref{L201} and Lemma \ref{L204}(b) that $V$ is consistent. Moreover, $V\subseteq V_\lambda$ for each $\lambda\in\gamma$.
If $V\subset V_\lambda$ for each $\lambda\in\gamma$, then the choice $V_\lambda=V$ implies that $(V_\lambda)_{\lambda\in\gamma\cup\{\gamma\}}$ satisfies (D) when $\alpha=\gamma\cup\{\gamma\}$. But this is impossible because of the choice of $(V_\lambda)_{\lambda\in\gamma}$. Thus
$V = \min\{V_\lambda\}_{\lambda\in\gamma}$, and $V$ is the last member of $(V_\lambda)_{\lambda\in\gamma}$ because this sequence is strictly decreasing. Since $G$ is order preserving, then $G(V) = \min\{G(V_\lambda)\}_{\lambda\in\gamma}=\underset{\lambda\in\gamma}{\bigcap}G(V_\lambda)$.
Thus $V = W\cap G(V)$, so that $V\subseteq G(V)$, i.e., $V$ is sound and is contained in $W$.
Assume that $U$ is consistent, that $U\subseteq G(U)$, and that $U\subseteq W$. Since $V_0=W$ by (D$_\gamma$), then $U\subseteq V_0$. If $0\in\mu\in\gamma$ and $U\subseteq V_\lambda$ for each $\lambda\in\mu$, then $G(U)\subseteq G(V_\lambda)$ for each $\lambda\in\mu$, whence
$U\subseteq W\cap G(U)\subseteq W\cap(\underset{\lambda\in\mu}{\bigcap}G(V_\lambda))=V_\mu$. Thus, by transfinite induction, $U\subseteq V_\lambda$ for each $\lambda\in\gamma$, so that $U\subseteq \min\{V_\lambda\}_{\lambda\in\gamma}=V$. Consequently, $V$ is the greatest sound and consistent subset of $D$ that is contained in $W$.
\end{proof}
The following results shows that if $W$ is any consistent set of G\"odel numbers of sentences of $L$, and $V$ is determined by Theorem \ref{T2}, then
the union of the $G$- sequences (C) of Theorem \ref{T1} is the smallest fixed point of $G$.
\begin{corollary}\label{C2} Let $W$ be a consistent subset of $D$ containing only G\"odel numbers of sentences of $L$. Then
(a) $V= W\cap G(W)$ is the greatest consistent solution of $V=V\cap G(V)$ contained in $W$.
(b) The union of the $G$- sequences (C) of $V= W\cap G(W)$ is the smallest fixed point of $G$.
\end{corollary}
\begin{proof} (a) Let $W$ be a consistent set of G\"odel numbers of sentences of $L$. $G(W)$ contains by rule (r1) G\"odel numbers of those sentences of $L$ which are true in $M$, and only those. Thus $V= W\cap G(W)$ is the set of G\"odel numbers of those sentences of $W$ which are true in $M$. Hence $V\subset G(\emptyset)\subseteq G(V)$, so that $V$ is sound. It is also consistent, as an intersection of two consistent sets. Consequently, $V= W\cap G(W)$ is a solution of $V=V\cap G(V)$, and by Theorem \ref{T2} the greatest solution contained in $W$.
\smallskip
(b) $V=W\cap G(W)$ is by (a) sound and consistent, and $V\subset G(\emptyset)$. If $U$ is a fixed point of $G$, then $V\subset G(\emptyset)\subset G(U)=U$. Thus $V$ is contained in the smallest fixed point of $G$. By Theorem \ref{T1}, the union of the $G$- sequences (C) is the smallest fixed point of $G$ that contains $V$. This proves (b).
\end{proof}
\begin{remark}\label{R51}
The smallest members of $(U_\lambda)_{\lambda\in\alpha}$ satisfying (C) are $n$-fold iterations $U_n=G^n(V)$, $n\in\mathbb N=\{0,1,\dots\}$. If they form a strictly increasing sequence, the next member $U_\omega$ is their union, $U_{\omega+n}= G^n(U_\omega)$, $n\in\mathbb N$, and so on.
\smallskip
If the set $W$ is finite, then
the longest sequence $(V_\lambda)_{\lambda\in\alpha}$ satisfying (D) is obtained by the finite algorithm:
$V_0=W$. For $n$ from $0$ while $V_n\ne W\cap G(V_n)$ do: $V_{n+1}=W\cap G(V_n)$.
Zorn's Lemma, together with Lemmas \ref{L203} and \ref{L204}, can be applied, e.g., as in \cite{[6],[10]}, to prove results of Theorem \ref{T1} and Corollary \ref{C1}, and also the existence of maximal fixed points of $G$. ZF set theory is no more sufficient framework for such proofs, since Zorn's Lemma is equivalent to the Axiom of Choice.
The result of Theorem \ref{T2} that gives a method to determine sound subsets of $D$ seems to be new.
As for generalizations of Theorems \ref{T1} and \ref{T2}, see, e.g., \cite{[3],[9]}.
\end{remark}
\smallskip
{\bf Acknowledgements:} The author is indebted to Prof. Hannes Leitgeb and to Ph.D. Tapani Hyttinen for valuable discussions on the subject. The present work is influenced by \cite{[10]}.
\smallskip
\baselineskip12pt
| 109,842
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As a faith-based health care provider, we understand the important relationship between spiritual and physical health. Because we know that a hospital stay can be an anxious and lonely time, we’re focused on providing a soothing environment to promote your healing. We invite our patients, their family members and our employees to visit our healing gardens and chapel, which are located near the main entrance and open 24 hours a day, seven days a week.
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We also invite you to contact our pastoral care team: Our chaplains are here to help you, whether you wish to participate in a sacrament or just need to talk or pray with someone. These are healthcare professionals who can help and encourage you and your loved ones during your stay, regardless of your faith.
| 284,025
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TITLE: Can D'Alembert's functional be derived from Cauchy functionas?
QUESTION [0 upvotes]: Is it possible to derive D'Alembert's functional equation from Cauchy's functional equations? If so, can somebody kindly point me to a reference?
Edit:
Can
$f(x + y) + f(x − y) = 2f(x)f(y)$
be derived from either of
$f(x+y)=f(x)+f(y)$,
$f(xy)=f(x)f(y)$,
$f(x+y)=f(x)f(y)$,
$f(xy)=f(x)+f(y)$.
PS: I am not a mathematician.
REPLY [4 votes]: It is quite an interesting observation that if $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(x+y)=f(x)f(y)$ and $f$ is even (about zero), then $f(x)$ also satisfies D'Alemberts functional equation. Since $f(x-y) = f(x)f(-y) = f(x)f(y)$ also.
So there are relationships between the equations!
| 92,121
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TITLE: Topology generated by a Family of Seminorms as a Initial Topology?
QUESTION [5 upvotes]: Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the coarsest topology on $X$ making all the maps $f_i$ continuous. It is not hard to show that the set $$\mathscr{S}:=\{f^{-1}_i(U): U\in\mathscr{T}_i\}$$ is a sub-basis for this topology.
I want to apply this in the context of topological vector spaces.
The topology generated by a family of semi-norms $\mathscr{P}=\{p_i\}_{i\in I}$ on a $\mathbb R$-vector space is the initial topology on $X$ with respect to the family of mappings $\{p_i\}_{i\in I}$.
Can anyone help me showing this topology makes the vector space operations continuous?
Sketch (for the vector addition): We also know, from the general theory of initial topologies, that given a topological space $Z$ and a map $g:Z\longrightarrow X$ then $$g\ \textrm{is continuous}\Leftrightarrow g\circ f_i\ \textrm{is continuous}\ \forall i\in I.$$ Once this holds it suffices showing $p_i\circ g$ is continuous for all $i\in I$ where $g$ is the vector adition, this is where I'm stuck.
REPLY [7 votes]: If $p$ is a semi-norm, you have $0 \leq p(x-x' + y-y') \leq p(x-x')+p(y-y')$ for all $x,y$, showing that the initial topology $\mathscr{T}_p$ for $p$ make the addition continous. Now, the results follows, as if $(p_i)_{i \in I}$ is a family of semi-norms, the initial topology they generate is simply $\cap_{i\in I} \mathscr{T}_{p_i}$.
| 70,516
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TITLE: Calculate the maximum probability of a result of rolling n dice of varying number of faces
QUESTION [2 upvotes]: Disclaimer: I'm a computer programmer more than a mathematician, so reading text like that of the answer to this question is a little (read: a lot) over my head.
I've written an algorithm (brute-force, O(nk) running time, where n is the number of dice, and k is the number of sides) that calculates the probability of rolling each possible value of a specified number of dice, each of may have a different number of faces. For example, rolling d8 + 2d20 (1 8-sided die and 2 20-sided dice) has a minimum roll of 3 and a maximum roll of 48, each of which have a probability of 0.0003125. The values 25 and 26 are most likely to show up, each with a probability of 0.045.
I'm wondering if there's a way, without doing the full calculation, to determine--or make a conservative estimate of--this maximum value of 0.045 in polynomial time. I know that this is also the probability of rolling the middle value, or ceiling((max - min) / 2) + min.
I've written a probability distribution graph using jQuery Flot. Since this is such a long running calculation for large numbers of dice with large numbers of sides (e.g. 4d100), I calculate the distribution in chunks, and update the graph periodically. I'd like to have a maximum probability calculated ahead of time so the axes don't change as I make updates to the data.
Calculating the total number of permutations is obviously easy--multiple each die's number of sides together--so calculating the probability of any one permutation is 1 / totalPermutations. That's about as far as I got. Help?
REPLY [2 votes]: The probability generating function for the sum of $n$ dice of $d_1, \ldots, d_n$ sides respectively is
$$ G(x) = \prod_{j=1}^n \sum_{i=1}^{d_j} \dfrac{x^i}{d_j}$$
The probability of a sum of $s$ is the coefficient of $x^s$ in this polynomial.
You can also write $$\sum_{i=1}^d x^i = x \dfrac{x^{d} - 1}{x - 1}$$
For example, for four dice, each with $100$ sides,
$$ G(x) = 100^{-4} x^4 \left(\dfrac{x^{100} - 1}{x - 1}\right)^{4}$$
Consider the numerator and denominator
$$\eqalign{N(x) &= 100^{-4} x^4 (x^{100} - 1)^4 = 100^{-4} (x^{404} - 4 x^{304} + 6 x^{204} - 4 x^{104} + x^4)\cr
D(x) &= (x - 1)^4 = x^4 - 4 x^3 + 6 x^2 - 4 x + 1\cr}$$
Now $G(x) = \sum_{s=4}^{400} p_s x^s$ where $p_s$ is the probability of getting a sum of $s$. Since $G(x) D(x) = N(x)$, the coefficient of $x^s$ in $N(x)$ is
$$ r_s = p_{s} - 4 p_{s-1} + 6 p_{s-2} - 4 p_{s-3} + p_{s-4}$$
which gives you a recurrence relation you can use to calculate all the $p_s$, starting with $p_0 = \ldots = p_3 = 0$, where $r_s = 0$ except for $$r_4 = r_{404} = 10^{-8}, r_{104} = r_{304} = -4 \times 10^{-8}, r_{204} = 6 \times 10^{-8}$$
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One domain blog(not DI) published that Afilias prevailed in private auction, in what looked derived from Afilias PR; the post was later removed from the blog. Adding the confirmation of the auction, perhaps this is a case of someone not honoring private auction agreements, which could be a can of worms that might end up in court, depending on the outcome of ICANN last resort auction.
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\begin{document}
\maketitle
\begin{abstract} Let $A$ be a positive self-adjoint linear operator acting on a real
Hilbert space $H$ and $\alpha , c$ be positive constants. We show that all solutions of the evolution equation $ u''+ Au+ c A^\alpha u' = 0$
with $u(0) \in D(A^{\frac{1}{2}}), \ u'(0) \in H$ belong for all $t>0$ to the Gevrey space
$ G(A, \sigma)$ with $\sigma = \min\{ \frac {1}{\alpha}, \frac {1}{1-\alpha}\}$. This result is optimal in the sense that $\sigma$ can not be reduced in general.
For the damped wave equation (SDW)$_\alpha$ corresponding to the case where $A = -\Delta$ with domain $D(A) = \{ w\in H^1_0(\Omega), \Delta w \in L^2(\Omega)\}$ with $\Omega$ any open subset of $\mathbb{R}^N$ and $(u(0), u'(0)) \in H^1_0(\Omega)\times L^2(\Omega)$, the unique solution $u$ of
(SDW)$_\alpha$ satisfies $ \forall t>0, \quad u(t) \in G^s(\Omega)$ with $ s = \min \{ \frac {1}{2\alpha}, \frac {1}{2(1-\alpha)}\} $, and this result is also optimal.
\vspace{4ex}
\noindent{\bf Mathematics Subject Classification 2010 (MSC2010):}
35L10, 35B65, 47A60.
\vspace{4ex}
\noindent{\bf Key words:} linear evolution equations, dissipative hyperbolic equation, fractional power, Gevrey regularity.\end{abstract}
\newpage
\section{Introduction}
Let $A$ be a positive self-adjoint linear operator acting on a real
Hilbert space $H$, let $A^\alpha$ be the fractional power of $A$
of order $\alpha > 0$ and let $c$ be a positive constant. We consider the following evolution equation :
\begin{equation}\label{main}
u''+ Au+ c A^\alpha u' = 0.
\end{equation}
The time regularity and smoothing effect at $t>0$ on solutions of \eqref{main} have been studied by quite a few authors, cf., e.g., \cite{CT1, CT2, CT3, G^2H}. In \cite{HO}, when $A$ is coercive, the authors of the present paper established by rather elementary means (avoiding complex analysis) that the semi-group generated by \eqref{main} is analytic if $\alpha \ge 1/2 $ and Gevrey of order $\frac{1}{2\alpha}$ if $\alpha < 1/2 $. But concerning spatial regularity, Theorem 5.1 from \cite{HO} only implies $C^\infty $ interior spatial regularity for $A = -\Delta$ or more generally for $A$ an elliptic operator with smooth coefficients. Now we shall study more specifically the problem of regularity of $u(t)$ for $t>0$ when the initial state $(u(0), u'(0))$ lies in the standard energy space $V\times H$ with $V = D(A^{1/2}).$ Before stating any precise result, an important remark will allow us to understand that the exponent $\alpha =1/2 $, corresponding to the so-called structural damping (cf. \cite{CR}), is very special. Indeed we notice that the time scaling $u(t): = v(kt)$ transforms the equation into $$ v''+ Bv + ck^{2\alpha-1} B^\alpha v' = 0$$ with $B= k^{-2}A$. Therefore whenever $\alpha \not=1/2 $, we can select $k$ in such a way that the coefficient of $B^{\alpha}v'$ becomes 1. On the contrary, if $\alpha =1/2 $, the coefficient of $B ^ {1/2}v'$ is equal to $c$. The equations with different values of $c$ are all different, and they indeed have different properties even if $H = \mathbb{R}$ and $A= I$. In that most elementary case, the value $c=2$ is the threshold deciding the oscillatory or non-oscillatory character of solutions. Finally, if $\alpha =1/2 $ and $A$ has compact inverse, it can be seen that solutions of \eqref{main} of the form $$u(t)= w(t)\varphi $$ with $A\varphi= \lambda \varphi $ are all given by $w(t) = z(\lambda t) $ where $z$ is a solution of the ODE
$$ z''+z+cz' = 0. $$ In simple terms, those solutions all have the same shape up to time scaling, and a larger eigenvalue gives rise to a ``faster" solution.
As for the level of spatial smoothing effect, one might have thought that it increases with $\alpha$. But Remark 5.2 from \cite{HO} completely disqualifies this idea, since for $\alpha \ge 1$ there is no spatial smoothing effect at all. As we shall see the regularity of solutions for $t>0$ culminates for $\alpha =1/2 $. In the case of the wave equation (i.e. $A = -\Delta$), the value $\alpha =1/2 $ is the only one for which all solutions with initial data in the energy space are analytic in space for all $t>0$. For any $\alpha\in (0,1)$ other than $1/2$, the spatial smoothing effect for the wave equation is best described by a local Gevrey regularity in the sense of \cite{G}.
Gevrey spaces have become rather popular when dealing with hyperbolic problems, and after the pioneering works on analyticity of solutions to PDE such as \cite {K1, K2} based on the method of \cite{Hor}, there are more recent papers dealing with interior Gevrey regularity of solutions to elliptic equations, cf., e.g., \cite{ N-Z, T}. Theorem 3 from \cite{ N-Z} will allow us to study rather easily the level of spatial smoothing effect at $t>0$ for equation \eqref{main} in the concrete PDE cases.
The plan of the paper is as follows: In Section 2 we state and prove the main abstract result, giving first a complete explicit proof in the coercive case and showing the additional necessary steps to obtain the result in the general, possibly non-coercive, case. In Section 3 we apply the general result to the case of the wave equation in any domain of $\mathbb{R}^N$, using a useful result from \cite{ N-Z} which connects local ultra-differentiability properties of solutions to Gevrey type estimates for powers of the main operator $A$. In Section 4 we establish several optimality results, both for the abstract theorem and for the local Gevrey class of solutions to the 1D wave equation with strong dissipation. Section 5 is devoted to higher order equations in space and various extensions of the results. Finally the appendix discusses equivalent definitions of the Gevrey classes for both operators and functions, and develops some useful tools used in the previous sections.
\section{An abstract regularity result} Before stating our main result we need to introduce some notation. Inspired by the notion of ``analytic vectors" for an operator $A$ defined by Nelson in \cite{N} and the Gevrey regularity class of functions (cf. \cite{G}) we define Gevrey vectors as follows
\begin{Def} Let $A$ be any positive self-adjoint operator on $H$. A vector $u\in H$ will be called Gevrey of order $s>0$ with respect to $A$ if $u\in D(A^n) $ for all n and for some $R>0$ we have $$ \forall n\in\mathbb{N}-\{0\}, \quad |A^n u|\le R^n n^{s n}.$$
In this case we write $ u\in G(A, s).$
\end{Def}
\begin{Rmk} The apparent divergence between this definition and those of \cite{N} and \cite{G} will be clarified in the appendix, Proposition 6.1.
\end{Rmk}
\begin{Rmk} It is clear that for any $ s>0$,$$ G( I+ A, s) = G(A, s) $$ and for any $\lambda>0, s>0$, $$ G(\lambda A, s) = G(A, s).$$
\end{Rmk}
\begin{Rmk} We shall prove in the appendix that for any positive self-adjoint operator $A$ and any positive numbers $\alpha, s$ we have
$$ G (A, s) = G(A^\alpha, \alpha s).$$
\end{Rmk}
\bigskip\noindent Our main result is the following.
\begin{Thm}\label{mainth} For any $\alpha\in (0,1)$ and any $(u_0, u_1)) \in V\times H$ , the unique solution $u$ of \eqref{main} with initial date $u(0) = u_0, \quad u'(0) = u_1$ satisfies
\begin{equation}\label{mainrs}
\forall t>0, \quad u(t) \in G(A, \sigma );
\quad \sigma = \min\{ \frac {1}{\alpha}, \frac {1}{1-\alpha}\}.
\end{equation}
\end{Thm}
\begin{proof} We consider successively the cases $0< \alpha\le \frac{1}{2}$ and $\frac{1}{2}<\alpha<1. $ In both cases we shall use the notation
$$ E = V \times H; \ U = (u,u') $$ and we shall occasionally drop $t$ to shorten some formulas.
We first observe that for any positive self-adjoint operator $B$ and any sufficiently smooth solution u of
$$
u'' + Au + Bu' = 0,
$$
we have the formal identity $$ \frac{d}{dt} \{|u'|^2 + |A^{\frac{1}{2}}u|^2 + {\frac{1}{2}}|Bu|^2 +(Bu, u')\} = - \{|B^{\frac{1}{2}}u'|^2 + (Au, Bu)\} ).$$
On the other hand $$ \Phi(u, u'):= |u'|^2 + |A^{\frac{1}{2}}u|^2 + {\frac{1}{2}}|Bu|^2 +(Bu, u')= {\frac{1}{2}}|u'|^2 + |A^{\frac{1}{2}}u|^2 +{\frac{1}{2}}|u'+Bu|^2 $$ yielding the convenient inequalities $$ 0\le {\frac{1}{2}}|u'|^2 + |A^{\frac{1}{2}}u|^2 \le \Phi(u, u') \le \frac{3}{2}|u'|^2 + |A^{\frac{1}{2}}u|^2 + |Bu|^2.
$$
In particular in this fairly general context we have the (formal) inequality
\begin{equation}\label{est:Bhalfuprime:above}
\int_0^t \{|B^{\frac{1}{2}}u'(s)|^2 + (Au(s), Bu(s))\} ds \le \frac{3}{2}|u'(0)|^2 + |A^{\frac{1}{2}}u(0)|^2 + |Bu(0)|^2.
\end{equation}
In order to make the proof easier to follow, we give first a complete proof when $A$ is coercive. Let us first apply this formula for $ B = cA\alpha$ with $0< \alpha \le \frac{1}{2}$. We obtain, since B commutes with all powers of $A$:
$$ \min \{c, c^2\} ~\! t | A^{\frac{\alpha}{2}} u(t), A^{\frac{\alpha}{2}} u'(t)|^{2}_ {V\times H}\le \int_0^t \{|B^{\frac{1}{2}}u'(s)|^2 + (Au(s), Bu(s))\} ds \le K E_0^2$$
with $ E_0 := |(u(0), u'(0))|_{V\times H}.$ Thus by iterating the process twice and replacing $t$ bu $t/2$ we obtain
$$ | A^{\alpha} u(t), A^{\alpha} u'(t)|_ {V\times H} \le K' t^{-1}E_0. $$
Replacing $t$ by $\frac{t}{m}$ and iterating $m$ times, we obtain with $K'= K^{\alpha}$
$$ | A^{m\alpha} u(t), A^{m\alpha} u'(t)|_ {V\times H}\le [ \frac {K' m} {t }]^{ m} E_0. $$
Hence both $u(t)$ and $u'(t)$ belong to $G(A^\alpha, 1) = G(A, \frac{1}{\alpha})$ for all $t>0$.
\\
For $ B = cA\alpha$ with $\frac{1}{2}\le \alpha <1$, we find by
\eqref{est:Bhalfuprime:above}
$$ t | A^{\frac{\alpha}{2}} u(t), A^{\frac{\alpha}{2}} u'(t)|^{2}_ {V\times H}\le K | A^{\alpha- \frac{1}{2}} u(0), A^{\alpha- \frac{1}{2}} u'(0)|^{2}_ {V\times H}.$$ Thus by setting $v(t)= A^{\alpha- \frac{1}{2}} u(t) $, we obtain
$$
| A^{1-\alpha} v(t), A^{1-\alpha} v'(t)|_ {V\times H} \le K' t^{-1}|(v(0), v'(0))|_{V\times H}.
$$
Then by iteration as before: we find that both $v(t)$ and $v'(t)$ belong to $G(A^{1-\alpha}, 1) = G(A, \frac{1}{1-\alpha})$ for all $t>0$, whence follows \eqref{mainrs}.
\bigskip
Let us now consider the general case, but only with $\alpha\not = \frac{1}{2}$ and then, using the remark of the introduction, we can drop the constant $c.$
1) The case $0< \alpha < \frac{1}{2}, c=1$. We start from the inequality
$$ t | A^{\frac{\alpha}{2}+ \frac{1}{2}} u(t), A^{\frac{\alpha}{2}} u'(t)|^{2}_ {H\times H}\le \frac{3}{2}|u'(0)|^2 + |A^{\frac{1}{2}}u(0)|^2 + |A^{\alpha} u(0)|^2.$$ When $A$ is non-coercive, the term $|A^{\alpha} u(0)|^2$ cannot be controlled by $|A^{\frac{1}{2}}u(0)|^2$ only. Instead we may use
$$
|A^{\alpha} u(0)|^2\le |u_0|^2+ |A^{\frac{1}{2}}u(0)|^2
,$$
which implies
$$
\frac{3}{2}|u'(0)|^2 + |A^{\frac{1}{2}}u(0)|^2 + |A^{\alpha} u(0)|^2 \le 2[|u'(0)|^2 + ||u(0)||^2],
$$
where the norm $|| . ||$ is defined on $V: = D(A^{1/2}$ by
$$
\forall x\in V, \quad ||x|| = ( |x|^2 + |A^{\frac{1}{2}} x|^2)^{1/2}.
$$ This will be the only norm used on $V$ later on and we set with $ V\times H: = E$ $$ \forall U = (u, v) \in E, \quad |U|_E = (||u||^2+|v|^2)^{1/2}.$$
A difference with the energy norm used in the coercive case is that now the modified norm is no longer non-increasing. Instead it is easy to prove that for any solution $U = (u, u') $ of \eqref{main}, we have $$\forall t\ge 0, \quad |U(t)|_E^2 \le e^t |U(0)|_E^2. $$
Next we have to handle powers of the operator $I+A$ instead of $A$. More precisely we want to estimate
$ | ((I+A)^{\frac{\alpha}{2}} u(t), (I+A)^{\frac{\alpha}{2}} u'(t))|^{2}_ {E}$ to compare it with $ | (u(0), u'(0))|^{2}_ {E}$. This is not really difficult but has to be done carefully. As a first step we observe that
$$|(I+A)^{\frac{\alpha}{2}} u'(t))|^{2} \le |u'(t)|^2 + |A^{\frac{\alpha}{2}} u'(t)|^2 $$ as a consequence of the operator inequality $$ (I+A)^\alpha \le I+ A^{\alpha}$$ is valid for any $\alpha \in [0, 1]$. Then we note
$$
|| (I+A)^{\frac{\alpha}{2}} u(t)||^2 \leq
| (I+A)^{\frac{\alpha}{2}} u(t)|^2 + | (I+A)^{\frac{\alpha+1}{2}} u(t)|^2.
$$
First by contraction we have
$$
| (I+A)^{\frac{\alpha}{2}} u(t)|^2 \le | (I+A)^{\frac{1}{2}} u(t)|^2 = ||u(t)||^2.
$$
Then
$$ | (I+A)^{\frac{\alpha+1}{2}} u(t)|^2\le |u(t)|^2 + | A^{\frac{\alpha +1}{2}} u(t)|^2 \le ||u(t)||^2 + | A^{\frac{\alpha +1}{2}} u(t)|^2$$ so that we obtain
$$ || (I+A)^{\frac{\alpha}{2}} u(t)||^2 \le 2 ||u(t)||^2 + | A^{\frac{\alpha +1}{2}} u(t)|^2$$ and finally
$$| ((I+A)^{\frac{\alpha}{2}} u(t), (I+A)^{\frac{\alpha}{2}} u'(t))|^{2}_ {E} \le | A^{\frac{\alpha +1}{2}} u(t)|^2 + |A^{\frac{\alpha}{2}} u'(t)|^2 + 2 |U(t)|^2_E.$$
Because $|U(t)|_E^2 \le e^t |U(0)|_E^2$ we now obtain
$$
| ((I+A)^{\frac{\alpha}{2}} u(t), (I+A)^{\frac{\alpha}{2}} u'(t))|^{2}_ {E}
\le 2 (e^t + \frac{1}{t}) |U(0)|_E^2.
$$
This is enough to conclude in a few easy steps.
\\
2) The case $\frac{1}{2}< \alpha < 1, c=1$. We start from the inequality
$$
t | A^{\frac{\alpha}{2}+ \frac{1}{2}} u(t), A^{\frac{\alpha}{2}} u'(t)|^{2}_ {H\times H}\le \frac{3}{2}|u'(0)|^2 + |A^{\frac{1}{2}}u(0)|^2 + |A^{\alpha} u(0)|^2.
$$
First we have
$$
|A^{\frac{1}{2}}u(0)|^2 \le |u_0|^2+ |A^{\frac{1}{2}} u(0)|^2
= |(I+A) ^{\frac{1}{2}}u(0)|^2
\le |(I+A) ^{\alpha}u(0)|^2.
$$
Moreover
$$
|A^{\alpha} u(0)|^2 \le |(I+A)^{\alpha} u(0)|^2;\quad |u'(0)|^2 \le |(I+A) ^{\alpha-\frac{1}{2}}u'(0)|^2
$$
and we obtain
$$
\frac{3}{2}|u'(0)|^2 + |A^{\frac{1}{2}}u(0)|^2 + |A^{\alpha} u(0)|^2
\le 2[|(I+A) ^{\alpha-\frac{1}{2}}u'(0)|^2 + ||(I+A) ^{ \alpha}u(0)||^2].
$$
So the RHS of the basic inequality is bounded by
$$
2 |(I+A) ^{\alpha-\frac{1}{2}}U(0)|^2_E.
$$
It remains to bound the quantity $ | ((I+A)^{\frac{\alpha}{2}} u(t), (I+A)^{\frac{\alpha}{2}} u'(t))|^{2}_ {E}.$ We first write
$$ |(I+A)^{\frac{\alpha}{2}} u'(t))|^{2} = |(I+A)^{\frac{1-\alpha}{2}} (I+A)^{\alpha -\frac{1}{2}}u'(t))|^{2} $$
$$ \le |(I+A)^{\alpha - \frac{1}{2}} u'(t))|^{2} + |A^{\frac{1-\alpha}{2}}(I+A)^{\alpha - \frac{1}{2}} u'(t))|^{2}$$
$$\le |(I+A)^{\alpha - \frac{1}{2}} u'(t))|^{2} + |A^{\frac{1-\alpha}{2}} u'(t))|^{2} + |A^{\frac{\alpha}{2}} u'(t))|^{2}$$
$$ \le |(I+A)^{\alpha - \frac{1}{2}} u'(t))|^{2} + |u'(t))|^{2} + 2|A^{\frac{\alpha}{2}} u'(t))|^2 $$
$$
\le 2 [ |(I+A)^{\alpha - \frac{1}{2}} u'(t))|^{2} + |A^{\frac{\alpha}{2}} u'(t))|^2 ].
$$
A quite similar calculation gives $$ |(I+A)^{\frac{\alpha+1}{2}} u(t))|^{2} \le 2 [ |(I+A)^{\alpha } u(t))|^{2} + |A^{\frac{\alpha +1}{2}} u(t))|^2 ] $$ and by addition we obtain
$$ |(I+A)^{\frac{\alpha}{2}} u'(t))|^{2} + |(I+A)^{\frac{\alpha+1}{2}} u(t))|^{2} \le |A^{\frac{\alpha}{2}} u'(t))|^2 + |A^{\frac{\alpha +1}{2}} u(t))|^2 +2 e^t |(I+A) ^{\alpha-\frac{1}{2}}U(0)|^2_E .$$ Finally we obtain
$$ |(I+A)^{\frac{\alpha}{2}} u'(t))|^{2} + |(I+A)^{\frac{\alpha+1}{2}} u(t))|^{2} \le 2( e^t + \frac{1}{t}) |(I+A) ^{\alpha-\frac{1}{2}}U(0)|^2_E . $$
Setting $ v(t):= (I+A) ^{\alpha-\frac{1}{2}}u(t) $ we obtain
$$ |(I+A)^{\frac{1-\alpha}{2}} V(t))|^{2} _E \le 2( e^t + \frac{1}{t}) |V(0)|^2_E . $$ Then the conclusion follows easily as previously. \\
3) The case $\alpha =\frac{1}{2}$ In this case we can follow either the method of case 1 or case 2 but the constant $c$ will appear in the estimate. Since the calculation is just a variant and has been done completely in the coercive case, for the sake of brevity we skip the details.
\end{proof}
\section{The case of the wave equation with strong damping}
\begin{Def} Let $\Omega$ be any open subset of $\mathbb{R}^N$. A function $f: \Omega \rightarrow \mathbb{R}$ will be called Gevrey of order $s>0$ in $\Omega$ if $f\in C^\infty (\Omega ) $ and for any $K$ compact subset of $\Omega$, there is $R= R(K)>0$ such that, for any differential monomial
$ D^p: = D_1^{p_1} D_2^{p_2} ...D_N^{p_N}$ we have
$$ ||D^p f||_{L^{\infty} (K)}\le R ^{|p|} |p|^{s|p|}. $$
In this case we write $ f\in G^s(\Omega).$
\end{Def}
\begin{Rmk} This definition is slightly different from the definition given in the historical literature, in particular in the seminal paper \cite{G}, but it is in fact equivalent and more condensed, cf. appendix, Proposition 6.1. \end{Rmk}
\begin{Thm}\label{wave} Let $\Omega$ be any open subset of $\mathbb{R}^N$ and $H= L^2(\Omega), V = H^1_0(\Omega)$.
Let $A = -\Delta$ with domain $D(A) = \{ w\in V, \Delta v\in H\}$ For any $\alpha\in (0,1)$ and any $(u_0, u_1) \in V\times H$ , the unique solution $u$ of \eqref{main} with initial data $u(0) = u_0, \,\,u'(0) = u_1$ satisfies
\begin{equation}\label{mainrs2}
\forall t>0, \quad u(t) \in G^s(\Omega) \end{equation} with\begin{equation}s = \min \{ \frac {1}{2\alpha}, \frac {1}{2(1-\alpha)}\} . \end{equation} In particular for $\alpha = 1/2$, $u(t)$ is analytic inside $\Omega$ for all $t>0$.
\end{Thm}
\begin{proof} By Theorem 3 of \cite{N-Z}, when $A $ is a second order elliptic operator, we have $ G(A, 2s)\subset G^s(\Omega)$. Therefore the result is an immediate consequence of Theorem \ref{mainth} .\end{proof}
\section{Optimality results}
The next two results provide a very strong optimality statement for Theorem \ref{mainth}.
\begin{Thm}\label{opt} Let $A$ be coercive with $A^{-1}$ compact and assume the two following conditions
i) For some $\varepsilon>0, \delta>0$ we have $$\forall n\ge 1, \quad \lambda_n\ge \delta n^\varepsilon. $$
ii) For some $C>1$ we have $$\forall n\ge 1, \quad \lambda_{n+1} \le C\lambda_n. $$
Assume $\alpha \in (1/2, 1).$ Then there is a solution of \eqref{main} with initial data $u(0) = u_0, u'(0) = u_1$ in $V\times H$ for which we have
$$ \forall k\ge 1, \forall t>0, \quad |A^k u(t)| \ge [\delta(t)k]^ {\frac{k}{1-\alpha}} $$
where $\delta(t)>0$ for all $t>0.$
\end{Thm}
\begin{proof} Assuming ${\lambda_{n_0}}^{2\alpha-1}> 4$ we look for a solution of \eqref{main} of the form $$ u(t) = \sum_{n\ge n_0} c_n e^{-\mu_n t} \varphi_n $$ with $\varphi_n$ a sequence of normalized eigenvectors corresponding to the eigenvalues $\lambda_n$ and $$ \mu_n: = \frac{\lambda_n^\alpha}{2}- \sqrt{\frac{\lambda_n^{2\alpha}}{4}- \lambda_n}= \frac {\lambda_n}{\frac{\lambda_n^\alpha}{2}+ \sqrt{\frac{\lambda_n^{2\alpha}}{4}- \lambda_n}}. $$
We observe that for all $n$ $$\lambda_n^{1-\alpha} \le \mu_n \le 2\lambda_n^{1-\alpha}. $$
It is clear that $u$ is indeed a solution of \eqref{main} if the coefficients $c_n$ tend to $0$ fast enough when $n$ grows to infinity. A sufficient condition for that is $$c_n = \lambda_n^{-K};\quad K> 1 +\frac{1}{2\varepsilon}.$$
Now for all k we have $$A^k u(t) = \sum_{n\ge n_0} c_n\lambda_n^k e^{-\mu_n t} \varphi_n $$ and as a consequence of orthogonality of the eigenvectors in $H$ we obtain
$$ \forall n\ge n_0, \forall t>0, \quad |A^k u(t)| \ge c_n \lambda_n^k e^{-\mu_n t}.$$
In that inequality we choose
$$ n= \inf \{m\ge n_0, \lambda_m \ge k^{\frac{1}{1-\alpha}}\}. $$
Then for $k$ large enough we must have $n>n_0$. In this case $ \lambda_{n-1}< k^{\frac{1}{1-\alpha}}$ and then $ \lambda_n \le Ck^{\frac{1}{1-\alpha}}$. It follows that $$ c_n \lambda_n^k e^{-\mu_n t} \ge \lambda_n^{-K} k^{\frac{k}{1-\alpha}} e^{-2C^{1-\alpha} kt }\ge C^{-K}k^{-\frac{K}{1-\alpha}}e^{-2C^{1-\alpha} kt }k^{\frac{k}{1-\alpha}}.$$ This concludes the proof.
\end{proof}
\begin{Thm}\label{opt2} Let $A$ be as in the statement of Theorem \ref{opt}. Assume $\alpha \in (0, 1/2).$ Then there is a solution of \eqref{main} with initial data $u(0) = u_0, u'(0) = u_1$ in $V\times H$ for which we have
$$ \forall k\ge n_0, \forall t>0, \forall \theta>t, \quad \int _t^{\theta} |A^k u(s)|^2ds \ge [\delta(t, s)k]^ {\frac{2k}{\alpha}}, $$
where $\delta(t,s)>0$.
\end{Thm}
\begin{proof} Assuming ${\lambda_{n_0}}^{1-2\alpha}> 1/4$, we shall find find a solution of \eqref{main} of the form $$ u(t) = \sum_{n\ge n_0} c_n e^{-\frac{\lambda_n^\alpha}{2}t} \cos \left(t \sqrt{\lambda_n- \frac{\lambda_n^{2\alpha}}{4}}\right)\varphi_n $$ with $\varphi_n$ a sequence of normalized eigenvectors corresponding to the eigenvalues $\lambda_n$.
It is clear that $u$ is indeed a solution of \eqref{main} if the coefficients $c_n$ tend to $0$ fast enough when $n$ grows to infinity. A sufficient condition for that is $$c_n = \lambda_n^{-K};\quad K> 1 +\frac{1}{2\varepsilon}.$$
Now for all k we have $$A^k u(t) = \sum_{n\ge n_0} c_n \lambda_n^k e^{-\frac{\lambda_n^\alpha}{2}t} \cos \left(t \sqrt{\lambda_n- \frac{\lambda_n^{2\alpha}}{4}}\right)\varphi_n $$ and as a consequence of orthogonality of the eigenvectors in $H$ we obtain
$$ \forall n\ge n_0, \forall t>0, \quad |A^k u(t)|^2 \ge c_n^2 \lambda_n^{2k} e^{-\lambda_n^\alpha t} \cos^2 \left(t \sqrt{\lambda_n- \frac{\lambda_n^{2\alpha}}{4}}\right).$$
In that inequality we choose
$$ n= \inf \{m\ge n_0, \lambda_m \ge k^{\frac{1}{\alpha}}\}. $$
Then for $k$ large enough we must have $n>n_0$. In this case $ \lambda_{n-1}< k^{\frac{1}{\alpha}}$ and then $ \lambda_n \le Ck^{\frac{1}{\alpha}}$. The end of the proof is now quite similar to the proof of the previous result, the only difference being integration in t to handle the oscillating term and the remark that the integral of the function $\cos^2 \left(t \sqrt{\lambda_n- \frac{\lambda_n^{2\alpha}}{4}}\right)$ on any time interval $J$ tends to $|J| /2$ as $n$ tends to infinity. We skip the details.
\end{proof}
\begin{Rmk}\label{opt3} Let $A$ be as in the statement of Theorem \ref{opt}. Assume $\alpha =1/2.$ Then there is a solution of \eqref{main} with initial data $u(0) = u_0, u'(0) = u_1$ in $V\times H$ for which we have
$$ \forall k\ge n_0, \forall t>0, \forall \theta>t, \quad \int _t^{\theta} |A^k u(s)|^2ds \ge [\delta(t, s)k]^ {4k}, $$
where $\delta(t,s)>0$. The proof follows the line of proof of either Theorem \ref{opt} if $c\ge 2$ or Theorem \ref{opt2} if $c<2$ . We skip the details.
\end{Rmk}
\begin{Cor}\label{opt3} Let $\Omega$ be a bounded interval of $\mathbb{R}$ and $H= L^2(\Omega); V = H^1_0(\Omega).$ Then for any $r<s=\min \{ \frac {1}{2\alpha}, \frac {1}{2(1-\alpha)}\}$ given by Theorem \ref{wave}, there is a solution of \eqref{main} with initial data $u(0) = u_0, u'(0) = u_1$ in $V\times H$ for which $u(t)$ never belongs to $G^r(\Omega)$. In particular the solutions are not analytic in general for $t>0$ if $\alpha\not= 1/2$.\end{Cor}
\begin{proof} By a translation and a space-scaling we can reduce the question to the case $ \Omega = (0, 3\pi)$ and show that for some solutions, the Gevrey estimates in the interior subset $ \omega = (0, \pi)$ are not better than those of the general theorems. The eigenfunctions of the Dirichlet-Laplacian in $ \Omega = (0, 3\pi)$ are the functions $\sin \frac{kx}{3}$. We choose the solutions with initial data spanned by the functions $\sin (mx) $ only, restricting ourselves to $k = 3m$. These solutions satisfy the same equation in $ \omega = (0, \pi)$ with homogeneous Dirichlet boundary conditions, and here the estimates of successive derivatives correspond exactly to the double exponents for the same powers of the Laplacian. Therefore the examples constructed in The two previous theorems and the remark provide solutions having in $ \omega = (0, \pi)$ for $t>0$ the exact Gevrey regularity allowed by Theorem \ref{wave}, and not more.\end{proof}
\section{Other examples and possible extensions}
The general Theorems apply also to plate (beam in 1D) equations, either clamped or simply supported.
\begin{Thm}\label{CPlate} Let $\Omega$ be any open subset of $\mathbb{R}^N$ and $H= L^2(\Omega); V = H^2_0(\Omega)$
Let $A = -\Delta$ with domain $D(A) = \{ w\in V, \Delta v\in H\}$ For any $\alpha\in (0,1)$ and any $(u_0, u_1) \in V\times H$, the unique solution $u$ of \eqref{main} with initial data $u(0) = u_0, \quad u'(0) = u_1$ satisfies
\begin{equation}\label{mainrs2}
\forall t>0, \quad u(t) \in G^s(\Omega) \end{equation} with \begin {equation} \label{mainrs3}
s = \min \{ \frac {1}{4\alpha}, \frac {1}{4(1-\alpha)}\} . \end{equation}
In particular for all $\alpha \in [1/4, 3/4] $, $u(t)$ is analytic inside $\Omega$ for all $t>0$.
\end{Thm}
\begin{proof} By Theorem 3 of \cite{N-Z}, when $A $ is a fourth order elliptic operator, we have $ G(A, 4s)\subset G^s(\Omega)$. Therefore the result is an immediate consequence of Theorem \ref{mainth} .\end{proof}
\begin{Thm}\label{SSPlate} Let $\Omega$ be any open subset of $\mathbb{R}^N$ and $H= L^2(\Omega); V = H^2\cap H^1_0(\Omega)$
Let $A = -\Delta$ with domain $D(A) = \{ w\in V, \Delta v\in H\}$ For any $\alpha\in (0,1)$ and any $(u_0, u_1) \in V\times H$, the unique solution $u$ of \eqref{main} with initial data $u(0) = u_0, \quad u'(0) = u_1$ satisfies \eqref{mainrs2} and \eqref{mainrs3}.
In particular for all $\alpha \in [1/4, 3/4] $, $u(t)$ is analytic inside $\Omega$ for all $t>0$.
\end{Thm}
\begin{proof} Same as for Theorem \ref{CPlate}.\end{proof}
\begin{Rmk} It is clear from the structure of the proof of Theorem \ref{mainth} that $B$ does not need to be an exact power of $A$ for the result to hold true. For instance, a linear combination with positive coefficients of an arbitrary number of powers of $A$ : $B= \sum c_i A^{\alpha_i} $ will give the same regularity result with $\alpha$ replaced by the highest exponent. Similarly $ B = c (dI+ A)^\alpha $ or a sum of such operators will give the same result as $ A^\alpha$. \end{Rmk}
\section{Appendix} In this appendix we establish a few properties of general interest that have been used in the proofs of our main results. \subsection{Equivalent formulations of Gevrey spaces} In this section, we clarify once and for all the connection between the various definitions of Gevrey regularity found in the literature. The original definition by Gevrey in \cite{G} involves a power of the multi-factorial $ p! : p_1!...p_N! $ when $p:= (p_1,...p_N)$ is an N-vector with integer coordinates. This was in fact motivated by the possibility of considering different regularity levels in the N different differentiation directions. When one is not interested in doing that, one might consider replacing the multi-factorial by the factorial of the total differentiation order $|p|: = p_1+...+ p_N$, i.e., consider $|p|!$ instead of $ p! $ Do we still find the same regularity class? Alternatively, many authors replaced in the definition the factorials $p_j !$ by $p_j^{p_j} .$ , justifying usually this change by Stirling's asymptotic formula. Then what about using simply the apparently larger number $|p|^{|p|}$? The next result shows that all those notions are equivalent, and we can quantify exactly the equivalence constants as a function of the dimension only.
\begin{Prop}\end{Prop} For any $N\in \mathbb{N}^*$ and any $p= (p_1,...p_N)\in {\mathbb{N}^*}^N$ we have
$$ p_1!...p_N!= p! \le p^p (= \prod _1^N p_i^{p_i}) \le |p|^{|p|} \le (4^{N-1})^{|p|}p^p \le (4^{N-1}e)^{|p|} p! $$
\begin{proof}The first 2 inequalities $ p_1!...p_N!= p! \le p^p (= \prod _1^N p_i^{p_i}) \le |p|^{|p|} $ are completely obvious. So we are left to check that $|p|^{|p|} \le (4^{N-1})^{|p|}p^p \le (4^{N-1}e)^{|p|} p! $
We do this in 3 steps. \\
Step 1. In the case of two components, we claim that $$ \forall (p,q)\in {\mathbb{N}^*}^2, \quad (p+q)^{p+q}\le 2^{2(p+q)} p^p q^q $$
Indeed assuming $p\leq$ with $2^r p\le q\le 2^{r+1} p$, we obtain first
$$ q^p\le (2^{r+1} p)^p = p^p 2^{p+rp} \le p^p 2^{p+q}$$ since $rp\le 2^r p \le q.$ Then
$$ (p+q)^{p+q} \le (2q)^{(p+q)}= 2^{p+q} q^{p+q}= q^q 2^{p+q} q^{p}\le q^q 2^{p+q} p^p 2^{p+q}$$ and the claim is justified. \\
Step 2. In the case of three or more components, we prove by induction that $$|p|^{|p|} \le (4^{N-1})^{|p|} p^p $$
Assuming the result to be true for N-1 components, we use the result for 2 components with $p_1$ and $ p_2+...p_N$ in place of $p$ and $q$, which gives
$$ (p_1+p_2 +...p_N)^{(p_1+p_2 +...p_N)} \le 2^{2|p|} p_1^{p_1}(p_2 +...p_N)^{(p_2 +...p_N)} $$
$$ \le 2^{2|p|} p_1^{p_1} 2^{2(N-2)(p_2+...+p_N)}{p_2}^{p_2} ...{p_N}^{p_N} \le 2^{2(N-1)|p|} p_1^{p_1} {p_2}^{p_2} ...{p_N}^{p_N} $$ and the result follows. \\
Step 3. The concavity of the function $\ln$ on $(0, 1)$ implies that for any integer $k$, we have $k^k\le e^k k!$ Hence $$ p^p \le e^{|p|}p! $$
The proof is concluded by combining Step 2 and Step 3.
\end {proof}
\subsection {Gevrey spaces for a power of an operator}
\begin{Prop} For any positive self-adjoint operator $A$ and any positive numbers $\alpha, s$ we have
$$ G (A, s) = G(A^\alpha, \alpha s).$$
\end{Prop}
\begin {proof} Assume that $u\in D(A^n) $ for all n and for some $R>0$ we have $$ \forall n\in\mathbb{N}, \quad |A^n u|\le R^n n^{s n}.$$
Then for any $\theta \in (0, 1)$, by the following interpolation inequality
\begin{equation*}
| A^\theta u | \leq | A u|^{\theta} |u|^{1-\theta} \qquad \forall u \in D(A),
\end{equation*}
we obtain
$$
\forall n\in\mathbb{N}, \quad
|A^{n\theta} u| \le ~\! R^{n\theta} n^{\theta s n} |u| ^{(1-\theta) n}
= ~\![R^{\theta} |u| ^{(1-\theta)}]^n n^{(\theta s) n} $$ which means exactly that $u \in G(A^\theta, \theta s).$ Hence for all $\alpha \in (0,1],$
$$
G (A, s) \subset G(A^\alpha, \alpha s).
$$
We also find
\begin{align*}
\forall m \in \mathbb{N}, \quad |A^{m+\theta} u|
& \le R^m m^{sm} ~\! |Au|^{\theta} |u|^{(1-\theta)}
\\
& \le |u|^{1-\theta} ~\!
[\max \{R, |Au|\}]^{m+\theta} (m +\theta)^{s(m +\theta)},
\end{align*}
which implies in particular
$$
\forall \tau \ge 1, \quad |A^{\tau} u| \le [\max \{R, |u|, |Au|, 1\}] ^{\tau} (\tau)^{s\tau}.
$$
In particular, taking $ \tau = n \alpha $, we have
$$ \forall n\in\mathbb{N}-\{0\}, \quad |A^{n\alpha} u|\le [\max \{R, |u|, |Au|, 1\}\alpha^s] ^{n\alpha} n^{sn\alpha}$$ which means exactly that $u \in G(A^\alpha, \alpha s).$ Hence for all $\alpha \ge 1,$ $$ G (A, s) \subset G(A^\alpha, \alpha s)$$ Finally $$ \forall \alpha>0, \forall s>0, \quad G (A, s) \subset G(A^\alpha, \alpha s)$$ The equality follows by exchanging the roles of $A$ and $A^\alpha$.
\end {proof}
\subsection {Some operatorial inequalities}
\begin{Prop} Let $A$ be any positive self-adjoint operator on a Hilbert space H. Then
$$ \forall \beta \in [0, 1], \quad (I+ A)^\beta \le I + A^\beta $$
$$ \forall \beta \in [0, 1], \forall u \in D(A),\quad |A^\beta u|^2 \le |u|^2 +|A u|^2 $$
\end{Prop}
\begin{proof}
The first inequality follows classically from the methods of operator calculus invoking the scalar inequality $$ \forall \beta \in [0, 1],\forall h>0, \quad (I+ h)^\beta \le I + h^\beta $$ As for the second inequality we just write, assuming $\beta<1$,
$$ |A^\beta u|^2 \le |Au|^{2\beta} |u|^{2(1-\beta )}\le [A|u|^{2\beta}]^{\frac{1}{\beta}} + [|u|^{2(1-\beta )}] ^{\frac{1}{(1-\beta)}} $$ as a consequence of interpolation and Young's inequality applied with the conjugate exponents $\frac{1}{\beta}$ and $\frac{1}{(1-\beta)}.$
\end {proof}
| 132,891
|
TITLE: Total internal reflection and waveguides
QUESTION [2 upvotes]: I'm looking at a model of a planar dielectric waveguide along the lines of this picture:
For the wave inside the $n_1$ dielectric slab to totally internally reflect $\theta_M$ needs to be smaller than $cos^{-1} (n_2 / n_1)$. In my textbook it says that the treatment of the planar dielectric waveguide is completely analogous to the case of two planar mirrors with perfect reflectivity SHORT OF ONE difference: Due to the evanescent wave that travels a very short length into the $n_2$ medium the reflected wave experiences an angle-dependent phase shift given by $tan(\phi /2) = \sqrt{\frac{sin^2(\theta_C)}{sin^2{\theta}} -1} $, where c denotes the critical incident angle and $\phi$ is the phase shift, that the reflected wave experiences.
There is no explanation of this in the book and I couldn't figure out how to get to that result. Anybody care to explain?
Cheers
REPLY [2 votes]: The phase-shift occurs due to the complex nature of the reflection coefficient. The so-called critical angle is given by:
$sin(\theta_M) = \frac{n_2}{n_1}$
As long as $\theta<\theta_M$ we have only partial reflection and a real valued reflection coefficient $R$.
As soon as the critical angle is exceeded $(\theta>\theta_M)$, we have $\mid R \mid=1$ and total reflection of the light occurs. $R$ is now complex and a phase shift is imposed on the reflected light, we write:
$R=e^{2j\phi}$
wehre $\phi$ is the phase shift between the incident and reflected wave.
Now, we know what the value of $R$ is, both for TE and for TM modes, i.e. the Fresnel coefficients for the two different states of polarization of the electric and magnetic fields. To get the formula for the phase-shift you only need to equal this complex expression for $R$ to the Fresnel formulas and solve for $\phi$. The formula you wrote is the one corresponding to TE modes, so it's the TE phase-shift (with electric fields perpendicular to the plane of incidence spanned by the wave normal and the normal to the interface). For TM modes you would get exactly the same result but multiplied by a factor of $n_{1}^{2}/n_{2}^{2}$.
| 47,555
|
V..
Copper Belle Drill Holes CB-18-38 and CB-18-36 Mineralized Intervals
* All assay values are uncut and intervals reflect drilled intercept lengths.
* True widths of the mineralization have not been determined....
Figure 1: Copper Belle Drill Hole Locations and Composite Mineralized Intervals
To view an enhanced version of Figure 1, please visit::.
| 129,956
|
\begin{document}
\maketitle
\begin{abstract}
We construct a new group model for fusion systems related to Robinson's models and study modules and a homology decomposition associated with it. Moreover we prove analogues of Glaubermann's and Thompson's theorems for $p-$local finite groups and a Kuenneth formula for fusion systems.
\end{abstract}
\section{Introduction}
\label{cIntro}
In the topological theory of $p-$local finite groups introduced by Broto, Levi and Oliver one tries to approximate the classifying space of a finite group via the $p-$local structure of the group, at least up to $\mathbb{F}_p-$cohomology. In this article we introduce a new group model related to Robinson's construction and study a homology decomposition and a family of modules associated to it. Building on the work of Diaz, Glesser, Mazza and Park we prove analogues of Glaubermann's and Thompson's theorems for $p-$local finite groups. Moreover we provide a Kuenneth formula independently of the existence of a classifying space.
\section{Preliminaries}
\subsection{Fusion Systems}
We review the basic definitions of fusion systems and centric linking systems and establish our notations. Our main references are \cite{BLO2}, \cite{BLO4} and \cite{IntroMarkus}.
Let $S$ be a finite $p$-group. A fusion system $\mathcal{F}$ on $S$ is a category whose objects are all the subgroups of $S$, and which satisfies the following two properties for all $P,Q\leq S$: The set $Hom_{\mathcal{F}}(P,Q)$ contains injective group homomorphisms and amongst them all morphisms induced by conjugation of elements in $S$ and each element is the composite of an isomorphism in $\mathcal{F}$ followed by an inclusion. Two subgroups $P,Q\leq S$ will be called $\mathcal{F}-$conjugate if they are isomorphic in $\mathcal{F}$.
Define $Out_{\mathcal{F}}(P)=Aut_{\mathcal{F}}(P)/Inn(P)$ for all $P\leq S$.
A subgroup $P\leq S$ is fully centralized resp. fully normalized in $\mathcal{F}$ if $|C_S(P)|\geq |C_S(P')|$ resp. $|N_S(P)|\geq |N_S(P')|$ for all $P'\leq S$ which is $\mathcal{F}$-conjugate to $P$.
$\mathcal{F}$ is called saturated if for all $P\leq S$ which is fully normalized in $\mathcal{F}$, $P$ is fully centralized in $\mathcal{F}$ and $Aut_S(P)\in Syl_p(Aut_{\mathcal{F}}(P))$ and moreover if $P\leq S$ and $\phi\in Hom_{\mathcal{F}}(P,S)$ are such that $\phi (P)$ is fully centralized, and if we set
$N_{\phi}=\{g\in N_S(P)|\phi c_g \phi^{-1}\in Aut_S(\phi (P))\}$,
then there is $\overline{\phi}\in Hom_{\mathcal{F}}(N_{\phi},S)$ such that $\overline{\phi}|_P=\phi$. A subgroup $P\leq S$ will be called $\mathcal{F}-$centric if $C_S(P')\leq P'$ for all $P'$ which are $\mathcal{F}-$conjugate to $P$. Denote $\mathcal{F}^c$ the full subcategory of $\mathcal{F}$ with objects the $\mathcal{F}-$centric subgroups of $S$.
Let $\mathcal{O}(\mathcal{F})$ be the orbit category of $\mathcal{F}$ with objects the same objects as $\mathcal{F}$ and morphisms the set $Mor_{\mathcal{O}(\mathcal{F})}(P,Q)=Mor_{\mathcal{F}}(P,Q)/Inn(Q)$.
Let $\mathcal{O}^c(\mathcal{F})$ be the full subcategory of $\mathcal{O}(\mathcal{F})$ with objects the $\mathcal{F}-$centric subgroups of $\mathcal{F}$.
A centric linking system associated to $\mathcal{F}$ is a category $\mathcal{L}$ whose objects are the
$\mathcal{F}$-centric subgroups of $S$, together with a functor
$\pi :\mathcal{L}\longrightarrow\mathcal{F}^c$,
and "distinguished" monomorphisms $\delta _P:P\rightarrow Aut_{\mathcal{L}}(P)$ for each $\mathcal{F}$-centric subgroup $P\leq S$ such that the following conditions are satisfied: $\pi$ is the identity on objects and surjective on morphisms. More precisely, for each pair of objects $P,Q\in\mathcal{L},Z(P)$ acts freely on $Mor_{\mathcal{L}}(P,Q)$ by composition (upon identifying $Z(P)$ with $\delta _P (Z(P))\leq Aut_{\mathcal{L}}(P)$), and $\pi$ induces a bijection
$Mor_{\mathcal{L}}(P,Q)/Z(P)\overset{\simeq}{\longrightarrow}Hom_{\mathcal{F}}(P,Q).$
For each $\mathcal{F}$-centric subgroup $P\leq S$ and each $x\in P$, $\pi (\delta _P(x))=c_x \in Aut_{\mathcal{F}}(P)$.
For each $f\in Mor_{\mathcal{L}}(P,Q)$ and each $x\in P, f\circ\delta_P(x)=\delta _Q(\pi f(x))\circ f$.
Let $\mathcal{F}$, $\mathcal{F'}$ be fusion systems on finite $p$-groups $S$, $S'$, respectively. A \texttt{}{morphism of fusion systems from $\mathcal{F}$ to $\mathcal{F'}$} is a pair $(\alpha, \Phi)$ consisting of a group homomophism $\alpha: S\rightarrow S'$ and a covariant functor $\Phi: \mathcal{F}\rightarrow\mathcal{F'}$ with the following properties: for any subgroup $Q$ of $S$ we have $\alpha (Q)=\Phi (Q)$ and for any morphism $\phi: Q\rightarrow R$ in $\mathcal{F}$ we have $\Phi(\phi)\circ\alpha |_Q=\alpha |_R\circ\phi$.
Let $G$ be a discrete group. A finite subgroup $S$ of $G$ will be called a Sylow $p-$subgroup of $G$ if $S$ is a $p-$subgroup of $G$ and all $p-$subgroups of $G$ are conjugate to a subgroup of $S$.
A group $G$ is called $p-$perfect if $H_1(BG;\mathbb{Z}_p)=0$.
Let $\mathcal{F}$ be a saturated fusion system over the finite $p-$group $S$.
Let $G_1,G_2$ be groups with Sylow $p-$subgroups and $\phi :G_1\rightarrow G_2$ a group homomorphism. $\phi$ will be called fusion preserving if the restriction to the respective Sylow $p-$subgroups induces an isomorphism of fusion systems $\mathcal{F}_{S_1}(G_1)\cong\mathcal{F}_{S_2}(G_2)$.
Let $S$ be a finite $p-$group and let $P_1,...,P_r,Q_1,...,Q_r$ be subgroups of $S$. Let $\phi _1,...,\phi _r$ be injective group homomorphisms $\phi _i:P_i\rightarrow Q_i$ $\forall i$. The fusion system generated by $\phi _1,...,\phi_r$ is the minimal fusion system $\mathcal{F}$ over $S$ containing $\phi _1,...,\phi _r$.
Let $\mathcal{F}$ be a fusion system on a finite $p-$group $S$. A subgroup $T\leq S$ is strongly closed in $S$ with respect to $\mathcal{F}$, if for each subgroup $P$ of $T$, each $Q\leq S$, and each $\phi\in Mor_{\mathcal{F}}(P,Q)$, $\phi (P)\leq T$.
Fix any pair $S\leq G$, where $G$ is a (possibly infinite) group and $S$ is a finite $p-$subgroup.
Define $\mathcal{F}_S(G)$ to be the category whose objects are the subgroups of $S$, and where
$Mor_{\mathcal{F}_S(G)}(P,Q)=Hom_G(P,Q)=\{c_g\in Hom(P,Q)|g\in G, gPg^{-1}\leq Q\}
\cong N_G(P,Q)/C_G(P).$
Here $c_g$ denotes the homomorphism conjugation by $g$ $(x\mapsto gxg^{-1})$, and $N_G(P,Q)=\{g\in G|gPg^{-1}\leq Q\}$ (the transporter set).
For each $P\leq S$, let $C'_G(P)$ be the maximal $p-$perfect subgroup of $C_G(P)$. Let $\mathcal{L}^c_S(G)$ be the category whose objects are the $\mathcal{F}_S(G)-$centric subgroups of $S$, and where $Mor_{\mathcal{L}^c_S(G)}(P,Q)=N_G(P,Q)/C_G'(P).$
Let $\pi:\mathcal{L}^c_S(G)\rightarrow\mathcal{F}_S(G)$ be the functor which is the inclusion on objects and sends the class of $g\in N_G(P,Q)$ to conjugation by $g$. For each $\mathcal{F}_S(G)-$centric subgroup $P\leq G$, let $\delta _P:P\rightarrow Aut_{\mathcal{L}^c_S(G)}(P)$ be the monomorphism induced by the inclusion $P\leq N_G(P)$.
A triple $(S,\mathcal{F},\mathcal{L})$ where $S$ is a finite $p-$group, $\mathcal{F}$ is a saturated fusion system on $S$, and $\mathcal{L}$ is an associated centric linking system to $\mathcal{F}$, is called a $p-$local finite group. It'
s classifying space is $|\mathcal{L}|\pcom$ where $(-)\pcom$ denotes the $p-$completion functor in the sense of Bousfield and Kan.
This is partly motivated by the fact that every finite group $G$ gives canonically rise to a $p-$local finite group $(S,\mathcal{F}_S(G),\mathcal{L}^c_S)$ and $BG\pcom\simeq |\mathcal{L}|\pcom$ \cite{BK}. In particular, all fusion systems coming from finite groups are saturated.
Let $\mathcal{F}$ be a fusion system on the the finite $p-$group $S$. $\mathcal{F}$ is called an Alperin fusion system if there are subgroups $P_1,P_2,\cdots P_r$ of $S$ and finite groups $L_1,\cdots ,L_r$ such that: $P_i\cong O_p(L_i)$(the largest normal $p-$subgroup of $L_i$) and $C_{L_i}(P_i)=Z(P_i)$, $L_i/P_i\cong Out_{\mathcal{F}}(P_i)$ for each $i$, $N_S(P_i)$ is a Sylow $p-$subgroup of $L_i$ for each $i$ and $P_1=S$, for each $i$ $\mathcal{F}_{N_S(P_i)}(L_i)$ is contained in $\mathcal{F}$, $\mathcal{F}$ is generated by all the $\mathcal{F}_{N_S(P_i)}(L_i)$. Recall that
every saturated fusion system is Alperin since let $\mathcal{F}$ be a saturated fusion system over a finite $p-$group $S$. Let $S=P_1,...,P_n$ be subgroups of $S$ which are representatives of isomorphism classes of centric radicals in $\mathcal{F}$. From Section 4 in \cite{BCGLO1} it follows that we can find corresponding groups $L_i$, $i=1,...,n$ which have all the properties.\\
One can define fusion systems and centric linking systems in a topological setting. We will need this when we make use of the fact that a group realizes a given fusion system if and only if its classifying space has a certain homotopy type. In particular we have for a $p-$local finite group $(S,\mathcal{F},\mathcal{L})$ and a group $G$ such that $\mathcal{F}_S(G)=\mathcal{F}$ that there is a map from the one-skeleton of the nerve of $\mathcal{L}$ to the classifying space : $|\mathcal{L}|^{(1)}\rightarrow BG$.
Fix a space X, a finite $p$-group $S$, and a map $f:BS\rightarrow X$.
Define $\mathcal{F}_{S,f}(X)$ to be the category whose objects are the subgroups of $S$, and whose morphisms are given by
$Hom_{\mathcal{F}_{S,f}(X)}(P,Q)=\{\phi\in Inj(P,Q)|f|_{BP}\simeq f|_{BQ}\circ B\phi\}$
for each $P,Q\leq S$.
Define $\mathcal{F}'_{S,f}(X)\subseteq\mathcal{F}_{S,f}(X)$ to be the subcategory with the same objects as $\mathcal{F}_{S,f}(X)$, and where $Mor_{\mathcal{F'}_{S,f}(X)}(P,Q)$ (for $P,Q \leq S$) is the set of all composites of restrictions of morphisms in $\mathcal{F}_{S,f}(X)$ between $\mathcal{F}_{S,f}(X)-$centric subgroups.
Define $\mathcal{L}^c_{S,f}(X)$ to be the category whose objects are the $\mathcal{F}_{S,f}(X)-$centric subgroups of $S$, and whose morphisms are defined by
$Mor_{\mathcal{L}^c_{S,f}}(P,Q)=\{(\phi,[H])|\phi\in Inj(P,Q), H:BP\times I\rightarrow X, H|_{BP\times 0}=f|_{BP}, H|_{BP\times 1}=f|_{BQ}\circ B\phi \}. $
The composite in $\mathcal{L}^c_{S,f}(X)$ of morphisms
$P\stackrel{(\phi,[H])}{\longrightarrow}Q\stackrel{(\psi,[K])}{\longrightarrow}R$,
where $H:BP\times I\rightarrow X$ and $K:BQ\times I\rightarrow X$ are homotopies as described above, are defined by setting
$(\psi,[K])\circ (\phi, [H])=(\psi\circ\phi,[(K\circ(B\phi\times ID))* H])$,
where $*$ denotes composition (juxtaposition) of homotopies. Let
$\pi : \mathcal{L}^c_{S,f}(X)\longrightarrow\mathcal{F}_{S,f}(X)$
be the forgetful functor: it is the inclusion on objects, and sends a morphism $(\phi,[H])$ to $\phi$. For each $\mathcal{F}_{S,f}(X)-$centric subgroup $P\leq S$, let
$\delta _P :P\longrightarrow Aut_{\mathcal{L}^c_{S,f}(X)}(P)$
be the "distinguished homomorphism" which sends $g\in P$ to $(c_g, [f|_{BP}\circ H_g])$, where $H_g:BP\times I\rightarrow BP$ denotes the homotopy from $Id_{BP}$ to $Bc_g$ induced by the natural transformation of functors $\mathcal{B}(G)\rightarrow\mathcal{B}(G)$ which sends the unique object $\circ _G$ in $\mathcal{B}G$ to the morphism $\widehat{g}$ corresponding to $g$ in $G$.
\begin{Theorem}[\cite{BLO4}, Theorem 2.1.]
Fix a space $X$, a $p-$group $S$, and a map $f:BS\rightarrow X$. Assume that $f$ is Sylow; $f|_{BP}$ is a centric map for each $\mathcal{F}_{S,f}(X)-$centric subgroup $P\leq S$; andevery $\mathcal{F}'_{S,f}(X)-$centric subgroup of $S$ is also $\mathcal{F}_{S,f}(X)$-centric.
Then the triple $(S,\mathcal{F}'_{S,f}(X),\mathcal{L}^c_{S,f}(X))$ is a $p-$local finite group.
\end{Theorem}
\subsection{Groups Realizing a Given Fusion System}
Given a fusion system $\mathcal{F}$ on a finite $p-$group $S$ it is not always true that there exists a finite group $G$ such that $\mathcal{F}_S=\mathcal{F}_S(G)$, (see \cite{BLO2}, chapter 9 for example).
However for every fusion system $\mathcal{F}$ there exists an infinite group $\mathcal{G}$ such that $\mathcal{F}_S(\mathcal{G})=\mathcal{F}$. We now describe the constructions by G. Robinson \cite{Robinson1}, and I. Leary and R. Stancu \cite{Ian+Radu}.
The groups of Robinson type are iterated amalgams of automorphism groups in the linking system, if it exists, over the $S-$normalizers of the respective $\mathcal{F}$-centric subgroups of $S$. Note that these automorphism groups exist and are known regardless of whether $\mathcal{L}$ exists or not.
\begin{Theorem}[\cite{Robinson1}, Theorem 2.]
Let $\mathcal{F}$ be an Alperin fusion sytem on a finite $p-$group $S$ and associated groups $L_1,...,L_n$ as in the definiton. Then there is a finitely generated group $\mathcal{G}$ which has $S$ as a Sylow $p-$subgroup such that the fusion system $\mathcal{F}$ is realized by $\mathcal{G}$.
The group $\mathcal{G}$ is given explicitely by $\mathcal{G}=L_1\underset{N_S(P_2)}{*}L_2\underset{N_S(P_3)}{*}...\underset{N_S(P_n)}*L_n$ with $L_i,P_i$ as in the definiton.
\end{Theorem}
Corresponding to the various versions of Alperin's fusion theorem (essential subgroups, centric subgroups, centric radical subgroups) there exist several canonical choices for the groups generating $\mathcal{F}$.
The group constructed by I. Leary and R. Stancu is an iterated HNN-construction.
\begin{Theorem}[\cite{Ian+Radu}, Theorem 2.]
Suppose that $\mathcal{F}$ is the fusion system on $S$ generated by $\Phi=\{\phi_1, \cdots, \phi_r\}$. Let $T$ be a free group with free generators $t_1, \ldots, t_r$, and define $G$ as the quotient of the free product $S*T$ by the relations $t_i^{-1}ut_i=\phi_i(u)$ for all $i$ and for all $u\in P_i$. Then $S$ embeds as a $p-$ Sylow subgroup of $G$and $\mathcal{F}_S(G)=\mathcal{F}$.
\end{Theorem}
\subsection{Graphs of Groups}
\label{cAppendix}
We give a short introduction to graphs of groups stating results we need.
A finite directed graph $\Gamma$ consists of two sets, the vertices $V$ and the directed edges $E$, together with two functions $\iota, \tau : E\rightarrow V$. For $e \in E,\iota (e)$ is called the initial vertex of $e$ and $\tau (e)$ is the terminal vertex of $e$. Multiple edges and loops are allowed in this definition. The graph $\Gamma$ is connected if the only equivalence relation on $V$ that contains all $(\iota (e),\tau (e))$ is the relation with just one class.
A graph $\Gamma$ may be viewed as a category, with objects the disjoint union of $V$ and $E$ and two non-identity morphisms with domain $e$ for each $e\in E$, one morphism $e\rightarrow \iota (e)$ and one morphism $e\rightarrow \tau (e)$.
A graph $\Gamma$ of groups is a connected graph $\Gamma$ together with groups $G_v, G_e$ for each vertex and edge and injective group homomorphims $f_{e,\iota}:G_e\rightarrow G_{\iota (e)}$ and $f_{e,\tau (e)}:G_e\rightarrow G_{\tau (e)}$ for each edge $e$.
\section{A new family realizing saturated fusion systems}
We inroduce a new group model realizing saturated fusion systems related to the construction of G. Robinson.
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group and \begin{eqnarray*}
\mathcal{G}=L_1\underset{N_S(P_2)}{*}L_2\underset{N_S(P_3)}{*}...\underset{N_S(P_n)}*L_n
\end{eqnarray*}
a model of Robinson type for $\mathcal{F}$. For each of the $L_i$, $i=1, \cdots , n$ choose subgroups $K_1, \cdots , K_m$ of $L_i$ such that each $K_j$ contains (an isomorphic copy of) the group $N_S(P_i)$ and $K_1, \cdots , K_m$ generate the group $L_i$. Assume as we can that $S\in Syl_p(K_1)$ and after reindexing let $G$ be the iterated amalgam
\begin{eqnarray*}
G=K_1\underset{N_S(P_2)}{*}K_2\underset{N_S(P_3)}{*}...\underset{N_S(P_n)}*K_n.
\end{eqnarray*}
Then $G$ contains $S$ as a Sylow $p-$subgroup and $\mathcal{F}_S(G)=\mathcal{F}$.
\end{Theorem}
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group and let $\Phi =\{\phi _1,\cdots ,\phi _n\}$ be a subset of the set of automorphisms of the fusion system which is chosen in a minimal way, i. e. we cannot omit any element without obtaining a proper subsystem. Moreover assume as we can that all the elements of $\Phi$ have order coprime to $p$. Then the group
\begin{eqnarray*}
\mathcal{G}:=S*F(\Phi)/<\phi u\phi ^{-1}=\phi (u), \phi ^{deg (\phi)}=1>
\end{eqnarray*}
has the following properties. The group $S\in Syl_p(\mathcal{G})$, $\mathcal{F}_S(\mathcal{G})=\mathcal{F}$, the classifying space $B\mathcal{G}$ is $p-$good and the cohomology of $B\mathcal{G}$ is $F-$isomorphic in the sense of Quillen to the stable elements.
\end{Theorem}
\subsection{Homology Decompositions}
We investigate the cohomology of our models for a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S$. In the following $\mathcal{G}$ will always be a model for $\mathcal{F}$ of Robinson type, i. e.
$\mathcal{G}=L_1\underset{N_S(P_2)}{*}L_2\underset{N_S(P_3)}{*}...\underset{N_S(P_n)}*L_n$
where $L_1,...,L_n$, $P_1,...,P_n$ can be chosen such that $P_1,...,P_n$ are representatives of isomorphism classes of centric radicals, of $\mathcal{F}-$centrics or of essential subgroups of $\mathcal{F}$, and $L_1,..., L_n$ are the corresponding automorphism groups of $P_1, ..., P_n$ in the linking system if it exists. Note that these groups are known and are unique and do exist regardless of whether $\mathcal{L}$ exists or not, see \cite[Theorem 4.6.]{IntroMarkus} following the discussion in \cite{BCGLO1}, Section 4.
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group and $
\mathcal{G}$
a discrete group such that $S\in Syl_p(\mathcal{G})$ and $\mathcal{F}=\mathcal{F}_S(\mathcal{G})$ . Then
there exist a natural map of unstable algebras $H^*(B\mathcal{G})\overset{q}{\rightarrow} H^*(\mathcal{F})$ making $H^*(\mathcal{F})$ a module over $H^*(B\mathcal{G})$.
\end{Theorem}
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group and
$\mathcal{G}$
a model of our type for $\mathcal{F}$. Then
there exist natural maps of unstable algebras over the Steenrod algebra $H^*(BG)\overset{q}{\rightarrow} H^*(|\mathcal{L}|)$ and $H^*(|\mathcal{L}|)\overset{r}{\rightarrow} H^*(BG)$ such that we obtain a split short exact sequence of unstable modules over the Steenrod algebra $0\rightarrow H^*(|\mathcal{L}|)\overset{\overset{q}{\leftarrow}}{\underset{r}{\longrightarrow}}H^*(BG)\overset{\pi}{\rightarrow}W\rightarrow 0$ where $W\cong Ker(Res^{G}_S)$.
\end{Theorem}
\underline{Proof:}
Let $\mathcal{C}$ be the following category. \\
\xymatrix@R=1pt@C=3pt{
{\bullet _2}&&&&&&{\bullet _3}&&\\
&&{\bullet _{n+1}}\ar[1,2]\ar[-1,-2]&&&{\bullet _{n+2}}\ar[1,-1]\ar[-1,1]&&&\\
&&&&{\bullet _1}&&&&\\
&&{\bullet _{2n-1}}\ar[1,-2]\ar[-1,2]&&&&{\bullet _{n+3}}\ar[-1,-2]\ar[1,2]&&&\\
{\bullet _n}&&&{\bullet _{2n-2}}\ar[-2,1]\ar[1,-1]&&{\cdots}&&&{\bullet _4}&\\
&&{\bullet _{n-1}}&&&&&&&\\
}
Denote by $\phi _{i,j}:\bullet_{i}\rightarrow \bullet _j$ the unique morphism in $\mathcal{C}$ between $\bullet _i$ and $\bullet _j$ if it exists.
Let $F:\mathcal{C}\rightarrow Spaces$ be a functor with $F(\bullet _i)=BL_i\text{ for }i=1,...,n, BN_S(P_i)\text{ for }i=n+1,...,2n-1$ and $F(\phi _{i,j})=Bincl : F(\bullet _i)\rightarrow F(\bullet _j)$ for all $\phi _{i,j}:\bullet _i\rightarrow \bullet _j$ in $\mathcal{C}$, $i=n+1,...,2n-1$, $j=i-n+1,1$.
Note that $\underset{\mathcal{C}}{hocolim(F)}$ is a $K(G,1)$. Since $K_i\leq L_i=Aut_{\mathcal{L}}(P_i)$ for all $i=1,...,n$ we have a functor $\mathcal{B}K_i\rightarrow\mathcal{L}$ which sends the unique object $\bullet$ to $P_i$ and a morphism $x$ to the corresponding morphism in $Aut_{\mathcal{L}}(P_i)$ for all $i=1,...,n$. Therefore we obtain a map $BK_i$ to $|\mathcal{L}|$ for all $i=1,...,n$. Note that all the diagrams
\xymatrix@R=3pt@C=3pt{
&{BK_1}\ar[1,1]^{}&{}\\
{BN_S(P_i})\ar[-1,1]^{Bincl}\ar[1,1]_{Bincl}&&{|\mathcal{L}|}\\
&{BK_i}\ar[-1,1]_{}\restore&{}\\
}
commute up to homotopy since the third axiom from the definition of the linking system guarantees that we can find a compatible system of lifts of the inclusion $\iota _{N_S(P_i),S}$ in $\mathcal{L}$ for all $i=1,...,n$ such that all the diagrams
\xymatrix@R=3pt@C=3pt{
&{\mathcal{B}K_1}\ar[1,1]^{}&{}\\
{\mathcal{B}N_S(P_i})\ar[-1,1]^{\mathcal{B}incl}\ar[1,1]_{\mathcal{B}incl}&&{\mathcal{L}}\\
&{\mathcal{B}K_i}\ar[-1,1]_{}\restore&{}\\
}
commute up to the natural transformation
which takes the object $\bullet\in Obj(\mathcal{B}N_S(P_i))$ to $\iota _{N_S(P_i),S}$ for $i=1,...n.$
We obtain a map from the 1-skeleton of the homotopy colimit of the functor $F$ over the category $\mathcal{C}$ to $|\mathcal{L}|$. Since $\mathcal{C}$ is a $1-$dimensional category we obtain a map from $BG$ to $|\mathcal{L}|$. This map will be denoted by $q$ inducing $H^*(|\mathcal{L}|)\overset{q^*}{\rightarrow } H^*(BG)$.
Denote the kernel of the map $f$ by $W$. We have the following commutative diagram of unstable algebras over the Steenrod algebra where the maps $q^*$ and $incl$ are injective
\xymatrix@R=9pt@C=9pt{ {H^*(|\mathcal{L}|)}\ar[1,1]_{incl}\ar[0,1]^{q^*}_{\underset{f}{\longleftarrow}}&{H^*(BG)}\ar[1,0]^{Res^{\mathcal{G}}_S}\\
&{H^*(BS).}\\
}
Commutativity implies that $W\cong Ker(Res^{G}_S)$ in the category of unstable modules over the Steenrod algebra.$\Box$
\begin{Theorem}
Let $\mathcal{F}$ be an Alperin fusion system and $G$ a model of our type for it. Then $BG$ is $p-$good.
\end{Theorem}
\underline{Proof:}
The group $G$ is a finite amalgam of finite groups. Note that each $K_i$ is generated by $N_S(P_i)$ and elements of $p'-$order. Therefore $G$ is generated by elements of $p'-$order and $S$. Let $K$ be the subgroup of $G$ generated by all elements of $p'-$order. Note that $K$ is normal in $G$ and $S$ surjects on $G/K$ and therefore $G/K$ is a finite $p-$group. We have $H^1(BK;\mathbb{F}_p)=0$ and therefore $K$ is $p-$perfect. Let $X$ be the cover of $BG$ with fundamental group $K$. Then $X$ is $p-$good and $X\pcom$ is simply connected since $\pi _1(X)$ is $p-$perfect as follows from \cite[VII.3.2]{BK}. Hence $X\pcom\rightarrow BG\pcom\rightarrow B(G/K)$ is a fibration sequence and $BG\pcom$ is $p-$complete by \cite[II.5.2(iv)]{BK}. So $BG$ is $p-$good. $\Box$
\begin{Theorem}
Let $(S, \mathcal{F}, \mathcal{L})$ be a $p-$local finite group and $G$ be a model of our type for it. Then $H^*(BG)$ is finitely generated.
\end{Theorem}
\underline{Proof:}
Note that we have a map $BG=\underset{\mathcal{C}}{hocolim(F)}\rightarrow |\mathcal{L}|$ where $F$ and $\mathcal{C}$ are as defined in the proof of Theorem 4.3. for the model of our type $G$. Note that $N_S(P_i)\in Syl_p(K_i)$ for all $i=1,...,n$. It follows from \cite[Lemma 2.3.]{BLO1} and \cite[Theorem 4.4.(a)]{BLO2} that $H^*(B(P_i))$ is finitely generated over $H^*(|\mathcal{L}|)$ for all $i=1,...,n$, and $H^*(|\mathcal{L}|)$ is noetherian as follows from \cite[Proposition 1.1. and Theorem 5.8.]{BLO2}. Therefore the Bousfield-Kan spectral sequence for $H^*(BG)$ is a spectral sequence of finitely generated $H^*(|\mathcal{L}|)-$modules, the $E_2$ term with $E_2^{s,t}= \underset{\mathcal{C}}{lim^s}H^t(F(-);\mathbb{F}_p)$ is concentrated in the first two columns and $E_{2}=E_{\infty}$ for placement reasons. Therefore $H^*(BG)$ is a finitely generated module over $H^*(|\mathcal{L}|)$ and in particular noetherian. $\Box$
\subsubsection{A stable retract}
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group and $G$ a model of our type for $\mathcal{F}$. Then
$|\mathcal{L}|\pcom$ is a stable retract of $BG\pcom$.
\end{Theorem}
\underline{Proof:}
The diagram
\xymatrix@R=9pt@C=9pt{&{\Sigma ^{\infty}BS\pcom}\ar[1,1]^{\Sigma ^{\infty}Bincl\pcom}\ar[1,-1]_{\Sigma ^{\infty}B(\delta _S)\pcom}&{}\\
{\Sigma ^{\infty}|\mathcal{L}|\pcom}&&{\Sigma ^{\infty}BG\pcom.}\ar[0,-2]^{\Sigma ^{\infty}q\pcom}\\
}
commutes where $q$ is the map constructed in the proof of Theorem 4.1.
By the work of K. Ragnarsson \cite{Ragnarsson} there is a map $\sigma _{\mathcal{F}}:\Sigma ^{\infty}|\mathcal{L}|\pcom\rightarrow\Sigma ^{\infty}BS \pcom$ such that the composition of maps $ \Sigma ^{\infty}|\mathcal{L}|\pcom\overset{\sigma _{\mathcal{F}}}{\longrightarrow}\Sigma ^{\infty}BS\pcom\overset{\Sigma ^{\infty}B(\delta _S)\pcom}{\longrightarrow}\Sigma ^{\infty}|\mathcal{L}|\pcom$ is the identity.
Since $\Sigma ^{\infty}B(\delta _S)\pcom\circ \sigma _{\mathcal{F}} =\Sigma ^{\infty}q\pcom\circ\Sigma ^{\infty}Bincl\pcom\circ \sigma _{\mathcal{F}}$ we have $|\mathcal{L}|\pcom$ is a stable retract of $BG\pcom$.$\Box$
\section{Modules and Euler characteristic}
Let $\mathcal{F}$ be an Alperin fusion system with associated groups $L_1,\cdots ,L_r$ and associated subgroups $K_1,\cdots ,K_n$ as described above and let
\begin{eqnarray*}
G=K_1\underset{N_S(P_2)}{*}K_2\underset{N_S(P_3)}{*}K_3\cdots \underset{N_S(P_n)}{*}K_n
\end{eqnarray*}
the group model discussed so far. Then we can see inductively that given a group $M$ and group homomorphisms $\phi _i:K_i\rightarrow M$ for $i=1,\cdots ,n$ with
\begin{eqnarray*}
Res^{K_i}_{N_S(P_i)}(\phi _i)=Res^{S}_{N_S(P_i)}(\phi _1)
\end{eqnarray*}
for each $i$, there is a unique group homomorphism $\phi :G\rightarrow M$ which extends each $\phi _i$.\\
We now want to study the finite-dimensional $kG-$modules. Notice that if $H$ is a finite group generated by subgroups $M_i$ for $i=1,\cdots ,n$ such that for each $i$ there is a group epimorphism $\alpha _i:K_i\rightarrow M_i$ with $Res^{K_i}_{N_S(P_i)}(\alpha _i)=Res^{S}_{N_S(P_i)(\alpha _1)}$, then $H$ is an epimorphic image of $G$.\\
We allow the possibility that $U=O_p(G)\neq 1$. We remark that $U\leq E$, for each $i$, since
\begin{eqnarray*}
N_U(E_i)E_iC_G(E_i)/E_iC_G(E_i)\leq O_p(N_G(E_i))/E_iC_G(E_i)\cong O_p(Out_{\mathcal{F}}(E_i))=1.
\end{eqnarray*}
\begin{Theorem}
For each $i$, let $H_i$ be a $p'-$subgroup of $K_i$, and let $t=lcm\{[K_i:H_i]:1\leq i\leq r\}$. Then there is a group homomorphism $\phi :G\rightarrow S_t$ whose kernel is a free group.
\end{Theorem}
\underline{Proof:} It is enough to construct the homomorphism $\phi$ so that $ker\phi$ has trivial intersection with $S$. Let $\Omega =\{1,2,\cdots ,t\}$ and let each $K_i$ act as it would on the direct sum of $t/[K_i:H_i]$ copies of the permutation module of $K_i$ on the cosets of $H_i$. Since $S$ acts semi-regularly, and each $N_S(P_i)$ does, we may label the points so that each $N_S(P_i)$ acts in the manner determined by regarding it as a subgroup of $S$. By the remarks preceding the Theorem, $\Omega$ now has the structure of a $G-$set. Since the action of $P$ is free, the kernel of the action is a free normal subgroup of finite index. $\Box$
\begin{Theorem}
\begin{enumerate}
\item For $1\leq i\leq r$ let $X_i$ be a finite-dimensional projective $kK_i-$module, and suppose that all $X_i$ have equal dimension. Then there is a $kG-$module $X$ such that $Res^G_{K_i}(X)\cong X_i $ for each $i$. Furthermore, $C_G(X)$ is a free normal subgroup of $G$ of finite index.
\item For each $i$, let $V_i$ be a simple $kK_i-$module. Then there is a finite dimensional projective $kG-$module $V$ such that for each $i$, $soc(Res^G_{K_i}(V))$ is isomorphic to a direct sum of copies of $V_i$.
\end{enumerate}
\end{Theorem}
\underline{Proof:}\begin{enumerate}\item Since $X_i$ is a free $kN_S(P_i)-$module for each $i$ we can suppose that $N_S(P_i)$ has the same action on $X_i$ as it does on $Res^S_{N_S(P_i)}(X_1)$ for each $i$. In that case, there is a unique way to extend the action of the $K_i$ on the underlying $k-$vector space to an action of $G$ on that space. We let $X$ denote the $kG-$module so obtained. Then $X$ is a free $kS$-module, so that $C_G(X)\cap S=1$ and $C_G(X)$ is a free normal subgroup of $G$ of finite index.
\item Let $Y_i$ denote the projective cover of $V_i$ as $kK_i-$module, so that we also have $soc(Y_i)\cong V_i$. Let $V$ be a simple $kG-$module of $X$. Then $soc(Res^G_{K_i}(V))$ is a submodule of $soc(Res^G_{K_i}(X))$ for each $i$, so the result follows. $\Box$
\end{enumerate}
\begin{Theorem}
Let $\mathcal{F}$ be a saturated fusion system over the finite $p-$group $S$ and \begin{eqnarray*}
G=K_1\underset{N_S(P_2)}{*}K_2\underset{N_S(P_3)}{*}...\underset{N_S(P_n)}*K_n
\end{eqnarray*}
a model of our type for $\mathcal{F}$. Then we have the following formula
\begin{eqnarray*}
\chi (G)=(\underset{1\leq i\leq n}{\sum }{\frac{1}{|K_i|}})-(\underset{2\leq i\leq n}{\sum }{\frac{1}{|N_S(P_i)|}})
\end{eqnarray*}
for the Euler characteristic of $\chi (G)$ of $G$. Moreover we have
\begin{eqnarray*}
\chi (G)=\frac{d_{\mathcal{F}}}{|S|lcm\{K_i:N_S(P_i),1 \leq i\leq n\}}
\end{eqnarray*}
for some negative integer $d_{\mathcal{F}}$.
\end{Theorem}
\underline{Proof:}
\underline{Proof:}
\begin{Theorem}
\begin{enumerate}
\item Any free subgroup of finite index of $X$ has index divisible by
\item
\end{enumerate}
\end{Theorem}
\section{A Kuenneth Formula for Fusion Systems}
We prove an analogue of the Kuenneth Formula for saturated fusion systems independently of the existence of a classifying space.
\begin{Theorem}
Let $\mathcal{F}_1,\mathcal{F}_2$ be saturated fusion systems over the finite $p-$groups $S_1,S_2$ respectively. Then $H^*(\mathcal{F}_1\times\mathcal{F}_2)\cong H^*(\mathcal{F}_1)\otimes H^*(\mathcal{F}_2)$.
\end{Theorem}
The proof requires a lemma.
\begin{Lemma}
Let $\mathcal{F}_1,\mathcal{F}_2$ be saturated fusion systems over the finite $p-$groups $S_1,S_2$ respectively. Let $\mathcal{F}=\mathcal{F}_1\times \mathcal{F}_2$ be the saturated fusion system over $S=S_1\times S_2$. Then $P=P_1\times P_2$ is $\mathcal{F}-$centric if and only if $P_1$ is $\mathcal{F}_1-$centric and $P_2$ is $\mathcal{F}_2-$centric.
\end{Lemma}
\underline{Proof:}
Assume $P_1$ is $\mathcal{F}_1-$centric and $P_2$ is $\mathcal{F}_2-$centric. Then we have for every $P'=P'_1\times P'_2$ which is $\mathcal{F}-$conjugate to $P$ that $|C_S(P)|=|C_{S_1\times S_2}(P_1\times P_2)|=|C_{S_1}(P_1)|\cdot|C_{S_2}(P_2)|\geq |C_{S_1}(P'_1)|\cdot |C_{S_2}(P'_2)|=|C_S(P')|$. From the precedent inequality it can be seen that the converse holds as well.$\Box$\\[0.3cm]
\underline{Proof of the Theorem:} The following diagram commutes where the vertical isomorphisms are given through the Kuenneth Formula for topological spaces $H^*(P_1\times P_2;\mathbb{F}_p)\cong H^*(P_1;\mathbb{F}_p)\otimes H^*(P_2;\mathbb{F}_p)$ and $H^*(Q_1\times Q_2;\mathbb{F}_p)\cong H^*(Q_1;\mathbb{F}_p)\otimes H^*(Q_2;\mathbb{F}_p)$.\\
\xymatrix@R=7pt@C=10pt{
&{H^*(\mathcal{F}_1\times\mathcal{F}_2)}\ar@{=}[1,0]&{}&\\
&{\underset{\mathcal{O}^c(\mathcal{F}_1\times\mathcal{F}_2)}{lim}H^*(-)}\ar[1,1]\ar[1,-1]&{}&\\
{H^*(P_1\times P_2)}\ar[0,2]^{(\phi _1\times\phi _2)^*}&{}&{}{H^*(Q_1\times Q_2)}&\\
{\cong}&&{\cong}&\\
{H^*(P_1)\otimes H^*(P_2)}\ar[0,2]^{\phi _1^*\otimes\phi _2^*}&{}&{H^*(Q_1)\otimes H^*(Q_2)}&\\
&{(\underset{\mathcal{O}^c(\mathcal{F}_1)}{lim}H^*(-))\otimes (\underset{\mathcal{O}^c(\mathcal{F}_2)}{lim}H^*(-))}\ar[-1,1]\ar[-1,-1]\ar@{=}[1,0]&{}&\\
&{H^*(\mathcal{F}_1)\otimes H^*(\mathcal{F}_2)}&&\\
}\\[0.3cm]
Since there are no finiteness issues we obtain via the universal property of inverse limits that $H^*(\mathcal{F}_1\times\mathcal{F}_2)\cong H^*(\mathcal{F}_1\otimes\mathcal{F}_2)$ in the category of unstable algebras over the Steenrod algebra.
$\Box$\\[0.3cm]
The Kuenneth formula is natural in the following way.
\begin{Theorem}
Let $(S_1,\mathcal{F}_1,\mathcal{L}_1)$ and $(S_2,\mathcal{F}_2,\mathcal{L}_2)$ be two $p-$local finite groups respectively. Then $H^*(\mathcal{F}_1\times\mathcal{F}_2)\cong H^*(|\mathcal{L}_1\times \mathcal{L}_2|\pcom ;\mathbb{F}_p)\cong H^*(|\mathcal{L}_1|\pcom;\mathbb{F}_p)\otimes H^*(|\mathcal{L}_2|\pcom;\mathbb{F}_p)\cong H^*(\mathcal{F}_1\otimes\mathcal{F}_2)$.
\end{Theorem}
\section{Glaubermann's and Thompson's theorems for $p$-local finite groups}
In \cite{dgmp} Diaz, Glesser, Mazza and Park prove analogues of Glaubermann's and Thompson's theorems for fusion systems.
We extend their results to $p-$local finite groups and give an algebraic criterion for the classifying space of a $p-$local finite group to be equivalent to $BS$ before completion.
\begin{Theorem} Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group. Then
\begin{eqnarray*}
(Z(S))^p\cap Z(N_{\mathcal{F}}(J(S)))\cap Z(N_{\mathcal{L}}(J(S))) \leq Z(\mathcal{F}).
\end{eqnarray*}
If $p$ is odd or $\mathcal{F}$ is $S_4$-free, then $Z(\mathcal{F}) = Z(N_{\mathcal{F}}(J(S)))= Z(N_{\mathcal{
L}}(J(S)))$.
\end{Theorem}
\underline{Proof:} In \cite{dgmp} the authors show that for
$(S,\mathcal{F},\mathcal{L})$ a $p-$local finite group we have $(Z(S))^p\cap Z(N_{\mathcal{F}}(J(S))) \leq Z(\mathcal{F})$ and if $p$ is odd or $\mathcal{F}$ is $S_4$-free, then $Z(\mathcal{F}) = Z(N_{\mathcal{F}}(J(S)))$. The statement follows from the fact that for every $p-$local finite group $(S,\mathcal{F},\mathcal{L})$ we have an equality $Z(\mathcal{F})=Z(\mathcal{L})$. $\Box$
\begin{Theorem}
Let $(S,\mathcal{F},\mathcal{L})$ be a $p-$local finite group.
Assume that $p$ is odd or that $\mathcal{F}$ is
$S_4$-free. If $C_{\mathcal{F}}(Z(S)) = N_{\mathcal{F}}(J(S)) = \mathcal{F}_S(S)$, and $N_{\mathcal{L}}(J(S))=\mathcal{L}^c_S(S)$ then $\mathcal{F} = \mathcal{F}_S(S)$ and $|\mathcal{L}|\simeq BS$.
\end{Theorem}
\underline{Proof:}
\section{Acknowledgements}
The author was supported by ANR BLAN08-2-338236, HGRT, an Erwin-Schroedinger-Institute Junior-Research-Fellowship, and a Mathematisches-Forschungsinstitut-Oberwolfach Leibniz-Fellowship.
| 147,576
|
There's nothing sweeter than a dad talking about his little girl (or girls). That's why I love this story about Obama rubbing elbows with French president Nicolas Sarkozy at the G20 summit.
As a father of two daughters, Obama was happy to congratulate Sarkozy on the birth of his new baby girl, Giulia, with model Carla Bruni, and to offer his own, experienced take on parenting ...
"So now we share one of the greatest challenges and blessings in life, and that is being fathers to our daughters," he said.
Aw, sweet. Then came Obama's petit plaisanter regarding Sarkozy's, uh, good looks ...
“I want to congratulate Nicolas and Carla on the birth of Giulia. And I informed Nicolas on the way in that I was confident Giulia inherited her mother’s looks, rather than her father’s, which I think is an excellent thing.”
Mon Dieu, Obama is at his most handsome when he's being effortlessly witty. Ah, just another day in politics! We joke, we talk about beautiful women, we look sophisticated and sexy in our well-tailored suits. Hey, if our politicians can't be effective, they should at least be glamorous. Both Michelle and Carla would look lovely in pillbox hats and oversized sunglasses.
All that's missing is the Serge Gainsbourg soundtrack.
Is your daughter a daddy's girl?
Share this Story
| 358,051
|
TITLE: Converting formula into linear expression
QUESTION [0 upvotes]: I recently conducted an experiment aimed at calculating the Earth's magnetic field.
A compass was placed inside a coil and the apparatus was aligned east-west. When current was applied to the coil, the deflection of the compass was the vector addition of the Earth's magnetic field and the magnetic field created by the coil.
To negate the error, I took multiple measurements with constant current and varied the compass position in relation to the coil. Recording the deflection angle and distance from the coil.
I have the following formulas the relate the deflection angle to the distance $x$ from the coil:
$$ \tan \theta = \frac{Bc}{Be} $$
$$ Bc = \frac{ \mu NIR^2}{2(R^2+x^2)^(3/2)} $$
I have found articles on similar experiments and it's mentioned that these two formulas can be combined into a linear equation: $$ Y = mX + A$$
My guess is that $ Y = \theta $ and $ X = x $. This function could then be graphed and the slope would give a more accurate result of $Be$
However, there was no guidance on how to combine these formulas into the linear form. Why does this offer better accuracy? Can anyone point me in the right direction on this transformation?
REPLY [0 votes]: You must be very careful with such a problem because what you measure is $\theta$ as a function of $x$ and not any possible transform of $\theta$.
This means that you model is
$$\theta=\tan ^{-1}\left(\frac{k}{\left(R^2+x^2\right)^{3/2}}\right) \qquad \text{where} \quad k=\frac{ \mu NIR^2}{2Be}$$
So, in a preliminary step, let $Y=\tan(\theta)$, $X=\frac{1}{\left(R^2+x^2\right)^{3/2}}$ and you have a linear regression
$$Y=k X \implies k=\frac{\sum_{i=1}^n X_i\,Y_i}{\sum_{i=1}^n X^2_i}$$ but this is only an estimate. Now, you need to start a nonlinear regression to obtain the most probable value of $k$ and then the best $Be$.
If you have no access to such a tool, I could show how to do it with algebra.
| 192,904
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\section{Feasibility and Rate Region}\label{section:results}
We present single-letter characterizations of securely computable randomized functions and the rate regions. Detailed proofs \iftoggle{paper}{can be found in an extended version of this paper.}{can be found in Appendix~\ref{appendix:proofs_omitted}.} {\color{black}The following theorem is about feasibility. Part~$(i)$ states that asymptotic secure computability (in $r$ rounds) of a function implies one-shot (i.e., $n=1$) perfectly secure computability (in the same number of rounds). Part~$(ii)$ shows that asymptotic secure computability depends on the input distribution $q_{XY}$ only through its support, $\text{supp}(q_{XY}):=\{(x,y): q_{XY}(x,y)>0\}$. In fact, asymptotic secure computability of a function $(q_{XY},q_{Z_1Z_2|XY})$ is preserved even with another input distribution $\tilde{q}_{XY}$ whose support is a subset of $\text{supp}(q_{XY})$.}
\begin{theorem}
\label{Feasibility_AB}
(i) $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy against both users using an $r$-round protocol in which
Alice starts the communication if and only if there exists a conditional p.m.f. $p(u_{[1:r]}|x,y,z_1,z_2)$ satisfying
\begin{align}
& U_i-(U_{[1:i-1]},X)-Y, \emph{if}\ i\ \emph{is} \ \emph{odd},\label{Eq_AB_Markov1} \\
& U_i-(U_{[1:i-1]},Y)-X, \emph{if} \ i \ \emph{is} \ \emph{even},\label{Eq_AB_Markov2}\\
& Z_1-(U_{[1:r]},X)-(Y,Z_2),\label{Eq_AB_Markov_Decod1} \\
& Z_2-(U_{[1:r]},Y)-(X,Z_1), \label{Eq_AB_Markov_Decod2}\\
& U_{[1:r]}-(X,Z_1)-(Y,Z_2),\label{Eq_AB_Markov_Secr1}\\
&U_{[1:r]}-(Y,Z_2)-(X,Z_1),\label{Eq_AB_Markov_Secr2}
\end{align}
$|\mathcal{U}_1|\leq |\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|+1$ and
$|\mathcal{U}_i|\leq
|\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|
\prod_{j=1}^{i-1}|\mathcal{U}_j|+1
$, $\forall i>1$.
{\color{black}\noindent (ii) If a function $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy against both users using an $r$-round protocol, then $(\tilde{q}_{XY},q_{Z_1Z_2|XY})$, where $\emph{supp}(\tilde{q}_{XY})\subseteq \emph{supp}(q_{XY})$, is also asymptotically securely computable with privacy against both users using an $r$-round protocol.}
\end{theorem}
\begin{remark}\label{remark:feasibility}
Notice that Alice can generate common randomness by sending some {\color{black}of her private randomness} along with the message in the first round. So, the presence or absence of common randomness should not affect the secure
computability of a function $(q_{XY},q_{Z_1Z_2|XY})$. As expected, the condition in part~$(i)$ does not depend on common randomness.
\end{remark}
{\color{black}\begin{remark}\label{remark:eavesdropper}
Part~$(i)$ of Theorem~\ref{Feasibility_AB} shows that, for our problem, asymptotic secure computability is equivalent to one-shot perfect secure computability. It is interesting to note that this is not the case for all secure function computation problems. Consider the problem of function computation with privacy against an eavesdropper \cite{TyagiNG11}. Tyagi et al. \cite{TyagiNG11} considers the asymptotic setting where a group of users with correlated inputs interact noiselessly to compute a common function. The privacy requirement is that the amount of information that an eavesdropper learns about the function from the communication vanishes asymptotically. \cite[Theorem~2]{TyagiNG11} states that a function $g$ is asymptotically securely computable by two users with privacy against an eavesdropper if $H(g(X,Y))<I(X;Y)$ (and only if $H(g(X,Y))\leq I(X;Y)$). In this setup, perfectly secure computability with privacy against an eavesdropper can be defined analogous to the asymptotic secure computability with privacy against an eavesdropper. Below, we give an example of a function which is computable with asymptotic security (with privacy from an eavesdropper) but not with perfect security. Furthermore, unlike part $(ii)$ of Theorem~\ref{Feasibility_AB}, asymptotic secure computability with privacy against an eavesdropper depends on the input distribution $q_{XY}$ not just through its support $\text{supp}(q_{XY})$ \cite[Theorem~2]{TyagiNG11}.
\begin{example}
Consider a doubly symmetric binary source $\text{DSBS}(a)$ with joint distribution $q_{XY}(x,y)=0.5(1-a)\mathbbm{1}_{\{x=y\}}+0.5a\mathbbm{1}_{\{x\neq y\}}$, $a\in [0,0.5]$ and $x,y\in \{0,1\}$. Let the function to be computed by both users is $g(x,y)=x\oplus y$, where `$\oplus$' is addition modulo-2. Choose $a\in (0,0.5]$ s.t. $h(a)<1-h(a)$ (where $h(\cdot)$ denotes the binary entropy function), so that $g$ is asymptotically securely computable with privacy against the eavesdropper (by \cite[Theorem 2]{TyagiNG11}). We show that there does not exist a protocol that perfectly securely computes $g$ with privacy from an eavesdropper. If $g$ is perfectly securely computable with privacy against an eavesdropper, then there exists some $r$ and a conditional p.m.f. $p(u_{[1:r]}|x,y)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Decod2} with $Z_1=Z_2=G:=g(X,Y)$ (for correctness), and $I(G;U_{[1:r]})=0$ (for privacy against the eavesdropper). For simplicity, we write $U$ for $U_{[1:r]}$. Suppose there exists a conditional p.m.f. satisfying the above conditions. In particular, we have
\begin{align}
(X\oplus Y)-(U,X)-Y\label{eqn:eaves1},\\
(X\oplus Y)-(U,Y)-X\label{eqn:eaves2},\\
I(X\oplus Y;U)=0\label{eqn:eaves3}.
\end{align}
\eqref{eqn:eaves1} implies that $I(X\oplus Y;Y|U,X)=0$, which in turn implies that $H(Y|U,X)=0$ (i.e., $Y$ is a function of $(U,X)$) since $H(Y|U,X,X\oplus Y)=0$. Similarly, \eqref{eqn:eaves2} implies that $H(X|U,Y)=0$, i.e., $X$ is a function of $(U,Y)$. Now if $p(u,x,y)>0$, then we claim that $p(u)=p(u,x,y)+p(u,\bar{x},\bar{y})$ ($\bar{x}$ denotes the compliment of $x$, i.e., $\bar{x}=1-x$). To see this, if $p(u,x,y)>0$, note that $p(u,x,\bar{y})=0$ since $Y$ is a function of $(U,X)$. Similarly, $p(u,\bar{x},y)=0$ since $X$ is a function of $(U,Y)$. Hence, when $p(u)>0$, since there exists $x,y$ s.t. $p(u,x,y)>0$, we can write $p(u)=p(u,x,y)+p(u,\bar{x},\bar{y})$. Hence, $X\oplus Y$ is a function of $U$ as $x\oplus y=\bar{x}\oplus \bar{y}$, $\forall x,y\in\{0,1\}$. This is a contradiction to \eqref{eqn:eaves3} as $a\in (0,0.5]$. Therefore, $g$ is not perfectly securely computable with privacy against an eavesdropper.
\end{example}
\end{remark}
}
\begin{remark}\label{remark:equv}
\RRB{Let us call the functions that are asymptotically securely computable in $r$ rounds, for some $r>0$, as {\em asymptotically securely computable} functions. Note that part $(i)$ of Theorem~\ref{Feasibility_AB} does not give a computable characterization of asymptotically securely computable functions since the number of auxiliary random variables to consider is unbounded.} This problem, which was partially addressed in \cite{MajiPR12,DataP17} for full support input distributions, remains open.
\end{remark}
{\color{black}\begin{proof}[Proof sketch of Theorem~\ref{Feasibility_AB}]
We give a proof sketch here. A detailed proof can be found in Appendix~\ref{appendix:proofs_omitted}. For part $(i)$, it is trivial to see the `if' part since \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Secr2} define an $r$-round perfectly secure protocol of blocklength one, i.e., the protocol satisfies \eqref{eqn:asymptotic_1}-\eqref{eqn:asymptotic_3} with $n=1$ and $\epsilon=0$. For the `only if' part, we first single-letterize the privacy constraints \eqref{eqn:asymptotic_2} and \eqref{eqn:asymptotic_3}. We then single-letterize \eqref{eqn:asymptotic_1} and the Markov chains that are implied by the joint distribution in \eqref{eqn:asymptoticdist} along the lines of two-way source coding of Kaspi \cite{Kaspi85}, interactive (deterministic) function computation of Ma and Ishwar \cite{MaI11}, and channel simulation of Yassaee et al. \cite{YassaeeGA15}. Then by using the continuity of mutual information and total variation distance in the probability simplex, we show that, if a function is computable with asymptotic security, it is also computable with perfect security. For part $(ii)$, we show that a protocol which securely computes $(q_{XY},q_{Z_1Z_2|XY})$ will also securely compute the function $(\tilde{q}_{XY},q_{Z_1Z_2|XY})$, where $\text{supp}(\tilde{q}_{XY})\subseteq \text{supp}(q_{XY})$.
\end{proof}
}
{\color{black}Next theorem characterizes the rate region $\mathcal{R}^{AB-\text{pvt}}_A(r)$ }
\begin{theorem}\label{Thm_Rate_Region_AB}
If a function $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy against both users, then $\mathcal{R}^{AB-\emph{pvt}}_A(r)$ is given by the set of all non-negative rate triples $(R_0,R_{12},R_{21})$ such that
\begin{align}
R_{12} &\geq I(X;Z_2|Y),\label{eqn_results_Thm_Rate_Region_AB_1}\\
R_{21} &\geq I(Y;Z_1|X),\label{eqn_results_Thm_Rate_Region_AB_2}\\
R_{0} + R_{12} &\geq I(X;Z_2|Y) + I(U_1;Z_1,Z_2|X,Y),\label{eqn_results_Thm_Rate_Region_AB_3}\\
R_{0} + R_{12} + R_{21} &\geq I(X;Z_2|Y) + I(Y;Z_1|X) + \nonumber\\
&\hspace{12pt}I(Z_1;Z_2|X,Y)\label{eqn_results_Thm_Rate_Region_AB_4},
\end{align}
for some conditional p.m.f. $p(u_{[1:r]}|x,y,z_1,z_2)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Secr2}, $|\mathcal{U}_1|\leq |\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|+5$ and
$
|\mathcal{U}_i|\leq
|\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|
\prod_{j=1}^{i-1}|\mathcal{U}_j|+4
$, $\forall i>1$.
\end{theorem}
\begin{remark}
Inequality \eqref{eqn_results_Thm_Rate_Region_AB_3} on $R_0+R_{12}$ makes the rate region $\mathcal{R}^{AB-\text{pvt}}_A(r)$ possibly asymmetric. This is, in fact, due to the assumption that Alice starts the communication.
This is similar to the possible asymmetry of the rate region observed in channel simulation \cite[Theorem 1]{YassaeeGA15}.
\end{remark}
\begin{remark}\label{remark:wang}
Substituting $X=Y=\emptyset$ in part~$(i)$ of Theorem~\ref{Feasibility_AB} recovers a result of \cite{WangI11} which states that a distribution $q_{Z_1,Z_2}$ is securely computable (i.e., securely sampleable as there are no inputs here) if and only if $C(Z_1;Z_2)=I(Z_1;Z_2)$, where $C(Z_1;Z_2):=\underset{Z_1-W-Z_2}{\min}I(Z_1,Z_2;W)$ is Wyner common information \cite{Wyner75}. To see this, note that $C(Z_1;Z_2)=I(Z_1;Z_2)+\underset{Z_1-W-Z_2}{\min}\left(I(Z_1;W|Z_2)+I(Z_2;W|Z_1)\right).$ Furthermore, when $R_0=0$, \iftoggle{paper}{it can be shown using part $(ii)$ of Theorem~\ref{Thm_Rate_Region_AB} and \eqref{eqn:results:simplification3} that the optimal sum-rate is $R_{12}+R_{21}=C(Z_1;Z_2)=I(Z_1;Z_2)$.}{Theorem~\ref{Thm_Rate_Region_AB} implies that the optimal sum-rate is $R_{12}+R_{21}=C(Z_1;Z_2)=I(Z_1;Z_2)$. {\color{black}This follows from \eqref{eqn:results:simplification3} (proved later) and the fact that $I(U_1; Z_1,Z_2|X,Y)\leq I(U_{[1:r]}; Z_1,Z_2|X,Y)$.}}
\end{remark}
{\color{black}\begin{remark}
For a function $(q_{XY},q_{Z_1Z_2|XY})$, which is asymptotically securely computable with privacy against both users using a one round protocol in which Alice starts the communication, \eqref{eqn_results_Thm_Rate_Region_AB_2} purports to give a lower bound on the rate of communication from Bob to Alice. However, note that this lower bound $I(Y;Z_1|X)$ is in fact zero. To see this, notice that if a function $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy against both users using a 1-round protocol in which Alice starts the communication, it follows from $U-X-Y$ and \eqref{eqn:results:simplification2} (proved later) that $I(Y;Z_1|X)=0$.
\end{remark}
}
{\color{black}\begin{proof}[Proof sketch of Theorem~\ref{Thm_Rate_Region_AB}]
We give a proof sketch here. A detailed proof can be found in Appendix~\ref{appendix:proofs_omitted}. Our proof of achievability is along similar lines as the achievability proof of channel simulation \cite[Theorem~1]{YassaeeGA15}. We modify this proof to give a protocol which also accounts for privacy. \RRB{Specifically, we show how the Markov chains \eqref{Eq_AB_Markov_Secr1} and \eqref{Eq_AB_Markov_Secr2} can be turned into privacy constraints~\eqref{eqn:asymptotic_2} and \eqref{eqn:asymptotic_3} retaining the correctness~\eqref{eqn:asymptotic_1}}. For the converse, we first single-letterize the privacy constraints \eqref{eqn:asymptotic_2} and \eqref{eqn:asymptotic_3}. The rest of the converse is in the spirit of two-way source coding of Kaspi \cite{Kaspi85}, interactive (deterministic) function computation of Ma and Ishwar \cite{MaI11}, and channel simulation of Yassaee et al. \cite{YassaeeGA15}. \RRB{Note that such a single-letterization of privacy constraints preserving the correctness may not always be possible for any secure function computation problem (see, e.g., Remark~\ref{remark:eavesdropper}).} This gives a rate region defined by the set of non-negative rate triples $(R_0,R_{12},R_{21})$ such that
\begin{align}
R_{12} &\geq I(X;U_{[1:r]}|Y),\label{eqn:ach1}\\
R_{21} &\geq I(Y;U_{[1:r]}|X),\label{eqn:ach2}\\
R_{0} + R_{12} &\geq I(X;U_{[1:r]}|Y) + I(U_1;Z_1,Z_2|X,Y),\label{eqn:ach3}\\
R_{0} + R_{12} + R_{21} &\geq I(X;U_{[1:r]}|Y) + I(Y;U_{[1:r]}|X)\nonumber\\
&\hspace{12pt} + I(U_{[1:r]}; Z_1,Z_2|X,Y)\label{eqn:ach4},
\end{align}
for conditional p.m.f. $p(u_{[1:r]},z_1,z_2|x,y)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Secr2}.
Notice that constraints \eqref{eqn:ach1}-\eqref{eqn:ach4} appear
in channel simulation \cite[Theorem 1]{YassaeeGA15} also, where the conditional p.m.f. $p(u_{[1:r]},z_1,z_2|x,y)$ satisfies \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Decod2}. The above region reduces to the form mentioned in Theorem~\ref{Thm_Rate_Region_AB} because of a simplification
possible here due to the additional privacy constraints \eqref{Eq_AB_Markov_Secr1}-\eqref{Eq_AB_Markov_Secr2}, which gives us (as shown in \iftoggle{paper}{the Appendix}{the detailed proof of Theorem \ref{Thm_Rate_Region_AB} in Appendix~\ref{appendix:proofs_omitted}})
\begin{align}
I(X;U_{[1:r]}|Y)&=I(X;Z_2|Y),\label{eqn:results:simplification1}\\
I(Y;U_{[1:r]}|X)&=I(Y;Z_1|X),\label{eqn:results:simplification2}\\
I(U_{[1:r]}; Z_1,Z_2|X,Y)&=I(Z_1;Z_2|X,Y)\label{eqn:results:simplification3}.
\end{align}
\end{proof}
}
\begin{remark}
\RRB{Note that the lower bounds in \eqref{eqn_results_Thm_Rate_Region_AB_1} and \eqref{eqn_results_Thm_Rate_Region_AB_2} are in fact the \emph{cut-set} lower bounds for {\em non-private} computation {(see Appendix~\ref{cutset_discussion} for details)}. Thus, Theorem~\ref{Thm_Rate_Region_AB} implies that for sufficiently large common randomness rates $R_0$, the cut-set bounds are met for securely computable functions. The intuition is as follows: from \eqref{eqn:ach1}-\eqref{eqn:ach4} in the proof of Theorem~\ref{Thm_Rate_Region_AB}, $R_{12}=I(X;U_{[1:r]}|Y)$ and $R_{21}=I(Y;U_{[1:r]}|X)$ are in the rate-region for sufficiently large $R_0$. Note that $I(X;U_{[1:r]}|Y)$ measures the rate of information Bob learns about Alice's input during the protocol. When privacy against Bob is required, the rate of information that Bob learns about Alice's input during the protocol must be equal to the rate of information about Alice's input that can be inferred just from his output, i.e., $I(X;U_{[1:r]}|Y)=I(X;Z_2|Y)$. To see this, first note that $I(X;U_{[1:r]}|Y)=I(X;U_{[1:r]},Z_2|Y)=I(X;Z_2|Y)+I(X;U_{[1:r]}|Y,Z_2)$, where the first equality follows from the correctness. Now privacy against Bob implies that $I(X,Z_1;U_{[1:r]}|Y,Z_2)=0$, which in turn implies that $I(X;U_{[1:r]}|Y,Z_2)=0$. Similarly, when privacy against Alice is required, we have $I(Y;U_{[1:r]}|X)=I(Y;Z_1|X)$.}
\end{remark}
Let us denote the minimum number of rounds required for secure computation by $r_{\text{min}}$, i.e., the smallest $r$ such that there exists auxiliary random variables $U_{[1:r]}$ which makes the function $(q_{XY},q_{Z_1Z_2|XY})$ one-shot perfectly securely computable with either Alice or Bob starting the communication. \RRB{For deterministic functions, it is known that $r_{\text{min}}<2\min\{|\mathcal{X}|,|\mathcal{Y}|\}$~\cite{Kushelvitz92}. No such bound is available for randomized functions in general~\cite{MajiPR12,DataP17}.} Note that Theorem \ref{Thm_Rate_Region_AB} is for any fixed number of rounds $r$. \RRB{As we remarked above, for securely computable functions, the cut-set lower bounds for non-private computation holds with equality for sufficiently large common randomness rate for every $r\geq r_{\text{min}}$. The following corollary shows that the optimal trade-off between the communication and common randomness rates is achieved in at most $r_{\text{min}}+1$ rounds.}
Notice that the expression for $\mathcal{R}^{AB-\text{pvt}}$ below does not involve any auxiliary random variables.
\begin{corollary}\label{corollary_1}
If $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy against both users,
then $\mathcal{R}^{AB-\emph{pvt}}$ is given by the set of all non-negative rate triples
$(R_0,R_{12},R_{21})$ such that
\begin{align}
R_{12} &\geq I(X;Z_2|Y),\label{eqn:optimal_region_1}\\
R_{21} &\geq I(Y;Z_1|X),\label{eqn:optimal_region_2}\\
R_{0} + R_{12} + R_{21} &\geq I(X;Z_2|Y) + I(Y;Z_1|X)\nonumber\\
&\hspace{12pt} + I(Z_1;Z_2|X,Y)\label{eqn:optimal_region_3}.
\end{align}
Furthermore, $\mathcal{R}^{AB-\emph{pvt}}(r_{\emph{min}}+1)=\mathcal{R}^{AB-\emph{pvt}}$.
\end{corollary}
\begin{proof}[\textbf{Proof of Corollary \ref{corollary_1}}]
\RRB{It suffices to prove that $\mathcal{R}^{AB-\text{pvt}}(r_{\text{min}}+1)=\mathcal{R}_{\text{opt}}$, where $\mathcal{R}_{\text{opt}}$ is defined to be the set of all non-negative rate triples $(R_0,R_{12},R_{21})$ such that \eqref{eqn:optimal_region_1}-\eqref{eqn:optimal_region_3} are satisfied. From Theorem~\ref{Thm_Rate_Region_AB} it is easy to see that $\mathcal{R}^{AB-\text{pvt}}(r_{\text{min}}+1)\subseteq\mathcal{R}_{\text{opt}}$. For the other direction, take a point $(R_0,R_{12},R_{21})\in\mathcal{R}_{\text{opt}}$. Without loss of generality, suppose that $r_{\text{min}}$ occurs when Alice starts the communication. Then, by Theorem~\ref{Thm_Rate_Region_AB}, there exists random variables $U_{[1:r_{\text{min}}]}$ with conditional p.m.f. $p(u_{[1:r_{\text{min}}]}|x,y,z_1,z_2)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Secr2}. We find new random variables $U^\prime_{[1:r_{\text{min}}+1]}$ so that $(R_0,R_{12},R_{21})$ becomes a point in $\mathcal{R}_B^{AB-\text{pvt}}(r_{\text{min}}+1)$. Define $U_1^\prime=\emptyset$ and $U_i^\prime=U_{i-1}$ for $i>1$. This gives us that $(R_0,R_{12},R_{21})\in\mathcal{R}_B^{AB-\text{pvt}}(r_{\text{min}}+1)$. Hence $(R_0,R_{12},R_{21})\in\mathcal{R}^{AB-\text{pvt}}(r_{\text{min}}+1)$.}
\end{proof}
As mentioned earlier, if a function is securely computable with privacy against both users, then the cut-set lower bounds (for function computation without any privacy requirement) are achievable. The converse is not true in general\footnote{\color{black}To see this, suppose $X=Y=\emptyset$, then any function $q_{Z_1Z_2}$ can be computed by using common randomness alone (see Wyner common information problem \cite{Wyner75}), i.e., by meeting the cut-set lower bounds which in this case are zero, $I(X;Z_2|Y)=0=I(Y;Z_1|X)$. Assume that $q_{Z_1Z_2}$ is such that $C(Z_1;Z_2)\neq I(Z_1;Z_2)$. Then $q_{Z_1Z_2}$ is not securely computable in view of Remark~\ref{remark:wang}.}. However, part $(ii)$ of the theorem below states that a converse holds for a class of functions including deterministic functions\footnote{\color{black}The characterization of all the functions for which the converse holds remains open.}.
Let the rate region $\mathcal{R}_A^{\text{No-privacy}}(r)$ be defined analogous to $\mathcal{R}^{AB-\text{pvt}}_A(r)$ (except that only correctness condition~\eqref{eqn:asymptotic_1} is required).
\begin{theorem}\label{cutset}
{\color{black} (i) If a function $(q_{XY},q_{Z_1Z_2|XY})$ is securely computable in $r$ rounds with privacy against both users, then there exists common randomness rate $R_0$ such that $\big(R_0,I(X;Z_2|Y),I(Y;Z_1|X)\big)\in \mathcal{R}_A^{\emph{No-privacy}}(r)$. Furthermore, a rate of $R_0=I(Z_1;Z_2|X,Y)$ suffices for this.
\noindent (ii) Suppose the function $(q_{XY},q_{Z_1Z_2|XY})$ is such that $H(Z_1|X,Y,Z_2)=0$ \RRB{and} $H(Z_2|X,Y,Z_1)=0$ (e.g., a deterministic function). If there exists $R_0$ such that $\big(R_0,I(X;Z_2|Y),I(Y;Z_1|X)\big)\in \mathcal{R}_A^{\emph{No-privacy}}(r)$, then the function is securely computable in $r$ rounds with privacy against both users.}
\end{theorem}
We prove Theorem~\ref{cutset} in \iftoggle{paper}{the Appendix}{Appendix~\ref{appendix:proofs_omitted}}. {\color{black}Part $(i)$ will follow from Theorem~\ref{Thm_Rate_Region_AB}. We prove part $(ii)$ by showing that, for the class of functions mentioned in Theorem~\ref{cutset}}, any protocol for computation without privacy that meets the cut-set bounds must satisfy the privacy conditions as well.
\subsection*{When privacy is required against only one user:}
{\color{black} Note that when privacy is required only against Alice, any function $(q_{XY},q_{Z_1Z_2|XY})$ can be securely computed using a 2-round protocol in which Alice starts the communication.}
Alice can transmit her input to Bob who can compute the function
according to $q_{Z_1Z_2|XY}$, and send $Z_1$ back to Alice. Part $(i)$ of the following theorem considers the feasibility of $1$ round protocols. Similar to part $(i)$ of Theorem~\ref{Thm_Rate_Region_AB}, it states that asymptotic secure computability implies one-shot perfectly secure computability. Part $(ii)$ characterizes the rate region for an arbitrary number of rounds $r$.
\begin{theorem}
\label{Thm_Rate_Region_A}
(i) $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy only against Alice using a 1-round protocol in which Alice starts the communication if and only if there exists
a conditional p.m.f. $p(u_1|x,y,z_1,z_2)$ satisfying $(a)$ $U_1-X-Y$, $(b)$ $Z_1-(U_1,X)-(Y,Z_2)$, $(c)$ $Z_2-(U_1,Y)-(X,Z_1)$, $(d)$~$U_1-(X,Z_1)-(Y,Z_2)$. {\color{black}Furthermore, if a function $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy only against Alice using a 1-round protocol, then $(\tilde{q}_{XY},q_{Z_1Z_2|XY})$, where $\emph{supp}(\tilde{q}_{XY})\subseteq \emph{supp}(q_{XY})$, is also asymptotically securely computable with privacy only against Alice using a 1-round protocol. }
\noindent(ii) $\mathcal{R}^{A-\emph{pvt}}_A(r)$ is given by the set of all non-negative rate triples $(R_0,R_{12},R_{21})$ such that
\begin{align*}
R_{12} &\geq I(X;U_{[1:r]}|Y),\\
R_{21} &\geq I(Y;Z_1|X),\\
R_{0} + R_{12} &\geq I(X;U_{[1:r]}|Y) + I(U_1;Z_1|X,Y),\\
R_{0} + R_{12} + R_{21} &\geq I(X;U_{[1:r]}|Y) + I(Y;Z_1|X) \nonumber\\
&\hspace{12pt}+ I(U_{[1:r]}; Z_1|X,Y),
\end{align*}
for some conditional p.m.f. $p(u_{[1:r]}|x,y,z_1,z_2)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Decod2}, \eqref{Eq_AB_Markov_Secr1}, \RRB{$|\mathcal{U}_1|\leq |\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|+4$ and
$
|\mathcal{U}_i|\leq
|\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|
\prod_{j=1}^{i-1}|\mathcal{U}_j|+3
$, $\forall i>1$.}
\end{theorem}
\iftoggle{paper}
{ Note that similar cardinality bounds on auxiliary random variables as in Theorem~\ref{Thm_Rate_Region_AB} and similar statements as in Remark~\ref{remark:feasibility} hold true for Theorem~\ref{Thm_Rate_Region_A} also. A theorem similar to Theorem~\ref{Thm_Rate_Region_A} holds for the case when privacy is required only against Bob and it can be found in the extended version.}
{When privacy is required only against Bob, any {\color{black}function} $(q_{XY},q_{Z_1Z_2|XY})$ is securely computable in at most $3$ rounds with Alice starting the communication. To see this, note that Alice may transmit nothing in the first round, Bob can transmit his input to Alice in the second round. {\color{black}She} can {\color{black}then} compute the function
according to $q_{Z_1Z_2|XY}$, and send $Z_2$ back to Bob in the third round. Part $(i)$ of the following theorem considers the feasibility of $1$ and $2$ round protocols. Similar to part $(i)$ of Theorems~\ref{Thm_Rate_Region_AB} and \ref{Thm_Rate_Region_A}, it states that asymptotic secure computability implies perfectly secure computability. Part $(ii)$ characterizes the rate region for an arbitrary number of rounds $r$.
\begin{theorem}
\label{Thm_Rate_Region_B}
(i) $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy only against Bob using an $r$-round protocol in which
Alice starts the communication if and only if there exists a conditional p.m.f. $p(u_{[1:r]}|x,y,z_1,z_2)$
satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Decod2} and \eqref{Eq_AB_Markov_Secr2}, for $r=1,2$. {\color{black}Furthermore, if a function $(q_{XY},q_{Z_1Z_2|XY})$ is asymptotically securely computable with privacy only against Bob using a 1(2, resp.)-round protocol, then $(\tilde{q}_{XY},q_{Z_1Z_2|XY})$, where $\emph{supp}(\tilde{q}_{XY})\subseteq \emph{supp}(q_{XY})$, is also asymptotically securely computable with privacy only against Bob using a 1(2, resp.)-round protocol. }
\noindent(ii) $\mathcal{R}^{B-\emph{pvt}}_A(r)$ is given by the set of all non-negative rate triples $(R_0,R_{12},R_{21})$ such that
\begin{align*}
R_{12} &\geq I(X;Z_2|Y),\\
R_{21} &\geq I(Y;U_{[1:r]}|X),\\
R_{0} + R_{12} &\geq I(X;Z_2|Y) + I(U_1;Z_2|X,Y),\\
R_{0} + R_{12} + R_{21} &\geq I(X;Z_2|Y) + I(Y;U_{[1:r]}|X)\nonumber\\
&\hspace{12pt} + I(U_{[1:r]}; Z_2|X,Y),
\end{align*}
for some conditional pmf $p(u_{[1:r]}|x,y,z_1,z_2)$ satisfying \eqref{Eq_AB_Markov1}-\eqref{Eq_AB_Markov_Decod2}, \eqref{Eq_AB_Markov_Secr2}, \RRB{$|\mathcal{U}_1|\leq |\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|+4$ and
$
|\mathcal{U}_i|\leq
|\mathcal{X}||\mathcal{Y}||\mathcal{Z}_1||\mathcal{Z}_2|
\prod_{j=1}^{i-1}|\mathcal{U}_j|+3
$, $\forall i>1$.}
\end{theorem}
}
| 94,253
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Henry, Marcia K. – Annals of Dyslexia, 1988
A discussion-oriented, direct approach to teaching decoding and spelling based on word origin and structure is proposed. The instruction leads students to a comparison and contrast of letter-sound correspondences, syllable patterns, and morpheme patterns in English words of Anglo-Saxon, Romance, and Greek origin. (Author/JDD)
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\begin{document}
\title {Fluctuation spectra of weakly driven nonlinear systems}
\author{Yaxing Zhang$^{1}$, Yukihiro Tadokoro$^{2}$, and M. I. Dykman$^{1}$}
\address{$^{1}$Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA}
\address{$^2$Toyota Central R\&D Labs., Inc., Nagakute, Aichi 480-1192, Japan}
\date{\today}
\begin{abstract}
We show that in periodically driven systems, along with the delta-peak at the driving frequency, the spectral density of fluctuations displays extra features. These can be peaks or dips with height quadratic in the driving amplitude, for weak driving. For systems where inertial effects can be disregarded, the peaks/dips are generally located at zero frequency and at the driving frequency.
The shape and intensity of the spectra very sensitively depend on the parameters of the system dynamics. To illustrate this sensitivity and the generality of the effect, we study three types of systems: an overdamped Brownian particle (e.g., an optically trapped particle), a two-state system that switches between the states at random, and a noisy threshold detector. The analytical results are in excellent agreement with numerical simulations.
\end{abstract}
\section{Introduction}
Fluctuation spectra and spectra of response to periodic driving are major tools of characterizing physical systems. The spectra are conventionally used to find system frequencies and relaxation rates and to characterize fluctuations in the system. For example, optical absorption spectra give the transition frequencies of atomic systems and the lifetimes of the excited states, and the spectrum of spontaneous radiation is a well-known example of the fluctuation (power) spectrum \cite{Mandel1995}. In macroscopic systems the spectra are often complicated by the effects of inhomogeneous broadening. Recent progress in nanoscience has made it possible to study the spectra of individual dynamical systems. A well-known example is provided by optically trapped Brownian particles and biomolecules \cite{Ashkin1997,Greenleaf2007}, where the power spectra are a major tool for characterizing the motion in the trap \cite{Berg-Sorensen2004,Mas2013}. Spectra of various types of individually accessible mesoscopic systems are studied nowadays in optics \cite{Walls2008,Carmichael2008}, nanomechanics and circuit quantum electrodynamics, cf.~\cite{Dykman2012b}, biophysics, cf.~\cite{Jaramillo1998,Bialek2012}, and many other areas; the technique based on spectral measurements has found various applications, photonic force microscopy being a recent example, see Ref.~\cite{Marago2013}.
A familiar effect of weak periodic driving is forced vibrations of the system. When ensemble-averaged, they are also periodic and occur at the driving frequency $\omega_F$. They lead to a $\delta$-shape peak at frequency $\omega_F$ in the system power spectrum. However, the driving also modifies the power spectrum away from $\omega_F$. A textbook example is inelastic light scattering and resonance fluorescence. In the both cases, the system driven by a periodic electromagnetic field emits radiation at frequencies that differ from the driving frequency \cite{Heitler2010}. This radiation is one of the major sources of information about the system in optical experiments.
In this paper we study the spectra of periodically driven nonlinear systems. We show that, in the presence of noise, along with the $\delta$-shape peak at the driving frequency $\omega_F$, these spectra display a characteristic structure. We are interested in the regime of relatively weak driving, where the driving-induced change of the power spectrum is quadratic in the amplitude of the driving, as in inelastic light scattering.
In view of the significant interest in the power spectra of systems optically trapped in fluids, we consider systems where inertial effects play no role. In the absence of driving the power spectra of such systems usually have a peak at zero frequency. In particular it is this peak that is used to characterize the dynamics of optically trapped particles.
For a linear system, like a Brownian particle in a harmonic trap, the $\delta$-shape peak at $\omega_F$ is the only effect of the driving on the power spectrum. This is because motion of such a system is a linear superposition of forced vibrations at $\omega_F$ and fluctuations in the absence of driving. The amplitude and phase of the forced vibrations depend on the parameters of the system and determine the standard linear susceptibility \cite{LL_statphys1}. In nonlinear systems forced vibrations become random, because the parameters of the system are fluctuating. The power spectrum of such random vibrations is no longer just a $\delta$-shape peak (although the $\delta$-shape peak is necessarily present). The driving-induced spectral features away from $\omega_F$ result from mixing of fluctuations and forced vibrations in a nonlinear system. We note the close relation of these features to inelastic light scattering, see \ref{App_susceptibility}.
\subsection{Qualitative picture}
The idea of the driving-induced change of the power spectrum can be gained by looking at a Brownian particle fluctuating in a confining potential, a typical situation for optical trapping. The motion of the particle, after proper rescaling of time and particle coordinate $q$, is described by the Langevin equation \cite{Langevin1908}
\begin{equation}
\label{eq:Brownian}
\dot q = - U'(q) + f(t), \qquad U'(q)\equiv dU/dq,
\end{equation}
where $U(q)$ is the scaled potential and $f(t)$ is thermal noise. If potential $U(q)$ is parabolic and the system is additionally driven by a force $F\cos\omega_F t$, forced vibrations are described by the textbook expression
\begin{equation}
\label{eq:trivial_susceptibility}
\langle q(t)\rangle = \frac{1}{2}F\chi(\omega_F)\exp(-i\omega_Ft) + {\rm c.c.}, \qquad \chi(\omega)=[U''(q_{\rm eq})-i\omega]^{-1},
\end{equation}
where $q_{\rm eq}$ is the equilibrium position [the minimum of $U(q)$] and $\chi(\omega)$ is the susceptibility.
For a nonlinear system the potential $U(q)$ is nonparabolic. Because of thermal fluctuations, the local curvature of the potential $U''(q)$ is fluctuating. Intuitively, one can think of the effect of thermal fluctuations on forced vibrations as if $U''(q_{\rm eq})$ in Eq.~(\ref{eq:trivial_susceptibility}) for the susceptibility were replaced by a fluctuating curvature, see Fig.~\ref{fig:fluct_potential}. If the driving frequency $\omega_F$ largely exceeds the reciprocal correlation time of the fluctuations $t_c^{-1}$, the fluctuations would lead to the onset of a structure in the power spectrum near frequency $\omega_F$ with typical width $t_c^{-1}$. The quantity $t_c^{-1}$ also gives the typical width of the peak in the power spectrum at zero frequency in the absence of driving [for a linear system, $t_c^{-1}=U''(q_{\rm eq})$].
\begin{figure}[h]
\begin{center}
\includegraphics[width=6cm]{fig1.eps}
\caption{Sketch of a potential of a nonlinear system near the potential minimum. Because of the interplay of nonlinearity and fluctuations, the curvature of the potential fluctuates. These fluctuations are shown as the smearing of the solid line, which represents the potential in the absence of fluctuations.}
\label{fig:fluct_potential}
\end{center}
\end{figure}
Another effect of the interplay of driving, nonlinearity, and fluctuations can be understood by noticing that the periodic force causes a periodic change in the system coordinate. For a nonlinear system, roughly speaking, this leads to a periodic modulation of the local curvature, and thus of $t_c^{-1}$. Since $t_c^{-1}$ determines the shape of the zero-frequency peak in the power spectrum, such modulation causes a change of this peak proportional to $F^2$, to the lowest order in $F$.
Even from the above simplistic description it is clear that the driving-induced change of the spectrum is sensitive to the parameters of the system and the noise and to the nonlinearity mechanisms. Explicit examples given below demonstrate this sensitivity and suggest that the effects we discuss can be used for characterizing a system beyond the conventional linear analysis. After formulating how the power spectrum can be evaluated in Section~\ref{sec:general}, we demonstrate the effects of the interplay of driving and fluctuations for three very different types of nonlinear systems: an overdamped Brownian particle (Section~\ref{sec:Brownian_particle}), a system that switches at random between coexisting stable states (Section~\ref{sec:two_state}), and a threshold detector (Section~\ref{sec:threshold}). All these systems are of broad interest, and all of them display a significant driving-induced change of the power spectrum.
\section{General formulation}
\label{sec:general}
We consider fluctuating systems driven by a periodic force $F\cos(\wF t)$ and assume that fluctuations are induced by a stationary noise, like in the case of an optically trapped Brownian particle, for example. After a transient time such system reaches a stationary state. The stationary probability distribution of the system with respect to its dynamical variable $q$, $\rho_{\rm st}(q,t)$, is periodic in time $t$ with the driving period $\tau_F=2\pi/\omega_F$. The two-time correlation function $\la q(t_1)q(t_2)\ra$ [$\la\cdot\ra$ implies ensemble averaging] is a function of $t_1-t_2$ and a periodic function of $t_2$ with period $\tau_F$. The power spectrum usually measured in experiment is of the form
\begin{eqnarray}
\label{eq:power_spectrum}
\Phi(\omega)&=2 \Re \int_0^\infty dt e^{i\omega t}\lal q(t+t')q(t')\rar,\nonumber\\
\lal q(t&+t')q(t')\rar = \frac{1}{\tau_F}\int_0^{\tau_F} dt'\la q(t+t')q(t')\ra.
\end{eqnarray}
The correlation function in Eq.~(\ref{eq:power_spectrum}) can be expressed in terms of $\rho_{\rm st}(q,t)$ and the transition probability density $\rho(q_1,t_1|q_2,t_2)$ that the system that was at position $q_2$ at time $t_2$ is at $q_1$ at time $t_1\geq t_2$,
\begin{eqnarray}
\label{eq:corr}
\la q(t_1)q(t_2)\ra = \int dq_1dq_2 \,q_1q_2\,\rho(q_1,t_1|q_2,t_2)\rho_{\rm st}(q_2,t_2).
\end{eqnarray}
For weak driving, function $\rho_{\rm st}(q,t_2)$ can be expanded in a series in $F\exp(\pm i\omega_Ft_2)$ with time-independent coefficients, whereas $\rho(q_1,t_1|q_2,t_2)$ can be expanded in $F\exp(\pm i\omega_Ft_2)$ with coefficients that depend on $t_1-t_2$. Therefore the power spectrum (\ref{eq:power_spectrum}) does not have terms linear in $F$. To the second order in $F$ for $\omega\geq 0$ we have
\begin{equation}
\label{eq:define_Phi_F}
\Phi(\omega)=\Phi_0(\omega)+ \frac{\pi}{2}F^2|\chi(\omega_F)|^2\delta(\omega-\omega_F) + F^2\Phi_F(\omega).
\end{equation}
The term $\Phi_0(\omega)$ describes the power spectrum of the system in the absence of driving. The term $\propto\delta(\omega-\omega_F)$ describes the conventional linear response, cf. Eq.~(\ref{eq:trivial_susceptibility}). However, the expression for the susceptibility $\chi(\omega)$ in nonlinear systems is far more complicated than Eq.~(\ref{eq:trivial_susceptibility}); generally, the susceptibility is determined by the linear in $F$ term in $\rho_{\rm st} (q,t)$. In the optical language, the term $\propto \delta(\omega - \omega_F)$ in (\ref{eq:define_Phi_F}) corresponds to elastic scattering of the field $F\cos\omega_Ft$ by the system.
Of primary interest to us is the term $\Phi_F(\omega)$. This term is often disregarded in the analysis of the power spectra of driven systems, while the major emphasis is placed on the $\delta$-function in Eq.~(\ref{eq:define_Phi_F}). Function $\Phi_F(\omega)$ describes the interplay of fluctuations and driving in a nonlinear system beyond the trivial linear response. In the considered lowest-order approximation in the driving amplitude, $\Phi_F$ does not contain a $\delta$-peak at $2\omega_F$. However, it may contain a $\delta$-peak at $\omega=0$, which corresponds to the static driving-induced shift of the average position of the system. In what follows we do not consider this peak, as the static equilibrium position can be measured independently.
Function $\Phi_F$ can be found from Eq.~(\ref{eq:corr}) by calculating the transition probability density and the stationary probability distribution. This can be done for Markov systems numerically and also, in the case of weak noise, analytically, see Secs.~\ref{sec:Brownian_particle} and \ref{sec:two_state}. Alternatively, function $\Phi_F$ can be related to fluctuations of linear and nonlinear response of the system and expressed in terms of the fluctuating linear and nonlinear susceptibility, see \ref{App_susceptibility}. We emphasize that the nonlinear response has to be taken into account when fluctuations are considered even though we are not interested in the behavior of the power spectrum near $2\omega_F$ or higher overtones or subharmonics of $\omega_F$.
\section{Power spectrum of a driven Brownian particle}
\label{sec:Brownian_particle}
A simple example of a system where $\Phi_F(\omega)$ displays a nontrivial behavior is a periodically driven overdamped Brownian particle in a nonlinear confining potential $U(q)$, see Eq.~(\ref{eq:Brownian}). This model immediately relates to many experiments on optically trapped particles and molecules. We will assume that thermal noise $f(t)$ is white and Gaussian and that it is not strong, so that it suffices to keep the lowest-order nontrivial terms in the potential,
\begin{eqnarray}
\label{eq:nonlinear_potential}
U(q)=\frac{1}{2}\kappa q^2 + \frac{1}{3}\beta q^3 + \frac{1}{4}\gamma q^4 + \ldots, \qquad \langle f(t)f(t')\rangle = 2D\delta(t-t'),
\end{eqnarray}
where $D\propto k_BT$ is the noise intensity. In the absence of driving the stationary probability distribution is of the Boltzmann form, $\rho_{\rm st}^{(0)}\propto \exp[-U(q)/D]$.
For small $D$ and weak driving force equation of motion $\dot q=-U'(q)+ f(t) + F\cos\omega_Ft$ can be solved directly by perturbation theory in the noise $f(t)$ and in $F$, as indicated in \ref{App_susceptibility}. Here we develop a different method, which is particularly convenient if one wants to go to high orders of the perturbation theory in $D$ and $F$.
\subsection{Method of Moments}
\label{subsec:method_of_moments}
Systems in which fluctuations are induced by white noise can be studied using the Fokker-Planck equation
\begin{equation}
\label{eq:FP_equation}
\partial_t\rho = -\partial_q\left\{\left[-U'(q) + F\cos\omega_Ft\right]\rho\right\} + D\partial^2_q\rho.
\end{equation}
This equation can be solved numerically. A convenient analytical approach is based on the method of moments, which are defined as
\begin{equation}
\label{eq:moments_defined}
M_n(\omega; t')=\int_0^\infty dte^{i\omega t}\int dq q^n\int dq' \rho(q,t+t'|q',t') q'\rho_{\rm st}(q',t').
\end{equation}
From Eq.~(\ref{eq:define_Phi_F}), the power spectrum is $\Phi(\omega)=(2/\tau_F){\rm Re}~\int_0^{\tau_F}dt' M_1(\omega;t')$.
The moments $M_n$ satisfy a set of simple linear algebraic equations
\begin{eqnarray}
\label{eq:moments_equation}
&&-i\omega M_n(\omega)+n\hat{\cal F}[M_n(\omega)]= Dn(n-1)M_{n-2}(\omega) \nonumber\\
&&+\frac{1}{2}F\left[e^{i\omega_F t'}nM_{n-1}(\omega+\omega_F) + e^{-i\omega_F t'}nM_{n-1}(\omega-\omega_F)\right] +Q_{n+1}(t').\end{eqnarray}
Here, we skipped the argument $t'$ in $M_n$ and introduced function
$
\hat{\cal F}[M_n]\equiv \kappa M_n +\beta M_{n+1} + \gamma M_{n+2}.
$.
Functions
\begin{equation}
\label{eq:Q_n}
Q_n(t)=\int dq q^n\rho_{\rm st}(q,t)
\end{equation}
in the right-hand side of Eq.~(\ref{eq:moments_equation}) can themselves be found from a set of linear equations similar to (\ref{eq:moments_equation}). They follow from Eq.~(\ref{eq:FP_equation}), if one sets $\rho = \rho_{\rm st}(q,t)$ and takes into account that $\rho_{\rm st}(q,t)$ is periodic in $t$. To the lowest order in $F$ it suffices to keep in $Q_n(t)$ only terms that are independent of $t$ or oscillate as $\exp(\pm i\omega_F t)$; respectively, in Eq.~(\ref{eq:Q_n}) $Q_n(t)\approx Q_n^{(0)} + \left[Q_n^{(1)}\exp(i\omega_F t)+ {\rm c.c.}\right]$, and
\begin{eqnarray}
\label{eq:Q_n_equations}
\hat{\cal F}[Q_n^{(0)}]=D(n-1)Q_{n-2}^{(0)}+F{\rm Re}\,Q_{n-1}^{(1)},\nonumber\\
i\omega_F Q_n^{(1)} + n\hat{\cal F}[Q_n^{(1)}] = Dn(n-1)Q_{n-2}^{(1)} + \frac{1}{2}nFQ_{n-1}^{(0)}.
\end{eqnarray}
The system of coupled linear equations for the moments $M_n$ and $Q_n$ can be quickly solved with conventional software to a high order in the noise intensity $D$. Nontrivial results emerge already if we keep terms $\propto DF^2$: these are the terms that contribute to the power spectrum $\Phi_F(\omega)$ to the lowest order in $D$. To find them it suffices to consider terms $M_n$ with $n\leq 3$ and $Q_n$ with $n\leq 4$. This gives
\begin{eqnarray}
\label{eq:Phi_F_overdamped}
\Phi_F(\omega)&\approx& \frac{2D}{(\kappa^2+\omega_F^2)(\kappa^2+\omega^2)^2}\left\{2\beta^2\frac{(4\kappa^2 + \omega_F^2)(\kappa^2+ \omega^2+\omega_F^2)}{[\kappa^2+(\omega - \omega_F)^2][\kappa^2+(\omega + \omega_F)^2]}\right.
\nonumber\\
&&\left. -3\gamma\kappa\right\}.
\end{eqnarray}
This expression refers to $|\omega|>0$; function $\Phi_F(\omega)$ contains also a $\delta$-peak at $\omega=0$, which comes from the driving-induced shift of the average static value of the coordinate.
The solution of the equations for the moments in the considered approximation gives a correction $\propto D^2$ to the power spectrum in the absence of driving $\Phi_0(\omega)$. To the lowest order in $D$ this function displays a Lorentzian peak at $\omega=0$, $\Phi_0(\omega)=2D/(\kappa^2+\omega^2)$. This peak is used in the analysis of optical traps for Brownian particles \cite{Berg-Sorensen2004,Mas2013}; with account taken of the term $\propto D^2$ the zero-frequency peak of $\Phi_0(\omega)$ becomes non-Lorentzian.
\subsection{Power spectrum for comparatively large driving frequency}
\label{subsec:overdamped_spectrum}
The interpretation of Eq.~(\ref{eq:Phi_F_overdamped}) is simplified in the case where the driving frequency exceeds the decay rate, $\omega_F\gg \kappa$. In this case, periodic driving leads to two well-resolved features in the spectrum $\Phi_F$. One is located at $\omega=0$ and has the form
\begin{equation}
\label{eq:small_frequency}
\Phi_F(\omega)\approx (2D/\omega_F^2)(2\beta^2-3\gamma\kappa)(\kappa^2+\omega^2)^{-2} \quad (\omega\ll \omega_F).
\end{equation}
This equation can be easily obtained directly by solving the equation of motion $\dot q=-U'(q) +F\cos\omega_Ft + f(t)$ by perturbation theory in which $q(t)$ is separated into a part oscillating at high frequency $\omega_F$ (and its overtones) and a slowly varying part. To the lowest order in $F$ and $D$, the fast oscillating part renormalizes the decay rate of the slowly varying part of $q(t)$, with $\kappa \to \kappa -(F/\omega_F)^2\left[\kappa^{-1}\beta^2 - (3/2)\gamma)\right]$. Using this correction in the expression for the power spectrum of a linear system $\Phi_0^{(0)}(\omega)=2D/(\kappa^2+\omega^2)$, one immediately obtains Eq.~(\ref{eq:small_frequency}) to the leading order in $\kappa/\omega_F$.
Interestingly, Eq.~(\ref{eq:small_frequency}) describes a peak or a dip depending on the sign of $2\beta^2-3\gamma\kappa$. That is, the sign of $\Phi_F$ is determined by the competition of the cubic and quartic nonlinearity of the potential $U(q)$. This shows high sensitivity of the spectrum to the system parameters. The typical width of the peak/dip of $\Phi_F$ near $\omega=0$ is $\kappa$; the shape of the peak/dip is non-Lorentzian.
\begin{figure}[ht]
\center
\includegraphics[width=6cm]{fig2.eps}
\caption{Scaled driving induced terms in the power spectrum of an overdamped Brownian particle moving in the quartic potential $U(q)$ given by Eq.~(\ref{eq:nonlinear_potential}), $\tilde\Phi_F(\omega)=10^2\kappa^2\Phi_F(\omega)/2D$.
Panels (a), (b), and (c) refer to the scaled cubic nonlinearity $\beta^2D/\kappa^3 = 0.002$ and quartic nonlinearity $\gamma D/\kappa^2$ = 0.0006, 0.00147, and 0.002, respectively. The black dots and red solid curves correspond to the numerical simulations and Eq.~(\ref{eq:Phi_F_overdamped}). The scaled driving frequency is $\omega_F/\kappa = 5$ and the driving strength is $\kappa F^2/\omega_F^2D = 20$. For this driving strength and the noise intensity, the simulation results in panels (b) and (c) deviate from the theoretical curve. The deviation decreases for weaker driving. This is seen from the simulation data in panel (b) that refer to $\kappa F^2/\omega_F^2D = 5$ (blue triangles) and 1.25 (green squares). The corresponding spectra are scaled up by factors 4 and 16, respectively.}.
\label{fig:overdamped}
\end{figure}
The other spectral feature is located at $\omega_F$ and near the maximum has the form of a Lorentzian peak, $\Phi_F(\omega)\approx (D\beta^2/\omega_F^4)[\kappa^2+(\omega - \omega_F)^2]^{-1}$. The height of this peak is smaller by a factor $\kappa^2/\omega_F^2\ll 1$ than the height of the feature near $\omega=0$. We note that the height of the peak at $\omega_F$ is proportional to the squared parameter of the cubic nonlinearity of the potential $\beta$, but is independent of the quartic-nonlinearity parameter $\gamma$, to the leading order in the noise intensity $D$.
In Fig.~\ref{fig:overdamped} we compare the analytical expression (\ref{eq:Phi_F_overdamped}) with the results of numerical simulations. The simulations were done by integrating the stochastic differential equation $\dot q = -U'(q) + f(t) + F\cos\omega_Ft$ using the Heun scheme (cf. \cite{Mannella2002a}). Panel (a) shows that the cubic nonlinearity of the potential leads to a peak at $\omega=0$ and a comparatively small peak at $\omega_F$. The spectrum becomes more interesting in the generic case where both cubic and quartic terms in the potential are present and $\beta^2$ is comparable to $\gamma\kappa$. Here, as seen from panel (b), as a result of the competition between these terms, $\Phi_F(\omega)$ can have a dip at $\omega=0$ and two peaks, one near $\omega_F$ and the other with the position determined by $\beta^2/\gamma\kappa$ and $\omega_F/\kappa$. Where the quartic nonlinearity dominates, $\gamma\kappa\gg \beta^2$, see panel (c), it is hard to detect the peak at $\omega_F$ for small noise intensity. Our analytical calculations and numerical simulations show that, for larger noise intensity, this peak becomes more pronounced.
A significant deviation of simulations and the asymptotic expression (\ref{eq:Phi_F_overdamped}) in panel (b) for small $\omega$ is a consequence of the near compensation of the contributions to $\Phi_F(\omega)$ from the cubic and quartic nonlinear terms in $U(q)$ to the lowest order in $F^2$ and $D$. The terms of higher-order in $D$ and $F^2$ become then substantial. Panel (b) illustrates how the difference is reduced if $F^2$ is reduced. We checked that by reducing also the noise intensity we obtain a quantitative agreement of simulations with Eq.~(\ref{eq:Phi_F_overdamped}).
In some cases the confining potential of an overdamped system has inversion symmetry, and then $\beta=0$ in Eq.~(\ref{eq:nonlinear_potential}). In such cases spectral features of $\Phi_F$ at the driving frequency are $\propto (\gamma D)^2$. They can be found by solving the equations for the moments $M_n$ with $n\leq 5$ and $Q_n$ with $n\leq 6$ or by solving the equations of motion by perturbation theory to the second order in $\gamma$, cf. \ref{App_susceptibility}.
\section{Power spectrum of a driven two-state system}
\label{sec:two_state}
We now consider the effect of driving on a two-state system. Various types of such systems are studied in physics, from spin-1/2 systems to two-level systems in disordered solids to classical Brownian particles mostly localized at the minima of double-well potentials. We will assume that the system dynamics are characterized by the rates $W_{ij}$ of interstate $i\to j$ switching, where $i,j=1,2$. In the case of quantum systems, this means that the decoherence rate largely exceeds $W_{ij}$; in other words, the typical duration of an interstate transition is small compared to $1/W_{ij}$. For classical systems, this description means that small fluctuations about the stable states are disregarded.
\subsection{The model: modulated switching rates}
A major effect of periodic driving is modulation of the switching rates. It can be quite strong already for comparatively weak driving. Indeed, if the rates are determined by the interstate tunneling, since the field changes the tunneling barrier, its effect can be exponentially strong. Similarly, it may be exponentially strong in the classical limit if the switching is due to thermally activated overbarrier transitions, because the driving changes the barriers heights. Nevertheless, for weak sinusoidal driving $F\cos\omega_Ft$ the modulated rates $W_{ij}^{(F)}(t)$ can still be expanded in the driving amplitude,
\begin{equation}
\label{eq:rates_with_driving}
W_{ij}^{(F)}\approx W_{ij} -\alpha_{ij}F\cos\omega_Ft, \qquad i,j=1,2.
\end{equation}
This equation is written in the adiabatic limit, where the driving frequency $\omega_F$ is small compared to the reciprocal characteristic dynamical times, like the imaginary time of motion under the barrier in the case of tunneling \cite{Leggett1995} or the periods and relaxation times of vibrations about the potential minima in the case of activated transitions. The rates $W_{ij}$ are also assumed to be small compared to the reciprocal dynamical times. The driving frequency $\omega_F$ is of the order of $W_{ij}$.
Parameters $\alpha_{ij}$ in Eq.~(\ref{eq:rates_with_driving}) describe the response of the switching rates to the driving. They contain factors $\sim W_{ij}$. Indeed, for activated processes $W_{ij}\propto \exp(-\Delta U_i/k_BT)$, where $\Delta U_i$ is the height of the potential barrier for switching from the state $i$. If $F\cos\omega_Ft$ is the force that drives the system, then $\alpha_{ij}\approx W_{ij}d_i/k_BT$, where $d_i$ is the position of the $i$th potential well counted off from the position of the barrier top \cite{Debye1929}. The terms $\propto F^2$, which have been disregarded in Eq.~(\ref{eq:rates_with_driving}), are $\propto W_{ij}(d_i/k_BT)^2$ in this case; a part of these terms that are $\propto \cos2\omega_Ft$ do not contribute to $\Phi_F(\omega)$ to the second order in $F$, whereas the contribution of the time-independent terms $\propto F^2$ comes to renormalization of the parameters $W_{ij}$ in $\Phi_0(\omega)$, see below. For incoherent interstate quantum tunneling, $\alpha_{ij}\propto W_{ij}$, too.
We will use quantum notations $|i\rangle$ ($i=1,2$) for the states of the system. One can associate these states with the states of a spin-1/2 particle by setting $|1\rangle \equiv \up$ and $|2\rangle \equiv \down$. The system dynamics is most conveniently described by the dynamical variable $q$ defined as
\begin{equation}
\label{eq:defined_variable}
q=|1\rangle\langle 1| - |2\rangle\langle 2|\equiv \sigma_z,
\end{equation}
where $\sigma_z$ is the Pauli matrix. For a particle in a double-well potential, $q$ is the coordinate that takes on discrete values $1$ and $-1$ at the potential minima 1 and 2, respectively.
The power spectra of driven two-state systems have been attracting much interest in the context of stochastic resonance, see \cite{Dykman1995d,Wiesenfeld1998,Gammaitoni1998,Thorwart1998} for reviews. By now it has been generally accepted that, for weak driving, the power spectrum of a system has a $\delta$-peak at the driving frequency with area $\propto F^2$, which is described by the standard linear response theory \cite{Dykman1990e}. This peak is of central interest for signal processing. However, as we show in this Section, along with this peak, the spectrum has a characteristic extra structure, which is also $\propto F^2$, to the leading order in $F$.
\subsection{Kinetic equation and its general solution}
It is convenient to write the analog of Eq.~(\ref{eq:corr}) for the correlation function of the discrete variable $q$ as
\begin{equation}
\label{eq:two_state_correlator}
\langle q(t_1)q(t_2)\rangle = \sum_{i,j}\langle i|\sigma_z\hat\rho(t_1|t_2)\sigma_z\hat\rho_{\rm st}(t_2)|j\rangle
\end{equation}
Here, $\hat\rho(t_1|t_2)$ is the transition density matrix, $\hat\rho(t_1|t_2)\equiv \sum|i\rangle \rho_{ij}(t_1|t_2)\langle j|$, and $\hat\rho_{\rm st}\equiv \sum |i\rangle (\rho_{\rm st})_{ii} \langle i|$ is the stationary density matrix. By construction (in particular, because of the decoherence in the quantum case) the stationary density matrix is diagonal. Its matrix elements $(\rho_{\rm st})_{ii}$ give the populations of the corresponding states and periodically depend on time, $\hat\rho_{\rm st}(t+2\pi/\omega_F)=\hat \rho_{\rm st}(t)$. The transition matrix elements $ \rho_{ij}(t_1|t_2)$ give the probability to be in state $i$ at time $t_1$ given that the system was in state $j$ at time $t_2$. At equal times we have $\hat\rho(t_2|t_2) = \hat I$, where $\hat I$ is the unit matrix.
Equation~(\ref{eq:two_state_correlator}) does not have the form of a trace over the states $|i\rangle$; rather it expresses the correlator in terms of the joint probability density to be in state $|j\rangle$ at time $t_2$ and in state $|i\rangle$ at time $t_1$, with summation over $i,j$ \cite{vanKampen_book}. In the quantum formulation, the applicability of this expression is a consequence of the decoherence and Markovian kinetics.
Matrix elements $\rho_{ij}(t|t')$ satisfy a simple balance equation, which in the presence of driving reads
\begin{equation}
\label{eq:balance_two_state}
\partial_{t}\rho_{1j}(t|t')= -W_{12}^{(F)}(t)\rho_{1j} + W_{21}^{(F)}(t)\rho_{2j},\quad \rho_{1j} + \rho_{2j} = 1,
\end{equation}
where $j=1,2$. Equation for the matrix elements of $\hat\rho_{\rm st}(t)$ has the same form, except that subscript $j$ has to be set equal to the first subscript.
From Eqs.~(\ref{eq:two_state_correlator}) and (\ref{eq:balance_two_state}) we obtain a general expression for the correlator of interest,
\begin{eqnarray}
\label{eq:two_state_general}
&&\langle q(t_1)q(t_2)\rangle = \exp\left[-\int_{t_2}^{t_1}dt W^{(F)}_+(t)\right]+ \langle\sigma_z(t_2)\rangle_{\rm st} \int_{t_2}^{t_1}dt \left\{W^{(F)}_-(t)
\nonumber \right.\\
&&\left. \times\exp\left[-\int_{t}^{t_1}dt' W^{(F)}_+(t')\right]\right\}, \qquad
W^{(F)}_\pm (t) = W_{21}^{(F)}(t) \pm W_{12}^{(F)}(t).
\end{eqnarray}
Here, $\langle \sigma_z(t)\rangle_{\rm st} \equiv \langle q(t)\rangle_{\rm st} \equiv {\rm Tr}~[\sigma_z \hat\rho_{\rm st}(t)]$ is the time-dependent difference of the state populations in the stationary state.
Generally , $\langle \sigma_z(t)\rangle_{\rm st}$ is nonzero even in the absence of driving unless the switching rates are equal, $W_{12}=W_{21}$. In the presence of driving there emerges a periodic term in $\langle \sigma_z(t)\rangle_{\rm st}$, which describes the linear response, for weak driving.
Disregarding terms oscillating as $\exp(\pm 2 i\omega_F t)$, to the second order in $F$ we obtain from the balance equation (\ref{eq:balance_two_state}) written for $(\rho_{\rm st})_{ii}$
\begin{eqnarray}
\label{eq:stationary_sigma}
&&\langle \sigma_z(t)\rangle_{\rm st} \approx \frac{W_-}{W_+} + \frac{F}{2}\left [\chi_1(\omega_F)e^{-i\omega_F t} + {\rm c.c.}\right] + \frac{\alpha_+ F^2}{2W_+} {\rm Re}\, \chi_1(\omega_F), \nonumber \\
&& \chi_1(\omega) = 2\left(\alpha_{12} W_{21} - \alpha_{21} W_{12}\right)/\left[W_+\left(W_+ - i\omega\right)\right].
\end{eqnarray}
Here we introduced notations
\begin{equation}
\label{eq:pm_notations}
\alpha_\pm = \alpha_{21} \pm \alpha_{12}, \qquad W_\pm = W_{21}\pm W_{12}.
\end{equation}
Function $\chi_1(\omega)$ gives the linear susceptibility. In the case of thermally activated transitions, Eq.~(\ref{eq:stationary_sigma}) for $\chi_1$ coincides with the classical result \cite{Debye1929}. The term $W_-/W_+$ gives the difference of the state populations in the absence of driving, whereas the term $\propto F^2$ gives the time independent part of the driving-induced correction to this difference.
\subsection{The driving-induced part of the power spectrum}
\label{subsec:power_spectrum_two_state}
Equation~(\ref{eq:two_state_general}) allows one to calculate the period-averaged correlator $\lal q(t_1)q(t_2)\rar$ in the explicit form and to obtain the power spectrum. As before, we will not consider the $\delta$-peak in $\Phi(\omega)$ for $\omega=0$. The spectrum is an even function of $\omega$, and we will consider it for $\omega > 0$:
\begin{eqnarray}
\label{eq:two_state_power}
&\Phi_0(\omega)= 8\frac{W_{12}W_{21}}{W_+^2}\,\frac{W_+}{W_+^2 + \omega^2}, \qquad \Phi_F(\omega) = \Phi_F^{\rm(r)}(\omega) + \Phi_F^{\rm (c)}(\omega), \nonumber\\
&\Phi_F^{\rm(r)}(\omega)=\alpha_+\sum_{\mu,\nu=\pm}\phi_F(\mu\omega,\nu\omega_F),\nonumber\\
&\phi_F(\omega,\omega_F) = - [W_+ - i(\omega-\omega_F)]^{-1}\left[\frac{\alpha_+ W_{12}W_{21}}{\omega_F^2W_+^2} + i\frac{W_-}{2\omega_FW_+}\chi_1^*(\omega_F)\right].
\end{eqnarray}
The term $\Phi_0$ is the familiar power spectrum of a two-state system in the absence of driving \cite{Debye1929}. It has a peak at $\omega=0$ with halfwidth $W_+$ equal to the sum of the switching rates. The term $\Phi_F^{\rm (c)}$ describes the driving-induced modification of the peak centered at $\omega=0$,
\begin{eqnarray}
\label{eq:zero_frequency_peak_two_state}
\Phi_F^{\rm (c)}(\omega) = (\alpha_+^2/2\omega_F^2)\Phi_0(\omega) -
|\chi_1(\omega_F)|^2W_+/(W_+^2+\omega^2) .
\end{eqnarray}
Of major interest to us is the part $\Phi_F^{\rm (r)}(\omega)$ of the driving-induced term in the power spectrum (\ref{eq:two_state_power}). For $\omega>0$, it shows a resonant peak (or a dip, depending on the parameters) at the driving frequency $\omega_F$. In contrast to the $\delta$-peak of the linear response, the peak has a finite halfwidth $\sim W_+= W_{12}+W_{21}$. It is well separated from the peak at $\omega=0$ for $\omega_F\gg W_+$ and generally is of a non-Lorentzian shape. We stress that, to the order of magnitude, the peak has the same overall area as the $\delta$-peak of the linear response (in the case of a dip, the absolute value of the area should be considered). Another important feature of the peak/dip seen from Eq.~(\ref{eq:two_state_power}) is that it is proportional to the parameter $\alpha_+=\alpha_{12}+\alpha_{21}$. This parameter describes the change of the sum of the switching rates due to the driving.
For activated switching between potential minima considered in the classical stochastic resonance theory, $\alpha_+ = (k_BT)^{-1}(W_{12}d_1+ W_{21} d_2)$. For a symmetric potential $\alpha_+=0$, since $W_{12}=W_{21}$ and $d_1 = -d_2$. Then $\Phi_F^{\rm (r)}=0$, in agreement with \cite{Mcnamara1989} where a symmetric potential was considered. On the other hand, for strong driving it was found \cite{Nikitin2007} that the power spectrum for an asymmetric potential displays peaks close to odd multiples of the driving frequency and dips close to even multiples of driving frequency. In our weak-driving analysis we do not consider peaks/dips near the overtones of $\omega_F$; however, as seen from Eq.~(\ref{eq:two_state_power}), the sign of $\Phi_F^{\rm (r)}(\omega)$ near $\omega_F$ can be positive or negative, depending on the parameters.
Examples of the driving-induced spectra $\Phi_F(\omega)$ are shown in Fig.~\ref{fig:ts}. One can clearly see the peaks or dips both at $\omega=0$ and at the driving frequency $\omega_F$. In agreement with Eqs.~(\ref{eq:two_state_power}) and (\ref{eq:zero_frequency_peak_two_state}), the signs of the features of $\Phi_F$ are determined by the interrelation between the parameters of the two-states system. For illustration purpose we chose the values of the ratio of the response parameters $\alpha_{21}/\alpha_{12}$ to lie between plus and minus the ratio of the switching rates in the absence of driving, $W_{21}/W_{12}$. As seen from Fig.~\ref{fig:ts}, the spectra are very sensitive to the ratio $\alpha_{21}/\alpha_{12}$. We have seen this sensitivity also for different values of $W_{21}/W_{12}$.
Unexpectedly, a finite-height spectrum $\Phi_F(\omega)$ emerges even where the linear susceptibility is equal to zero, which happens for $\alpha_{12}W_{21}=\alpha_{21}W_{12}$. This is seen from Eq.~(\ref{eq:two_state_power}) and also from Fig.~\ref{fig:ts}. The red line with $\alpha_{21}/\alpha_{12}=7/3$ refers to this case, and the area of $\delta$-peak in the spectrum is zero. As seen from the figure, numerical simulations are in excellent agreement with the analytical expressions.
\begin{figure}[h]
\center
\includegraphics[width=11cm]{fig3.eps}
\caption{The driving induced terms in the power spectrum of the two-state system for the ratio of the switching rates $W_{21}/W_{12} = 7/3$. The scaled driving frequency and amplitude are $\omega_F/W_+ = 5$ and $F\alpha_{12}/W_{12} = 1.$ On the thick solid (red), dot-dashed (black), long-dashed (blue), short-dashed (green), and thin solid (purple) lines the ratio $\alpha_{21}/\alpha_{12}$ is 7/3, 7/6, 0 , $-7/6$, and $-7/3$. The vertical line at $\omega_F$ shows the position of the $\delta$-peak at $\omega_F$. The areas of the $\delta$-peaks for different $\alpha_{21}/\alpha_{12}$ are given by the heights of the vertical segments. The heights are counted off from the lines to the symbols of the same color, i.e., to the circle, triangle, and open and full square, in the order of decreasing $\alpha_{21}/\alpha_{12}$; there is no symbol for $\alpha_{21}/\alpha_{12}=7/3$ as there is no $\delta$-peak in this case. The inset shows the full spectrum with (red) and without (black) driving for $\alpha_{21}/\alpha_{12} = 7/3$. The curves and the dots show the analytical theory and the simulations, respectively.}
\label{fig:ts}
\end{figure}
The structure of the spectrum near $\omega=0$ will be modified if one takes into account terms $\propto F^2$ in the expressions for the switching rates (\ref{eq:rates_with_driving}). In the considered leading-order approximation in $F$ these terms have to be averaged over the driving period and are thus independent of time. The correction due to these terms can be immediately found from Eq.~(\ref{eq:two_state_power}) for $\Phi_0(\omega)$ by expanding $\Phi_0$ to the first order in the corresponding increments of $W_{ij}$; this correction is of a non-Lorentzian form.
\section{Threshold detector}
\label{sec:threshold}
An insight into the dynamical nature of the driving-induced change of the power spectrum can be gained from the analysis of the spectrum of a threshold detector. Such detectors are broadly used in science and engineering, and their analogs play an important role in biosystems. We will employ the simplest model where the output of a threshold detector is $q=-1$ if the signal at the input is below a threshold value $\eta$, whereas $q=1$ otherwise, and will consider the case where the input signal is a sum of the periodic signal $F\cos\omega_Ft$ and noise $\xi(t)$,
\begin{eqnarray}
\label{eq:threshold_model}
q(t) = 2\Theta\left[F(t)+\xi(t) - \eta\right] - 1,
\end{eqnarray}
where $\Theta(x)$ is the Heaviside step function. To avoid singularities related to non-differentiability of the $\Theta$-function, we will model the output by
\begin{equation}
\label{eq:tanh_model}
q(t)=\tanh\left[\Lambda\bigl(F(t)+\xi(t)-\eta\bigr)\right], \qquad \Lambda \gg 1,
\end{equation}
and in the final expressions will go to the limit $\Lambda\to\infty$. Much work on the interplay of noise and driving in threshold detectors has been done in the context of stochastic resonance, cf. \cite{Wiesenfeld1994,Gingl1995a,Jung1995}. In these papers of primary interest was the signal to noise ratio; the issues we are considering here, i.e., the occurrence of the effective ``inelastic scattering" and ``fluorescence" as a result of interplay of nonlinearity and noise, have not been addressed, to the best of our knowledge.
In the absence of noise, the power spectrum of $q(t)$ is a series of $\delta$-peaks at $\omega_F$ and its overtones (including $\omega=0$), provided the driving amplitude $F>\eta$, whereas for $F<\eta$ we have $q=-1$ and the power spectrum is just a $\delta$-peak at $\omega=0$. On the other hand, if $F=0$ and $\xi(t)$ is white noise, in the limit $\Lambda\to\infty$ in Eq.~(\ref{eq:tanh_model}) the correlator $\langle q(t)q(t')\rangle =0$ for $t\neq t'$, since the values of $q(t)$ at different instants of time are uncorrelated and $\langle q\rangle \to 0$, whereas $\langle q^2\rangle \to 1$.
The singular behavior of the correlator $\langle q(t)q(t')\rangle$ in the case of white noise persists also in the presence of driving. This is a consequence of the absence of {\it dynamics}, i.e., any memory effects in the variable $q(t)$ (\ref{eq:threshold_model}), and the singular distribution of white noise, where the intensity $\langle \xi^2(t)\rangle$ diverges.
Dynamics can be brought into the system by the noise color. Such noise can be thought of as coming from a dynamical system with retarded response, which is driven by white noise. We will be interested in the correlator $\langle q(t)q(t')\rangle$ and the power spectrum $\Phi(\omega)$ for weak driving, where the driving amplitude is $F\ll \eta$ (subthreshold driving), and for a simple colored noise, the Ornstein-Uhlenbeck noise. This is Gaussian noise with correlator
\begin{equation}
\label{eq:color_noise}
\langle \xi(t)\xi(t')\rangle = (D/\kappa) \exp(-\kappa|t-t'|).
\end{equation}
Parameter $\kappa$ characterizes the decay rate of noise correlations.
Because the threshold detector has no dynamics on its own, the value of the variable $q(t)$ is determined by the instantaneous value of the noise $\xi(t)$. We can write $q(t)\equiv \tilde q\bigl(t,\xi(t)\bigr)$, where $\tilde q(t,\xi)$ is given by Eqs.~(\ref{eq:threshold_model}), (\ref{eq:tanh_model}) with $\xi(t)$ replaced with $\xi$. Then the general expression for the correlator of $q(t)$, Eq.~(\ref{eq:corr}), can be rewritten as
\begin{eqnarray}
\label{eq:td_corr}
\la q(t_1)q(t_2) \ra = \int d\xi_1 d\xi_2\, \tilde q(t_1, \xi_1)\tilde q(t_2,\xi_2)\,
\rho^{(\xi)}(\xi_1,t_1|\xi_2,t_2)\rho^{(\xi)}_{\rm st}(\xi_2,t_2).
\end{eqnarray}
Here, the superscript $\xi$ indicates that the corresponding transition probability density and the stationary distribution refer to the random process $\xi(t)$.
The form of the transition probability for the process (\ref{eq:color_noise}) is well-known \cite{Risken1989},
\begin{eqnarray}
\label{eq:transition_prob_xi}
\fl
\rho^{(\xi)}(\xi_1,t_1|\xi_2,t_2) = \sqrt{\frac{\kappa}{2\pi D(1-e^{-2\kappa|t_1-t_2|})}}
\exp\left\{-\frac{\kappa(\xi_1-\xi_2 e^{-\kappa|t_1-t_2|})^2}{2D(1-e^{-2\kappa|t_1-t_2|})}\right\}.
\end{eqnarray}
The stationary distribution $\rho^{(\xi)}_{\rm st}(\xi_1,t_1)$ is given by the same expression with $t_2\to -\infty$. Substituting these expressions into Eq.~(\ref{eq:td_corr}) and expanding $\tilde q(t,\xi)$ in $F(t)$, after averaging over the driving period we obtain to second order in $F(t)$ for $t_1>t_2$
\begin{eqnarray}
\label{eq:threshold_correlator}
\fl
\lal q(t_1)q(t_2) \rar =C + & 4\int_{\eta }^{\infty} d\xi_1 \int_{\eta}^{\infty} d\xi_2\left[ \rho^{(\xi)}(\xi_1,t_1|\xi_2,t_2) -
\rho^{(\xi)}_{\rm st}(\xi_1)\right] \rho^{(\xi)}_{\rm st}(\xi_2) \nonumber\\
&+ 2F^2 \cos\wF(t_1-t_2) \rho^{(\xi)}(\eta,t_1|\eta,t_2) \rho^{(\xi)}_{\rm st}(\eta)\nonumber\\
&- 2F^2\int_{\eta}^{\infty}d\xi_2\rho^{(\xi)}_{\rm st}(\xi_2) \frac{d}{d\eta}\left[\rho^{(\xi)}(\eta,t_1|\xi_2,t_2) -
\rho^{(\xi)}_{\rm st}(\eta)\right].
\end{eqnarray}
Here, $C$ is a constant independent of time; it leads to a $\delta$ peak at $\omega = 0$ in the power spectrum and will not be considered in what follows. The remaining terms are time-dependent. They decay with increasing $|t_1-t_2|$, except for the term that oscillates as $\exp[\pm i\omega_F(t_1-t_2)]$ and describes the standard linear response to periodic driving. As seen from Eq.~(\ref{eq:threshold_correlator}), this term has the form
\begin{eqnarray}
\label{eq:susceptibility_threshold}
&2F^2 \cos\wF(t_1-t_2)\left[\rho^{(\xi)}_{\rm st}(\eta)\right]^2 \equiv
\frac{1}{2}F^2|\chi(\omega_F)|^2\cos\omega_F(t_1 - t_2) , \nonumber\\
&\chi(\omega)= 2\rho^{(\xi)}_{\rm st}(\eta)\equiv (2\kappa/\pi D)^{1/2}\exp[-\kappa\eta^2/2D),
\end{eqnarray}
where $\chi(\omega)$ is the standard linear susceptibility \cite{LL_statphys1} of the threshold detector. Interestingly, this susceptibility is independent of frequency. This is because the detector has no dynamics, its response to the driving is instantaneous. An alternative derivation of the expression for the susceptibility, which provides a useful insight into the response of the threshold detector, is given in \ref{App_susceptibility}. It also shows how to deal with the singularities in Eq.~(\ref{eq:threshold_correlator}) for $t_1\to t_2$, which emerge after the transition $\Lambda\to\infty$ in Eqs.~(\ref{eq:tanh_model}) and (\ref{eq:td_corr}).
The power spectrum $\Phi(\omega)$ is obtained from Eq.~(\ref{eq:threshold_correlator}) by a Fourier transform. The $F$-independent term in Eq.~(\ref{eq:threshold_correlator}) gives the power spectrum $\Phi_0(\omega)$ in the absence of driving. It has a peak at $\omega=0$. The term $\propto \cos\omega_F(t_1 - t_2)$ gives a $\delta$-peak and also a finite-width peak $F^2\Phi_F^{\rm (r)}(\omega)$ at frequency $\omega_F$. The last term in Eq.~(\ref{eq:threshold_correlator}) gives a driving-induced feature in the power spectrum at zero frequency $F^2\Phi^{\rm (c)}(\omega)$.
The shape of the spectra is determined by the dimensionless parameter that characterizes the ratio of the threshold to the noise amplitude $\eta(\kappa/D)^{1/2}$. For weak noise, where $\eta(\kappa/D)^{1/2}\gg 1$, the peak near $\omega_F$ has the form
\begin{equation}
\label{eq:threshold_asymptotic_resonant}
\Phi_F^{\rm (r)}(\omega) \approx \frac{1}{D\sqrt{2\pi}}\,{\rm Re}\,\left(\frac{\kappa\eta^2}{4D}+i\frac{\omega - \omega_F}{\kappa}\right)^{-1/2}e^{-\kappa\eta^2/2D}.
\end{equation}
Here we assumed that $\omega_F/\kappa$ is sufficiently large, so that the features of $\Phi_F$ centered at $\omega_F$ and $\omega=0$ are well separated; Eq.~(\ref{eq:threshold_asymptotic_resonant}) applies for $|\omega-\omega_F|\ll \omega_F$. The spectrum (\ref{eq:threshold_asymptotic_resonant}) has a characteristic non-Lorentzian form with typical width $\kappa^2\eta^2/4D$. However, its area is small.
In the opposite limit of low threshold, $\eta(\kappa/D)^{1/2}\ll 1$, to the leading order
\begin{equation}
\label{eq:threshold_resonant_small}
\Phi_F^{\rm (r)}(\omega)\approx \frac{1}{2\sqrt{\pi}D}\,{\rm Re}\left[\Gamma\left(i\frac{\omega-\omega_F}{2\kappa}\right)/\Gamma\left(\frac{1}{2}+i\frac{\omega-\omega_F}{2\kappa}\right) \right]
\end{equation}
near $\omega_F$. This spectrum falls off slowly away from the maximum, as $|\omega-\omega_F|^{-1/2}$ for $|\omega-\omega_F|\gg \kappa$. Equation~(\ref{eq:threshold_resonant_small}) does not contain the threshold $\eta$. The small-$\eta$ correction to (\ref{eq:threshold_resonant_small}) for $\omega=\omega_F$ is $(1-\ln 2)\kappa\eta^2/\pi D^2$. It is positive. From the comparison of Eqs.~(\ref{eq:threshold_asymptotic_resonant}) and (\ref{eq:threshold_resonant_small}), one sees that the height of the peak at $\omega_F$ first increases with the increasing $\eta(\kappa/D)^{1/2}$, but then starts decreasing.
In Fig.~\ref{fig:td} we show analytical results for the power spectra obtained from Eq.~(\ref{eq:threshold_correlator}) for several parameter values and compare them with the results of simulations. Immediately seen from this figure is that the driving modifies the overall spectrum most significantly near $\omega=0$ and near $\omega_F$ for large $\omega_F/\kappa$. There emerges a finite-width peak at $\omega_F$. As seen from the inset in panel (b), the width of this peak increases with decreasing noise intensity, that is, with increasing $\eta (\kappa/D)^{1/2}$. This is a counterintuitive consequence of the unusual interplay of noise and driving in a threshold detector. The height of the peak displays a nonmonotonic dependence on $\eta (\kappa/D)^{1/2}$.
\begin{figure}[h]
\includegraphics[width=74mm]{fig4a.eps}
\includegraphics[width=74mm]{fig4b.eps}
\caption{Power spectrum of the threshold detector. (a): The full power spectrum; the scaled frequency and the intensity of the driving are $\wF /2\pi\kappa = 100$ and $F^2\kappa/D = 0.0025$. The scaled threshold is $\eta(\kappa/D)^{1/2} = 0.5$. Inset: the spectrum near the driving frequency. The delta peak has been subtracted. The curves and black dots refer to the theory and simulations, respectively. (b): The low-frequency part of the driving-induced term in the power spectrum for $\omega_F/\kappa = 50$ as given by Eq.~(\ref{eq:threshold_correlator}). The solid (black), long-dashed (red), short-dashed (blue) and dot-dashed (green) curves correspond to the scaled value of the threshold $\eta( \kappa/D)^{1/2} = 0.1, 0.8, 1.2$, and 2. Inset: the spectrum near the driving frequency, $\wF /\kappa =50 $. }
\label{fig:td}
\end{figure}
The low-frequency spectrum $\Phi_F(\omega)\approx \Phi_F^{\rm (c)}(\omega)$ also displays a pronounced feature near $\omega=0$. One can show from the analysis of the last term in Eq.~(\ref{eq:threshold_correlator}) that, for small $\eta(\kappa/D)^{1/2}$, this feature is a dip, with $\Phi_F^{\rm (c)}(0)=-1/D$ for $\eta(\kappa/D)^{1/2}\to 0$. The shape of the dip is non-Lorentzian, with typical width $\kappa$. As $\eta(\kappa/D)^{1/2}$ increases, the depth of the dip decreases. Ultimately the shape of the spectrum changes completely. For large $\eta(\kappa/D)^{1/2}$ the spectrum $\Phi_F^{\rm (c)}$ becomes broad and shallow. To the leading order in $[\eta(\kappa/D)^{1/2}]^{-1}$, it can be written as $(2/\pi D)(D/ \kappa\eta^2)^{1/2} \exp(-\kappa\eta^2/2D)\tilde\Phi_F^{\rm (c)}(2D\omega/\kappa^2\eta^2)$, where the dimensionless function $\tilde\Phi_F^{\rm(c)} (x)$ is zero for $x=0$, has a minimum at $x\approx 1.7$, where it is $\approx -0.6$, and then approaches zero with increasing $x$ as $x^{-1/2}$.
\section{Conclusions}
The results of this paper demonstrate that the interplay of driving and fluctuations leads to the onset of specific spectral features in the power spectra of dynamical systems. Such features are analogs of inelastic light scattering and fluorescence in optics, where an electromagnetic field can excite radiation at a frequency shifted from its frequency and also at the characteristic system frequency. Our results show that, in classical systems and in incoherent quantum systems, the spectral features emerge as a result of the fluctuation-induced modulation of the response to the driving. Such modulation is common to nonlinear systems.
Since nonlinearity and noise are always present in real systems, the occurrence of the driving-induced spectral features in the power spectra should be also generic. However, these features are specific for particular systems, which allows using them for system characterization.
We have studied three types of systems, all of which are attracting significant interest in mesoscopic physics and in several other areas of science. The first one is an overdamped Brownian particle fluctuating in a nonparabolic potential well. This model describes, in particular, small particles and molecules optically trapped in a liquid. We find that, when the particle is periodically driven, the nonparabolicity of the potential leads to an extra spectral peak or a dip at zero frequency. For comparatively weak noise, the sign of the driving-induced term in the spectrum at small $\omega$ is determined by the competition of the cubic and quartic nonlinearity of the potential. The overall shape of the low-frequency spectrum strongly depends on the form of the confining potential as well. In addition, along with a $\delta$-peak at the driving frequency, the driving-induced spectrum displays a peak at this frequency with a width of the order of the relaxation rate of the system.
We have also studied a two-state system that at random switches between the states. We assumed that the driving modulates the rates of interstate switching. The driving-induced spectrum has a rich form. Depending on the interrelation between the switching rates without driving and the driving-induced corrections to the rates, it can have peaks or dips both at $\omega=0$ and at the driving frequency. The typical width of the peaks/dips is given by the sum of the interstate switching rates without driving. Interestingly, these finite-width spectral features can emerge even where the $\delta$-peak at the driving frequency has very small (or zero) intensity.
The third system we studied is a threshold detector. Here the dynamical nature of the driving-induced spectral change is particularly pronounced, as this change does not occur if the noise in the detector is white, except for the $\delta$-peak at the driving frequency. On the other hand, for colored noise driving does change the power spectrum nontrivially. As in other systems, we find a driving-induced spectral feature near zero frequency. It can be a peak or a dip depending on the ratio of the threshold to the appropriately scaled noise intensity. Also, the height of the finite-width peak at the driving frequency displays a nonmonotonic dependence on this ratio, as does the width of the peak, too, i.e., noise can both increase or decrease the width.
In all studied systems inertial effects played no role: the peaks of the power spectra are located at zero frequency in the absence of driving. Therefore driving-induced spectral features near the driving frequency and zero frequency correspond to inelastic scattering and fluorescence, respectively. However, in contrast to the conventional fluorescence, driving can induce a dip in the spectrum at zero frequency, as we have seen in all studied systems (the total power spectrum remains positive, of course). The occurrence of the dip looks as if the driving were decreasing the noise in the system, although in fact the dip has dynamical nature.
The power spectra of weakly damped nonlinear systems should also display extra features in the presence of weak periodic driving. The effect should be most pronounced where the driving is resonant. Along with the features near the driving frequency and near $\omega=0$, there should arise features near the eigenfrequencies of slowly decaying vibrations about the stable states. Several features of the power spectra have been studied for nonlinear oscillators in the regime of strong driving, see recent papers \cite{Leyton2012,Dykman2011} and references therein. Interestingly, the results do not immediately extend to the weak-driving regime, and the features of the interplay of nonlinearity and driving where they are of comparable strength remain to be explored. However, it is clear from the presented results that the driving-induced change of the spectra is a general effect that provides a sensitive tool for characterizing fluctuating systems and their parameters.
The research of YZ and MID was supported in part by the US Army Research Office (grant W911NF-12-1-0235), US Defense Advanced Research Agency (grant FA8650-13-1-7301), and by TOYOTA Central R\&D Labs., Inc.
\appendix
\section{Formulation in terms of fluctuating susceptibilities}
\label{App_susceptibility}
The change of the power spectrum induced by the driving can be analyzed in terms of the fluctuating linear and nonlinear susceptibility of the system. If the dynamical variable that describes the state of the system is $q(t)$, to the second order in the driving $F(t)$ we have
\begin{eqnarray}
\label{eq:susceptibility}
\fl
q(t) \approx q_0(t)+\int_{-\infty}^tdt' \chi_1(t,t')F(t') + \int\!\!\!\int_{-\infty}^t dt' dt''\chi_2(t,t',t'')F(t')F(t''),
\end{eqnarray}
where $q_0(t)$ is the (random) value of $q(t)$ in the absence of driving. Functions $\chi_1$ and $\chi_2$ describe the linear and nonlinear response. We emphasize that these functions themselves are random, there is no ensemble averaging in Eq.~(\ref{eq:susceptibility}). This equation is merely a consequence of the causality principle. Spatial and temporal fluctuations of the linear susceptibility $\chi_1$ are standardly considered in the context of light scattering \cite{Einstein1910,Reichl2009}. However, the analysis of the power spectrum of nonlinear systems in the presence of driving requires also taking into account the fluctuating nonlinear susceptibility $\chi_2$.
The linear and nonlinear fluctuating susceptibilities lead to two terms in the driving-induced part of the power spectrum defined by Eq.~(\ref{eq:define_Phi_F}), $\Phi_F(\omega)= \Phi_F^{(1)}(\omega) + \Phi_F^{(2)}(\omega)$. Substituting into (\ref{eq:susceptibility}) $F(t) = F\cos\omega_Ft \exp(\epsilon t)$ with $\epsilon\to +0$, we obtain \cite{Zhang2014}
\begin{eqnarray}
\label{eq:Phi_1_general}
\Phi_F^{(1)}(\omega)&=& \frac{1}{2}{\rm Re}\int_0^\infty dt e^{i(\omega-\omega_F)t}
\int\!\!\!\int_{-\infty}^0 dt' dt'' e^{i\omega_F(t'' - t')}\nonumber\\
&&\times
\Big\langle\chi_1(t,t+ t')[\chi_1(0,t'')-\langle \chi_1(0,t'')\rangle]\Big\rangle,
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:Phi_2_general}
\Phi_F^{(2)}(\omega)&=& {\rm Re}\int_0^\infty dt e^{i\omega t}\int\!\!\!\int_{-\infty}^0 dt' dt'' \cos[\omega_F(t'-t'')]
\nonumber\\
&&\times \left[\langle \chi_2(t,t+t',t+t'')q_0(0)\rangle + \langle q_0(t)\chi_2(0,t',t'')\rangle\right],
\end{eqnarray}
where we assumed $\langle q_0(t)\rangle = 0$.
The general form of Eqs.~(\ref{eq:Phi_1_general}) and (\ref{eq:Phi_2_general}) immediately shows two distinct effects of the driving, which are pronounced where the driving frequency $\omega_F$ largely exceeds the typical relaxation rate of the system. The term $\Phi_F^{(1)}(\omega)$ is a function of the detuning of frequency $\omega$ from the driving frequency $\omega_F$. Therefore one may expect that $\Phi_F^{(1)}(\omega)$ displays features like peaks or dips near $\omega_F$. In contrast, $\Phi_F^{(2)}(\omega)$ should display features near the characteristic frequencies of the system. In particular, for overdamped systems that we consider here such features occur around $\omega=0$. We note that $\Phi_F^{(1)}$ may also display features at the system eigenfrequency, since the integrand in Eq.~(\ref{eq:Phi_1_general}) depends on $\omega_F$.
A convenient way to calculate the fluctuating susceptibilities $\chi_{1,2}$ is based on solving dynamical equations of motion of the system. For example, for an overdamped Brownian particle described by the Langevin equation $\dot q=-U'(q)+ f(t) + F\cos\omega_Ft$ with nonlinear potential (\ref{eq:nonlinear_potential}), one can proceed by rewriting this equation in the integral form,
\begin{eqnarray}
\label{eq:integral_equation}
q(t)=&&\int_{-\infty}^t dt'e^{-\kappa(t-t')}\exp\left\{-\int_{t'}^tdt''\left[\beta q(t'')+\gamma q^2(t'')\right]\right\}\nonumber\\
&&\times\left[F\cos\omega_Ft' + f(t')\right].
\end{eqnarray}
For small $f$ and $F$, one can then expand the $q$-dependent exponential in the right-hand side and use successive approximations in $F$ and $f$. The fluctuating susceptibility $\chi_1$ is given by linear in $F$ terms, whereas $\chi_2$ is given by the terms quadratic in $F$. The advantageous feature of this method is that it is not limited to white noise. However, the method becomes impractical if the noise intensity is not weak, and even for weak noise it becomes cumbersome if one goes to high-order terms in the noise intensity.
We have checked that the calculation based on Eq.~(\ref{eq:integral_equation}) gives the same result for the driving-induced part of the power spectrum $\Phi_F(\omega)$ as the method of moments. We have also found that, in the second order in the noise intensity $D$, the term $\gamma q^4/4$ in $U(q)$ leads to the onset of a peak in $\Phi_F(\omega)$ at $\omega_F$.
\subsection{Fluctuating susceptibility of a threshold detector}
\label{subsec:suscept_threshold}
Fluctuating linear susceptibility has a particularly simple form for a threshold detector. By linearizing in $F(t)$ expression (\ref{eq:threshold_model}) for the output of the detector, we obtain from the definition of the susceptibility (\ref{eq:susceptibility})
\begin{equation}
\label{eq:linear_fluct_suscept_threshold}
\chi(t,t') = 2\delta (t-t'-0)\delta\bigl(\xi(t)-\eta\bigr),
\end{equation}
where $\eta$ is the threshold and $\xi(t)$ is the noise. Zero in $\delta (t-t'-0)$ reflects causality: the detector output $q(t)$ is determined by the value of the driving just before the observation time; the very $\delta$-function indicates that the effect of the driving is not accumulated over time, the response is instantaneous (but causal).
The standard linear susceptibility $\chi(\omega)$ is given by expression
\[\chi(\omega) = \int_0^\infty dt e^{i\omega t}\langle \chi(t,0)\rangle.\]
From (\ref{eq:linear_fluct_suscept_threshold}), $\chi(\omega)=2\rho^{\xi}_{\rm st}(\eta)$. where $\rho^{\xi}_{\rm st}(\eta)$ is the stationary probability density of the noise $\xi(t)$, cf. Eq.~(\ref{eq:susceptibility_threshold}). It applies for an arbitrary noise $\xi(t)$, not just for the exponentially correlated noise considered in Sec.~\ref{sec:threshold}.
Similarly, the fluctuating nonlinear susceptibility of the detector is
\begin{equation}
\label{eq:nonlinear_susc_threshold}
\chi_2(t,t',t'')= -\delta(t-t'-0) \delta(t-t''-0)\partial_\eta \delta\bigl(\xi(t)-\eta\bigr).
\end{equation}
Substituting Eqs.~(\ref{eq:linear_fluct_suscept_threshold}) and (\ref{eq:nonlinear_susc_threshold}) into the general expressions for the power spectrum in terms of fluctuating susceptibilities, Eqs.~(\ref{eq:Phi_1_general}) and (\ref{eq:Phi_2_general}), we obtain the power spectrum in the same form as what follows from Eq.~(\ref{eq:threshold_correlator}).
\section*{References}
\providecommand{\newblock}{}
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\begin{document}
\baselineskip=17pt
\title{Weighted generalization of the Ramadanov theorem and further considerations.}
\author{by
\sc Zbigniew Pasternak-Winiarski \rm (Warsaw) \\
and
\sc Pawe{\l} M. W\'ojcicki \rm (Warsaw)
}
\date{}
\maketitle
\renewcommand{\thefootnote}{}
\footnote{2010 \emph{Mathematics Subject Classification}: Primary 32A36; Secondary 32A25.}
\footnote{\emph{Key words and phrases}: Weighted Bergman kernel; Admissible weight; Sequence of domains.}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\begin{abstract}
We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space $\CC^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given in \cite[p. 38]{Skwarczy\'nski}, highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover we will show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of Ramadanov's theorem holds.
\end{abstract}
\section{Introduction}
The Bergman kernel (see for instance \cite{Bergman, Jarnicki; Pflug, Krantz1, Krantz2, Shabat, Skwarczy\'nski; Mazur} ) has become a very important tool in geometric function theory, both in one and several complex variables. It turned out that not only classical Bergman kernel, but also weighted one can be useful. Let $D\subset\CC^N$ be a bounded domain. For example (see \cite{Englis}), if we denote by $\Pi : L^2(D)\rightarrow L^2_H(D)$ (the Bergman projection), we may define for any $\psi\in L^{\infty}(D)$, the Toeplitz operator $T_{\psi}$ as a (bounded linear) operator on $L^2_H(D)$ by $T_{\psi}f:=\Pi(\psi f)$. In particular, for $\psi>0$ on $D$ we have that $T_{\psi}$ is positive definite (so injective), so there exists an inverse $T_{\psi}^{-1}$. Taking positive continuous weight function $\mu\in L^{\infty}(D)$, $T_{\mu}$ extends to a bounded operator from $L^2_H(D,\mu)$ into $L^2_H(D)$, and $K_{D,\,\mu}(\cdot,x)=T_{\mu}^{-1}K_{D}(\cdot,x)$, where $K_{D,\,\mu}(\cdot,x)$ denotes the weighted Bergman kernel (associated to weighted Bergman space $L^2_H({D,\,\mu})$) at $x\in D$. Another practical application of weighted Bergman kernel may bo found in quantum theory (see \cite{Englis1} and \cite{Odzijewicz}, and \cite{Pasternak-Winiarski})- we may consider a K\"ahler manifold $\Omega$ as a classical phase space of a physical system (many leading quantized classical systems have such a phase space). The Hilbert space $H$ of quantum states of such a system consists of the holomorphic sections of some Hermitian line bundle $E$ over $\Omega$, which belong to $L^2(\Omega,\,\mu)$ (for the Liouville measure $\mu$ on $\Omega$). One of the most interesting and important objects of this model is the reproducing kernel $K$ of $H$ (that is the kernel $K_{\Omega,\,\mu}$ ). This kernel makes the quantization of classical states possible as follows : one can assign to any classical state $z\in\Omega$ the quantum state $$v_z:=[K(\cdot,z)/||K(\cdot,z)||]\in H.$$ Using this embedding one can calculate the transition probability amplitude from one point to another : $$a(z,w):=|<v_z | v_w>|,\quad z,w\in\Omega.$$ Then the calculation of the Feynman path integral for such a system is equivalent to finding the reproducing kernel $K$ (that is $K_{\Omega,\,\mu}$). \\
But in general, it is difficult to say anything about the unweighted (regular) or weighted kernel of a given domain. One of the classic results for unweighted Bergman kernels is Ramadanov's theorem (see \cite{Ramadanov}) :
\begin{thm}[Ramadanov]
Let $D_1\Subset D_2\Subset D_3\ldots$ be an increasing sequence of domains and set $D:=\bigcup_j D_j$. Then, $K_{D_j}\rightarrow K_D$ uniformly on compact subsets of $D\times D.$
\end{thm}
It is very natural to ask whether the similar theorem for weighted Bergman kernels is true. Let's recall the Forelli-Rudin construction (see \cite{Forelli; Rudin} and \cite{Ligocka}) :
If $\mu$ is a continuous weight on $D$ and $\Omega$ denotes the Hartogs domain $$\Omega=\{(z,w)\in D\times \CC^n : ||w||^{2n}<\mu(x)\}$$ in $\CC^{N+n} $, then $$K_{D,\mu}(z,p)=\frac{\pi^n}{n!}K_{\Omega}((z,0),(p,0))$$ (that is the weighted Bergman kernel $K_{D,\mu}(z,p)$ of $D$ is the restriction of the unweighted Bergman kernel $K_{\Omega}((z,w),(p,s))$ of $\Omega$ to the hyperplane $w=s=0$). Thus using Ramadanov's th. for the kernels $K_{\Omega_j}((z,0),(p,0))$ we can derive (under some conditions on weights - monotonicity for instance) the weighted analog of this theorem. And in fact, we may find some versions of this theorem in
({{\rm\cite[Prop. 3.17; Th. 3.18]{Jacobson}}} for instance), but considered weights are in the special form, as a moduli of holomorphic functions or $C^2$ functions, or as a product of one of those with the given weight $\psi$. Additionally, unweighted generalization of Ramadanov th. was given in \cite{Krantz} (in weighted case we can't proceed similarly, since we would have to strictly restrict weights of the kernels (we would need further assumptions for instance that $\mu_{\Omega_j}\circ\Theta_j=\mu_\Omega$ for any diffeomorphism $\Theta_j : \Omega\rightarrow\Omega_j$)). We can easily see that continuity of weight $\mu$ in the Forelli-Rudin construction provides basically that $\Omega$ is an open set. In this paper we will derive a weighted version of Ramadanov's theorem for so called "admissible weights" $\mu$ (we don't require $\mu$ to be continuous) without using Forelli - Rudin construction. It is very natural to consider such kind of weights, just by their definition (see below). We will prove the inverse of this theorem as well (see also \cite[p. 37]{Skwarczy\'nski} for an unweighted situation). In the second part of the paper we will show that density of holomorphic functions on a considered domain is very related to the convergence of the weighted Bergman kernels. In fact, we will get an equivalence in the unweighted case. This will provide with characterization of the domains, for which the inverse of Ramadanov's theorem holds. We shall start from the definitions and basic facts used in this paper.
\section{Definitions and notations}
Let $D\subset\CC^N$ be a domain, and let $W(D)$ be the set
of weights on $D$, i.e., $W(D)$ is the set of all Lebesque measurable, real - valued,
positive functions on $D$ (we consider two weights as equivalent if they are equal almost
everywhere with respect to the Lebesque measure on $D$).
If $\mu\in W(D)$, we denote by $L^2(D,\mu)$ the space of all Lebesque measurable, complex-valued,
$\mu$-square integrable functions on $D$, equipped with the norm $||\cdot||_{D, \mu}:=||\cdot||_{\mu}$
given by the scalar product
\begin{align*}
<f|g>_{\mu}:=\int_Df(z)\overline{g(z)}\mu(z)dV,\quad f, g\in L^2(D,\mu).
\end{align*}
The space $L^2_H(D,\mu)=H(D)\cap L^2(D,\mu)$ is called the \textbf{weighted Bergman space}, where $H(D)$ stands the space of all holomorphic functions on the domain $D$. For any $z\in D$ we define the evaluation functional $E_z$ on $L^2_H(D,\mu)$ by the formula
\begin{align*}
E_zf:=f(z),\quad f\in L^2_H(D,\mu).
\end{align*}
Let us recall the definition [Def. 2.1] of admissible weight given in \cite{Pasternak-Winiarski1}.
\begin{defin}[Admissible weight]
A weight $\mu\in W(D)$ is called an{ \textit{admi\-ssible weight} }, an a-weight for short, if $L^2_H(D,\mu)$
is a closed subspace of $L^2(D,\mu)$ and for any $z\in D$ the evaluation functional $E_z$ is
continuous on $L^2_H(D,\mu)$. The set of all a-weights on $D$ will be denoted by $AW(D)$.
\end{defin}
The definition of admissible weight provides us basically with existence and uniqueness of related Bergman kernel and completeness of the space $L^2_H(D,\mu)$.
In \cite{Pasternak-Winiarski} the concept of a-weight was introduced, and in \cite{Pasternak-Winiarski1} several theorems concerning admissible
weights are given. An illustrative one is :
\begin{thm}\cite[Cor. 3.1]{Pasternak-Winiarski1}
Let $\mu\in W(D)$. If the function $\mu^{-a}$ is locally integrable on $D$ for some $a>0$ then $\mu\in AW(D)$.
\end{thm}
Now, let 's fix a point $t\in D$ and minimize the norm $||f||_{\mu}$ in the class $E_t=\{f\in L^2_H(D,\mu); f(t)=1\}$.
It can be proved in a similar way as in the classical case, that if $\mu$ is an admissible weight then there exists exactly one function minimizing the norm. Let us denote it by $\phi_{\mu}(z,t)$.
\textbf{Weigted Bergman kernel function $K_{D,\,\mu}$} is defined as follows :
\begin{align*}
K_{D,\,\mu}(z,t)=\frac{\phi_{\mu}(z,t)}{||\phi_{\mu}||_{\mu}^2}.
\end{align*}
\section{Variations on the Ramadanov theorem and domain dependence.}
In this section we study the limit behavior of weighted Bergman kernels for admissible weights. Moreover we give a weighted characterization of the Bergman kernel (see also \cite[p. 36]{Skwarczy\'nski}) by means of which we prove kind of converse of Ramadanov theorem. We show that density of holomorphic functions is very related to the convergence of weighted Bergman kernels, and in the case of $\mu_n\equiv 1$ we even have an equivalence (see also \cite{Skwarczy\'nski; Iwi\'nski}).
\subsection{Weighted generalization of the Ramadanov theorem }
\begin{mainthm}[Weighted generalization of the Ramadanov theorem]
Let $\{D_i\}_{i=1}^{\infty}$ be a sequence of domains in $\CC^N$
and set $D:=\bigcup_jD_j$. Let $\mu\in AW(D),\mu_k\in AW(D_k)$ (extend $\mu_k$ by $\mu$ on $D$).
Assume moreover that
\begin{enumerate}
\item[a)]For any $n\in\NN$ there is $N=N(n)$ s.t. $D_n\subset D_m$ and $\mu_n(z)\leq\mu_m(z)\leq\mu(z)$ for $m\geq N(n),\,\,z\in D_n$.
\item[b)] $\displaystyle\mu_k\xrightarrow[k\to\infty]{}\mu$ pointwise a.e. on $D$.
\end{enumerate}
Then
\begin{align*}
\lim_{k\to\infty}K_{D_k, \mu_k}=K_{D, \mu}
\end{align*}
locally uniformly on $D\times D$.
\end{mainthm}
The first step in the proof is to show the monotonicity property for the weighted kernels.
Then we should check that the limit of the sequence of weighted kernels of the domains $D_n$,
if exists, is equal to $K_{D,\mu}$.
\begin{lem}[Monotonicity property]
For any $n\in\NN,\,\,t\in D_n$ the inequality $K_{D_n, \mu_n}(t,t)\geq K_{D_m, \mu_m}(t,t)$ holds for $m\geq N(n)$.
\end{lem}
\begin{proof}
Let us fix $n\in\NN,\,\,t\in D_n$. Let $m\geq N(n)$.
The inequality in the statement of the lemma is true if $K_{D_m,\,\mu_m}(t,t)=0$. Then suppose that
$K_{D_m,\,\mu_m}(t,t)>0$.
In the proof we will use the simple remark that
\begin{align*}
\frac{1}{K_{D_n,\,\mu_n}(t,t)}= \int_{D_n}\left|\frac{K_{D_n,\,\mu_n}(s,t)}{K_{D_n,\,\mu_n}(t,t)}\right|^2\mu_n(s)dV
\end{align*}
since
$K_{D_n,\,\mu_n}(t,t)>0$ and
\begin{align*}
K_{D_n,\,\mu_n}(t,t)=\displaystyle\int_{D_n}\overline{K_{D_n,\,\mu_n}(z,t)}
K_{D_n,\,\mu_n}(z,t)\mu_n(z)dV
\end{align*}
by the reproducing property (\cite{Pasternak-Winiarski}) for $f(\cdot)=K_{D_n,\,\mu_n}(\cdot,t)$. Moreover the term
$\displaystyle\frac{K_{D_n,\,\mu_n}(\cdot,t)}{K_{D_n,\,\mu_n}(t,t)}$
is the only element in the class $\{f\in L^2_H(D_n,\mu_n), f(t)=1 \}$ with the minimal norm.
Thus for $m\geq N(n)$ we have
\begin{align*}
\frac{1}{K_{D_n,\,\mu_n}(t,t)}
&\leq \int_{D_n}\left|\frac{K_{D_m,\,\mu_m}(s,t)}{K_{D_m,\,\mu_m}(t,t)}\right|^2\mu_n(s)dV\\[12pt]
&\leq \int_{D_n}\left|\frac{K_{D_m,\,\mu_m}(s,t)}{K_{D_m,\,\mu_m}(t,t)}\right|^2\mu_m(s)dV \\[12pt]
&\leq \int_{D_m}\left|\frac{K_{D_m,\,\mu_m}(s,t)}{K_{D_m,\,\mu_m}(t,t)}\right|^2\mu_m(s)dV=
\frac{1}{K_{D_m,\,\mu_m}(t,t)}.
\end{align*}
\end{proof}
\begin{rem}
One can show similarly that $K_{D_n,\,\mu_n}(t,t)\geq K_{D,\,\mu}(t,t)$ for \\$n\in\NN$.
\end{rem}
\begin{lem}[Uniqueness of the limit]
If $\displaystyle\lim_{n\to\infty}K_{D_n,\,\mu_n}=k$ locally uniformly on $D\times D$, then
$k=K_{D,\,\mu}$.
\end{lem}
\begin{proof}
Since the sequence $(K_{D_n,\,\mu_n})$ converges locally uniformly on $D\times D$ and any function
$K_{D_n,\,\mu_n}$ is continuous we obtain that $k$ is continuous on $D\times D$.
Let's recall that
\begin{align}\label{E:1}
\int_{D_m}\overline{K_{D_m,\,\mu_m}(z,t)}K_{D_m,\,\mu_m}(z,t)
\mu_m(z)dV=K_{D_m,\,\mu_m}(t,t).
\end{align}
Fix a compact set $E\subset D$, and $t\in E$. For $m$ large enough $E\subset D_m$ and $t\in D_m$.
By Fatou's lemma
\begin{align*}
\int_E|k(z,t)|^2\mu(z)dV
&\leq \liminf_{m\to\infty}\int_{E}|K_{D_m,\,\mu_m}(z,t)|^2\mu_m(z)dV \\[12pt]
&\leq \liminf_{m\to\infty}\int_{D_m}|K_{D_m,\,\mu_m}(z,t)|^2\mu_m(z)dV\\[12pt]
&= \liminf_{m\to\infty}K_{D_m,\,\mu_m}(t,t)=k(t,t).
\end{align*}
Since $E$ is an arbitrary compact set,
\begin{align}\label{E:1}
&\int_D|k(z,t)|^2\mu(z)dV\leq k(t,t).
\end{align}
By Weierstrass theorem $k(\cdot,t)\in H(D)$, so $k(\cdot,t)\in L^2_H(D,\mu)$.
\newline By Lemma $5$ we get
\begin{align*}
K_{D_n,\,\mu_n}(t,t)\geq K_{D,\,\mu}(t,t)
\end{align*}
for $n=1, 2,\ldots,\quad t\in D$. In the limit $n\to\infty$ we obtain
\begin{align*}
k(t,t)\geq K_{D,\,\mu}(t,t).
\end{align*}
It suffices to show that $k(z,t)=K_{D,\,\mu}(z,t)$.
We should consider two cases :
\\
\\
$1.$ $K_{D,\,\mu}(t,t)=0$, for some $t\in D.$
\\
\\
Then for $z\in D$,
$K_{D,\,\mu}(z,t)=0$ since $\displaystyle K_{D,\,\mu}(t,t)=\int_D|K_{D,\,\mu}(z,t)|^2\mu(z)dV,$
and $K_{D,\,\mu}$ is continuous with respect to $z.$ Thus for any $f\in L^2_H(D,\mu)$
\begin{align*}
f(t)=\int_Df(w)K_{D,\,\mu}(t,w)\mu(w)dV=0
\end{align*}
and we have that $k(t,t)=0$, since $f(\cdot):=k(\cdot,t)\in L^2_H(D,\,\mu).$ \\But $\displaystyle\int_D|k(z,t)|^2\mu(z)dV\leq k(t,t),$
so $k(z,t)=0$, for $z\in D.$
\\
\\
$2.$ $K_{D,\,\mu}(t,t)>0$, for some $t\in D$.
\\
\\
Then $k(t,t)>0,$ since $k(t,t)\geq K_{D,\,\mu}(t,t)>0.$ We will use once more the well known fact, that in the set $\{f\in L^2_H(D,\mu), f(t)=1 \}$ (for some fixed $t\in D$) function
$\displaystyle\frac{K_{D,\,\mu}(\cdot,t)}{K_{D,\,\mu}(t,t)}$ is the only minimal element.
It is easy to see, that $\displaystyle\frac{k(\cdot,t)}{k(t,t)}$ belongs to this set
(since $k(\cdot,t)\in L^2_H(D,\mu)$) and moreover by (2) $||k(\cdot,t)||_{\mu}\leq\sqrt{k(t,t)}.$ \\Thus
\begin{align*}
\left|\left|\frac{k(\cdot,t)}{k(t,t)}\right|\right|_{\mu}
&\leq \frac{\sqrt{k(t,t)}}{k(t,t)}=\frac{1}{\sqrt{k(t,t)}}\leq \frac{1}{\sqrt{K_{D,\,\mu}(t,t)}}\\[12pt]
&= \left|\left|\frac{K_{D,\,\mu}(\cdot,t)}{K_{D,\,\mu}(t,t)}\right|\right|_{\mu}.
\end{align*}
By the minimality property of $\displaystyle\frac{K_{D,\,\mu}(\cdot,t)}{K_{D,\,\mu}(t,t)}$
we get from the above, that :
\begin{align*}
\left|\left|\frac{k(\cdot,t)}{k(t,t)}\right|\right|_{\mu}=
\frac{1}{\sqrt{k(t,t)}}= \frac{1}{\sqrt{K_{D,\,\mu}(t,t)}}=
\left|\left|\frac{K_{D,\,\mu}(\cdot,t)}{K_{D,\,\mu}(t,t)}
\right|\right|_{\mu}.
\end{align*}
So $k(t,t)=K_{D,\,\mu}(t,t)$, and $k(z,t)=K_{D,\,\mu}(z,t)$ for $z, t\in D$.
\end{proof}
\begin{proof}[Proof of the main theorem]
We will show that for $n\in\NN$ the sequence $\{K_{D_m,\,\mu_m}\}_{m\geq N(n)}$ is locally bounded on $D_n\times D_n$.
\\
Using well known version of Schwarz inequality for reproducing kernels and Lemma $5$
we obtain for any $z, t\in D_n$.
\begin{align*}
|K_{D_m,\,\mu_m}(z,t)|
&\leq \sqrt{K_{D_m,\,\mu_m}(z,z)}\sqrt{K_{D_m,\,\mu_m}(t,t)}\\[12pt]
&\leq \sqrt{K_{D_n,\,\mu_n}(z,z)}\sqrt{K_{D_n,\,\mu_n}(t,t)},\quad m\geq N(n).
\end{align*}
The term in the right hand side of the estimation above is locally bounded on $D_n\times D_n.$
By Montel's property, any subsequence of $\{K_{D_m,\,\mu_m}\}$ has a subsequence convergent locally
uniformly on $D\times D$. By Lemma $7$ the limit does not depend on a subsequence and is
identically equal to $K_{D,\,\mu}$. Thus
\begin{align*}
\lim_{m\to\infty}K_{D_m,\,\mu_m}(z,t)=K_{D,\,\mu}(z,t)
\end{align*}
locally uniformly on $D\times D$.
\end{proof}
\begin{rem}
Look that the case of increasing sequence of domains is a subcase of the Main Theorem $4$ (see \cite{Skwarczy\'nski; Mazur} for the very interesting considerations, and unweighted kind of Lemma $7$).
\end{rem}
\subsection{Characterization of the weighted Bergman kernel and further remarks on "decreasing-like"
sequence of domains}
In \cite[p. 36]{Skwarczy\'nski} some characterization lemma for unweighted Bergman kernels is given.
One can easily conclude similar one for weighted Bergman kernels, as the following Lemma $9$ shows.
The proof is attached only for the convenience of the reader.
\begin{lem}
Denote by $S_{\mu,\,t}\subset L^2_H(D,\mu)$ the set of all functions $f$ such that $f(t)\geq 0$
and $||f||_{\mu}\leq\sqrt{f(t)}$, where $t\in D$ is fixed. Then the weighted Bergman
function $\varphi_{\mu,\,t}(\cdot):=K_{D,\,\mu}(\cdot,t)$ is uniquelly characterized by the proper\-ties :
\begin{enumerate}
\item[(i)] $\varphi_{\mu,\,t}\in S_{\mu,\,t}$
\item[(ii)] if $f\in S_{\mu,\,t}$ and $f(t)\geq \varphi_{\mu,\,t}(t)$, then $f\equiv\varphi_{\mu,\,t}$.
\end{enumerate}
\end{lem}
\begin{proof}
One can easily see, that there exists at most one element $\varphi_{\mu,\,t}\in L^2_H(D,\mu)$
which satisfies (i) and (ii) (if $\varphi_1, \varphi_2$ satisfies (i) and (ii), then both $\varphi_1(t)$ and $\varphi_2(t)$ are nonnegative, and either $\varphi_1(t)\geq\varphi_2(t)$ and then $\varphi_1\equiv\varphi_2$ or $\varphi_2(t)\geq\varphi_1(t)$ and then $\varphi_2\equiv\varphi_1$).
We shall show $\varphi_{\mu,\,t}(\cdot)=K_{D,\,\mu}(\cdot,t)$ has both properties.
\newline We have
\begin{align*}
\varphi_{\mu,\,t}(t)=K_{D,\,\mu}(t,t)\geq 0
\end{align*}
and
\begin{align*}
||K_{D,\,\mu}(\cdot,t)||_{\mu}^2=K_{D,\,\mu}(t,t)
\end{align*}
(see (1)).
Now let $f\in S_{\mu,\,t}$. If $f(t)=0$, then $\varphi_{\mu,\,t}(t)=0$. Hence $||f||_\mu =
||\varphi_{\mu,\,t}||_\mu=0$, so $f\equiv 0\equiv \varphi_{\mu,\,t}$.
\\
Assume now $f(t)>0$. By the definition of weighted Bergman kernel function $\displaystyle\frac
{\varphi_{\mu,\,t}(\cdot)}{\varphi_{\mu,\,t}(t)}$ is uniquely characterized as an element in the set
$\{h\in L^2_H(D,\mu), \newline h(t)=1\}$ with the minimal norm. But $\displaystyle\frac{f(\cdot)}{f(t)}$
belongs to this set as well, moreover
\begin{align*}
\Bigg\|\frac{f(\cdot)}{f(t)}\Bigg\|_\mu=
\frac{||f||_\mu}{\sqrt{f(t)}\sqrt{f(t)}}
\leq\frac{1}{\sqrt{f(t)}}\leq\frac{1}{\sqrt{\varphi_{\mu,\,t}(t)}}=
\Bigg\|\frac{\varphi_{\mu,\,t}(\cdot)}{\varphi_{\mu,\,t}(t)}\Bigg\|_\mu.
\end{align*}
Thus (by minimality)
\begin{align*}
\frac{1}{\sqrt{f(t)}}=\frac{1}{\sqrt{\varphi_{\mu,\,t}(t)}}
\end{align*}
and for any $z\in D$
\begin{align*}
\frac{f(z)}{f(t)}=\frac{\varphi_{\mu,\,t}(z)}{\varphi_{\mu,\,t}(t)}.
\end{align*}
So $f\equiv\varphi_{\mu,\,t}$.
\end{proof}
Let's assume that $D={\rm{int}}(\overline{D})$ to exclude slit domains from our considerations (a disc with
one radius removed for instance) and consider "decreasing - like" version of the Ramadanov th. Let us recall the definition of "approximation from outside" given in \cite[Def. V.6; p. 38]{Skwarczy\'nski}.
\begin{defin}
We say that a sequence of domains $\{D_n\}_{n=1}^{\infty}$ approximates $D$ from outside if $D\subset D_n$ for all $n$ and for each open $G$ such that $\overline{D}\subset G$ the inclusion $D\subset D_m\Subset G$ holds for all sufficiently large $m$.
\end{defin}
\begin{mainthm}
Let $\{D_n\}_{n=1}^{\infty}$ be a sequence of domains in $\CC^N$ which approximates $D$ from outside and $\mu\in AW(D),\mu_k\in AW(D_k)$ (extend $\mu_k$ by $\mu_n$ on $D_n$ for $k\geq n$, and $\mu$ by $\mu_n$ on $D_n$). Assume moreover that
\begin{enumerate}
\item[a)]$\mu(z)\leq\mu_m(z)$\quad for \quad $m\in\NN,\,\,z\in D$.
\item[b)]$\displaystyle\mu_k\xrightarrow[k\to\infty]{}\mu$ pointwise a.e. on $D$.
\end{enumerate}
Then $\{K_{D_m,\,\mu_m}\}_{m=1}^{\infty}$ converges to $K_{D,\,\mu}$ locally uniformly
on $D\times D$ iff for any fixed $t\in D$
\begin{align*}
\lim_{m\to\infty}K_{D_m,\,\mu_m}(t,t)=K_{D,\mu}(t,t).
\end{align*}
\end{mainthm}
\begin{proof}
We shall only make sure, that the converse implication is true, since the necessity is obvious.
Let $F\subset D$ be a compact set. Then there is a constant $M=M(F)$ such that $\displaystyle\max_{z\in F}|K_{D,\,\mu}(z,z)|\leq M$.
By Schwarz inequality
\begin{align*}
|K_{D_m,\,\mu_m}(z,t)|
&\leq \sqrt{K_{D_m,\,\mu_m}(z,z)}\sqrt{K_{D_m,\,\mu_m}(t,t)}\\[12pt]
&\leq \sqrt{K_{D,\,\mu}(z,z)}\sqrt{K_{D,\,\mu}(t,t)}\leq M.
\end{align*}
for any $z,\,t\in F$. Thus $\{K_{D_m,\,\mu_m}\}_{m=1}^{\infty}$ is a Montel family on $D\times D$. It suffices
to show that every convergent subsequence of this family converges to $K_{D,\,\mu}$.
With no loss of generality let us consider $\{K_{D_m,\,\mu_m}\}$ itself and assume
that it does converge to some $k$. For $t\in D$, by Fatou's lemma
\begin{align*}
\int_F|k(z,t)|^2\mu(z)dV
&\leq \liminf_{m\to\infty}\int_F|K_{D_m,\,\mu_m}(z,t)|^2\mu_m(z)dV\\[12pt]
&\leq \liminf_{m\to\infty}\int_{D_m}|K_{D_m,\,\mu_m}(z,t)|^2\mu_m(z)dV \\[12pt]
&= \liminf_{m\to\infty}K_{D_m,\,\mu_m}(t,t)=K_{D,\,\mu}(t,t)=k(t,t).
\end{align*}
Since $F\subset D$ is an arbitrary compact set,
\begin{align*}
||k(\cdot,t)||_{\mu}^2\leq k(t,t)=K_{D,\,\mu}(t,t)<\infty.
\end{align*}
Thus taking in Lemma $9$ $f(\cdot)=k(\cdot,t)$ we obtain $K_{D,\,\mu}(z,t)=k(z,t)$ for any $z,\,t\in D$.
\end{proof}
\begin{rem}
Look that decreasing sequence of domains satisfies assumptions of the Main Theorem $11$.
This theorem for classical Bergman kernels and decreasing case of domains could be found in \cite[p. 37]{Skwarczy\'nski}. Main Theorem $4$ could be proved in the same fashion using Lemma $9$ (look in \cite{W\'ojcicki}).
\end{rem}
\subsection{Domain dependence}
In this paragraph, among others, we will give a generalization of \cite[p. 38]{Skwarczy\'nski} for weighted Bergman kernels. Moreover we will show that the converse of this theorem holds as well. We shall start with notation used in this paragraph. The first thing is to extend the weights outside its natural domain (the domain may intersect). Let $\{D_n\}_{n=1}^{\infty}$ be an approximating sequence for $D$. Let $\mu_n\in AW(D_n)$ for $n\in\NN$. Then for $k\geq 2$ we define :
\begin{displaymath}
\widetilde{\mu_k(z)} = \left\{ \begin{array}{ll}
\mu_k(z) & \textrm{ for $z\in D_k$ }\\
\widetilde{\mu_{k-1}(z)} & \textrm{ for $z\in D_1\cup\ldots\cup D_{k-1}\setminus D_k$ }
\end{array} \right.
\end{displaymath}
See that $\widetilde{\mu_k(z)}$ is well defined on $D_1\cup\ldots\cup D_k$.
For example
\begin{displaymath}
\widetilde{\mu_3(z)} = \left\{ \begin{array}{ll}
\mu_3(z) & \textrm{ for $z\in D_3$ }\\
\widetilde{\mu_2(z)} & \textrm{ for $z\in D_1\cup D_{2}\setminus D_3$ }
\end{array} \right.
= \left\{ \begin{array}{ll}
\mu_3(z) & \textrm{ for $z\in D_3$ }\\
\mu_2(z) & \textrm{ for $z\in D_{2}\setminus D_3$ }\\
\mu_1(z) & \textrm{ for $z\in(D_1\setminus D_{2})\setminus D_3$ }
\end{array} \right.
\end{displaymath}
We will now define the extension of $\mu(z)$ outside $D$.
Since $\{D_n\}_{n=1}^{\infty}$ is an approximating sequence for $D$ then for large $s\in\NN$ we have $D\subset D_s\Subset D_1$.
We define
\begin{displaymath}
\widetilde{\mu(z)} = \left\{ \begin{array}{ll}
\mu_1(z) & \textrm{ for $z\in D_1$ }\\
\mu_2(z) & \textrm{ for $z\in D_2\setminus D_1$ }\\
\mu_3(z) & \textrm{ for $z\in D_3\setminus(D_1\cup D_2)$ }\\
\ldots \\
\mu_{s-1}(z)&\textrm{ for $z\in D_{s-1}\setminus(D_1\cup D_2\ldots\cup D_{s-2})$ }
\end{array} \right.
\end{displaymath}
Let $E\subset\CC^N$ be Lebesque measurable, $\mu$ be a-weight on $E$ (we set that $\mu\in AW(E)$ if for some open set $W\subset\CC^N$ s.t. $E\subset W$ there is $\nu\in AW(W)$ s.t. $\nu_{|E}=\mu$) and $L^2(E,\mu)$ be the Hilbert space of all complex-valued functions which are square $\mu-$ integrable on a set $E$ and holomorphic in the interior of $E$. Let's moreover $H(E,\mu)$ be the subset of $L^2(E,\mu)$ consisting of all functions possessing holomorphic extension to an open neighborhood of $E$. We will need the following :
\begin{propp}
$H(E,\mu)$ is dense in $L^2(E,\mu)$.
\end{propp}
\begin{mainthm}
Suppose that $\overline{D}$ has Property $13$, and
a sequence $D_m$ approximates $D$ from outside.
Let $\mu\in AW(\overline{D}),\mu_k\in AW(D_k)$ (extend $\mu_k$ and $\mu_n$ as mentioned above).
Assume moreover, that for some $p\in\NN$
\begin{enumerate}
\item[a)]$\mu(z)\leq\mu_m(z)\leq\mu_{p}(z)$\,\,for\,\,$p\leq m$ and $z\in D_{p}$
\item[b)]$\mu_{p}\in L^1(D_{p})$
\item[c)]$\displaystyle\mu_k\xrightarrow[k\to\infty]{}\mu$ pointwise a.e. in $D_{p}$
\end{enumerate}
Then
\begin{align*}
\lim_{m\to\infty}K_{D_m,\mu_m}=K_{D,\mu}
\end{align*}
locally uniformly on $D\times D$.
\end{mainthm}
\begin{proof}
Let $t\in D$ and $f\in L^2_H(D,\mu)$ be fixed. We can extend $f$ by $0$ on $\partial D$, to provide
$f\in L^2(\overline{D},\mu)$. Consider any $h\in H(\overline{D},\mu)$. Then for $m$ large enough, $h\in L^2_H(D_m,\mu_m)$ (because of $b)$).
We have
\begin{align*}
|h(t)|
&= \left|\int_{D_m}h(z)K_{D_m,\,\mu_m}(z,t)\mu_m(z)dV\right|\\[12pt]
&= \left|\int_{D_m}h(z)\mu_m(z)^{1/2}K_{D_m,\,\mu_m}(z,t)\mu_m(z)^{1/2}dV\right| \\[12pt]
&\leq ||h||_{\mu_m}\left(\int_{D_m}|K_{D_m,\mu_m}(z,t)|^2\mu_m(z)dV\right)^{1/2}\\[12pt]
&= ||h||_{\mu_m} K_{D_m,\mu_m}(t,t)^{1/2}.
\end{align*}
In the limit $m\to\infty$ we get (by Domin. Conv. Th.) $|h(t)|\leq k(t,t)^{1/2}||h||_{E,\mu}$\,,
\newline where $\displaystyle k(t,t)=\lim_{m\to\infty}
K_{D_m,\mu_m}(t,t)$. By density Property $13$ there is a sequence $\{h_m\}$ of functions in $H(\overline{D},\mu)$ such that $h_m\xrightarrow{L^2(\overline{D},\mu)}f$ (thus locally uniformly on $D$). So
\begin{align*}
|f(t)|\leq k(t,t)^{1/2}||f||_{\overline{D},\mu}= k(t,t)^{1/2}||f||_{D,\mu}.
\end{align*}
So for $f(t)=K_{D,\mu}(t,t)$ we have
\begin{align*}
|K_{D,\mu}(t,t)|=K_{D,\mu}(t,t)\leq k(t,t)^{1/2}||K_{D,\mu}||_{\mu}=k(t,t)^{1/2}K_{D,\mu}(t,t)^{1/2}.
\end{align*}
Thus $K_{D,\mu}(t,t)\leq k(t,t)$. Obviously $K_{D_m,\mu_m}(t,t)\leq K_{D,\mu}(t,t)$. In the limit
$m\to\infty$ we get $k(t,t)\leq K_{D,\mu}(t,t)$. Therefore
\begin{align*}
K_{D,\mu}(t,t)=k(t,t)=\lim_{m\to\infty}K_{D_m,\mu_m}(t,t).
\end{align*}
The hypothesis follows from the Main Theorem $11$.
\end{proof}
What is interesting, it turns out that some kind of the converse of Main Theorem $14$ holds as well, namely :
\begin{mainthm}
Let $D\subset\CC^N$ be a domain and let $\mu$ be a weight on the closure $\overline{D}$ of $D$ in $\CC^N$ s.t. the measure of the boundary $\partial D$ given by $\mu$ is equal to $0$ and $\mu_{|D}\in AW(D)$. Suppose that for some sequence $\{D_n\}$ approximating $D$ from outside, and some sequence of admissible weights $\{\mu_n\}$
(where $\mu_n\in AW(D_n)$)
\begin{align*}
\lim_{n\to\infty}{K_{D_n, \mu_n}}_{|D}=K_{D, \mu}
\end{align*}
holds locally uniformly on $D\times D$; for any $t\in D, K_{D_n,\mu_n}(\cdot,t)\in L^2_H(D,\mu)$ and
\begin{align*}
\lim_{n\to\infty}||K_{D_n,\mu_n}(\cdot,t)||^2_{\mu}=||K_{D,\mu}(\cdot,t)||^2_{\mu}=K_{D,\mu}(t,t).
\end{align*}Then Property $13$ holds.
\end{mainthm}
\begin{proof}
For any $t\in D$ we have
\begin{align*}
& ||{K_{D_n, \mu_n}}_{|D}(\cdot,t)-K_{D,\mu}(\cdot,t)||^2_{\mu} \\[12pt]
&= \int_D(K_{D_n,\mu_n}(z,t)-K_{D,\mu}(z,t))\overline{(K_{D_n,\mu_n}(z,t)-K_{D,\mu}(z,t))}\mu(z)dV\\[12pt]
&= \int_D|K_{D_n,\mu_n}(z,t)|^2\mu(z)dV - \int_D K_{D,\mu}(t,z)K_{D_n,\mu_n}(z,t)\mu(z)dV\\[12pt]
&- \overline{\int_D K_{D,\mu}(t,z)K_{D_n,\mu_n}(z,t)\mu(z)dV}+\int_D|K_{D,\mu}(z,t)|^2\mu(z)dV \\[12pt]
&= ||K_{D_n,\mu_n}(\cdot,t)||^2_{\mu}-K_{D_n,\mu_n}(t,t)-\overline{K_{D_n,\mu_n}(t,t)}+K_{D,\mu}(t,t)\\[12pt]
&= ||K_{D_n,\mu_n}(\cdot,t)||^2_{\mu}-2K_{D_n,\mu_n}(t,t)+K_{D,\mu}(t,t).
\end{align*}
By assumptions
\begin{align*}
\lim_{n\to\infty}||{K_{D_n,\mu_n}}_{|D}(\cdot,t)-K_{D,\mu}(\cdot,t)||^2_{\mu}=0.
\end{align*}
which means that the closure in $L^2-$ norm
\begin{align*}
cl\{K_{D,\,\mu}(\cdot,t),t\in D\}\subset cl\{K_{D_n,\,\mu_n}(\cdot,t), t\in D, n\in\NN \}\subset L^2_H(\overline{D},\mu).
\end{align*}
On the other hand, by reproducing property (\cite{Pasternak-Winiarski})
\begin{align*}
cl \{K_{D,\,\mu}(\cdot,t),t\in D\}=L^2_H(D,\mu)=L^2_H(\overline{D},\mu).
\end{align*}
Taking into account that
\begin{align*}
\{K_{D_n,\mu_n}(\cdot,t),t\in D, n\in\NN \}\subset H(\overline{D},\mu)
\end{align*}
we obtain desired result.
\end{proof}
\begin{rem}
Look also in \cite{Skwarczy\'nski; Iwi\'nski} for some considerations concerning unweighted, decreasing case of Main Thm. $15$ and very interesting remarks. Look that taking for any $n,\,\,\mu_n\equiv 1$ we get in fact that Property $13$ and hypothesis of the Main Theorem $14$ are equivalent, which gives us a description of the domains, for which "decresing-like" version of Ramadanov theorem holds. Moreover, using Main Theorem $4$ we can prove a weighted version of counterexample to the Lu Qi-Keng conjecture given in \cite{Boas}.
\end{rem}
\section*{Acknowledgements}
We are very grateful to Prof. Harold Boas for his very helpful advices on the Main Theorem $4$ and other suggestions. We would like to express our thanks to Prof. Maciej Skwarczy\'nski $(\dagger)$ for his passion and a great book for students, and his contribution to the theory of Bergman kernels. We would like to thank our families for supporting us through their presence.
| 106,638
|
\begin{document}
\title{Clusters, currents and Whitehead's algorithm}
\author{\Large Ilya Kapovich}
\date{}
\maketitle
\begin{abstract}
Using geodesic currents, we provide a theoretical justification
for some of the experimental results, obtained by Haralick,
Miasnikov and Myasnikov via pattern recognition methods, regarding
the behavior of Whitehead's algorithm on non-minimal inputs. In
particular we prove that the images of ``random" elements of a
free group $F$ under the automorphisms of $F$ form ``clusters"
that share similar normalized Whitehead graphs and similar
behavior with respect to Whitehead's algorithm.
\end{abstract}
\section{Introduction}\label{intro}
The \emph{automorphism problem} for a free group $F=F(a_1,\dots,
a_k)$, where $k\ge 2$, asks, given two arbitrary elements $u,v\in
F$, whether there exists $\phi\in Aut(F)$ such that $\phi(u)=v$.
In a classic 1936 paper~\cite{Wh} Whitehead provided an algorithm
solving the automorphism problem. He introduced a special finite
generating set of $Aut(F)$, consisting of the so-called
\emph{Whitehead automorphisms}. He proved that if $u\in F$ is a
cyclically reduced word that is not shortest in its
$Aut(F)$-orbit, then there exists a Whitehead automorphism $\tau$
such that $\tau(u)$ has smaller cyclically reduced length than
$\tau$. This provides a quadratic time algorithm for finding a
\emph{minimal} element in the orbit $Aut(F)f$ for any $f\in F$,
that is, the element of smallest length in $Aut(F)f$. Namely,
first cyclically reduce $f$ to get $f'\in F$, and then check if
there is a Whitehead automorphism $\tau$ that decreases the
cyclically reduced length of $f'$. If not, then $f'$ is minimal.
If yes, replace $f'$ by $\tau(f)$ and then repeat the entire step.
Whitehead also proved that if $u,v\in F$ are cyclically reduced
minimal elements of the same length, then $v\in Aut(F)u$ if and
only if there exists a chain of Whitehead automorphisms taking $u$
to $v$ and such that the cyclically reduced length is constant
throughout the chain. Together with the above procedure for
computing minimal representatives, this provides an algorithm for
solving the automorphism problem that runs in at most exponential
time in terms of $|u|+|v|$. The second, ``hard'' part of
Whitehead's algorithm, has an a priori exponential time upper
bound for the running time, although in practice the algorithm
appears to always terminate much faster.
Since this 1936 paper of Whitehead there has been a great deal of
work on the study of the automorphism problem and of Whitehead's
algorithm (e.g. see the recent paper of Lee~\cite{Lee}). However,
even now, 70 years later, it is still not known what the precise
complexity of Whitehead's algorithm is or if there exists a
polynomial time algorithm for solving the automorphism problem in
a free group. The only well-understood case is $k=2$, where it is
known that the automorphism problem is indeed solvable in
polynomial time~\cite{MS,Khan}.
A recent paper of Kapovich, Schupp and Shpilrain~\cite{KSS} proves
that for any $k\ge 2$ Whitehead's algorithm has linear time
generic-case complexity. It turns out that ``random'' cyclically
reduced elements of $F$ are already minimal, so that the first
(minimization) part of Whitehead's algorithm terminates in a
single step. Moreover, even the second ``hard'' part of the
algorithm is also proved in \cite{KSS} to run in at most linear
time in this case.
It is therefore interesting to understand the behavior of Whitehead's
algorithm on non-minimal inputs that are also generated via some
natural probabilistic process. A.~D.~Miasnikov, A.~G.~Myasnikov and
R.~Haralick~\cite{HMM1,HMM2,HMM3}, via pattern recognition methods,
experimentally discovered some interesting features of the behavior of
Whitehead's algorithm in this set-up. Before discussing their
observations, we need to fix some notations.
\begin{conv}
For the remainder of the paper let $F=F(A)$ be a free group with a
fixed free basis $A=\{a_1,\dots, a_k\}$, where $k\ge 2$. Let
$X=\Gamma(F,A)$ be the Cayley graph of $F$ with respect to $A$, so
that $X$ is a $(2k)$-regular tree.
Denote $\Sigma=A\cup A^{-1}=\{a_1,\dots, a_k, a_1^{-1}, \dots,
a_k^{-1}\}$.
For a word $w$ in $\Sigma^*$ we will denote the length of $w$ by
$|w|$. A word $w\in \Sigma^*$ is said to be \emph{reduced} if $w$ is
freely reduced in $F$, that is $w$ does not contain subwords of the
form $a_ia_i^{-1}$ or $a_i^{-1}a_i$. A word $w$ is \emph{cyclically
reduced} if all cyclic permutations of $w$ are reduced. (In
particular $w$ itself is reduced.) We denote by $C$ the set of all
nontrivial cyclically reduced words in $F$.
Since every element of $F$ can be uniquely represented by a freely
reduced word, we identify elements of $F$ and freely reduced words.
Any freely reduced element $w$ can be uniquely decomposed as a
concatenation $w = vuv^{-1}$ where $u$ is a cyclically reduced word.
The word $u$ is called the \emph{cyclically reduced form of $w$} and
$||w||:=|u|$ is the \emph{cyclic length} of $w$.
\end{conv}
Some of the experimental conclusions of A.~D.~Miasnikov,
A.~G.~Myasnikov and R.~Haralick, described in detail in \cite{HMM3},
can be summarized as follows. First take a large sample of long
random cyclically reduced words $W_1$ in $F$. If there are any
non-minimal elements, apply Whitehead's algorithm and replace them by
their minimal representatives. The resulting set $W_2$ consists of
only minimal words. By the results of~\cite{KSS} most of elements of
$W_1$ are already minimal and therefore the difference between $W_1$
and $W_2$ will be very small and can be disregarded.
Then some of the elements $w$ of $W_2$ (again usually chosen at
random) are replaced by $\phi_w(w)$ where $\phi_w$ comes from some
finite collection $\Phi$ of automorphisms chosen so that
$||w||<||\phi_w(w)||$. The resulting set $W_3$ thus contains both
minimal and non-minimal elements. Some of the observed results were
that:
\begin{itemize}
\item The non-minimal elements of the set $W_3$ formed several
``clusters".
\item For each ``cluster" $\mathcal C$ all the elements of
$\mathcal C$
had approximately the same normalized Whitehead graphs.
\item Moreover, for each ``cluster" $\mathcal C$ there was a
Whitehead
automorphism $\tau$ such that for all $w\in \mathcal C$
$$
||\tau(w)||<||w||.
$$
(In fact, often, depending on how $\Phi$ is constructed, one can
choose $\tau$ to be a Nielsen automorphism).
\end{itemize}
In the present paper we provide a theoretical justification of these
experimental results. It turns out that the explanation comes from
exploring the action of $Out(F)$ on the space of geodesic currents on
$F$, analyzed by the author in~\cite{Ka,Ka1}.
Recall that $C$ denotes the set of all nontrivial cyclically reduced words in $F$.
Our main result is:
\begin{theor}\label{A}
Let $F=F(A)$ be a free group where $A=\{a_1,\dots, a_k\}$ and $k\ge
2$. Let $\phi\in Aut(F)$ be an arbitrary automorphism that is not a
composition of a relabelling and an inner automorphisms.
Then there exist a Whitehead automorphism $\tau$ of $F$ and a cyclic
word $w$ with the following properties:
\begin{enumerate}
\item For $m_A$-a.e. point $\omega\in \partial F$ we have
$$
||\tau\phi(\omega_n)||<||\phi(\omega_n)||
$$
as $n\to\infty$ and
$$
\lim_{n\to\infty} [\Gamma_{\phi(\omega_n)}]=[\Gamma_{\phi(w)}],
$$
where $[\Gamma_g]$ is the normalized Whitehead graph corresponding to
the conjugacy class of $g\in F$.
\item For every $\epsilon>0$ there is a $C$-exponentially generic
subset $U\subseteq C$ such that for each $f\in C$
$$
||\tau\phi(f)||<||\phi(f)||
$$
and
$$
d\big([\Gamma_{\phi(f)}],[\Gamma_{\phi(w)}]\big)\le \epsilon.
$$
\item For every $\epsilon>0$ there is an $F$-exponentially generic
subset $W\subseteq F$ such that for every $f\in W$
$$
||\tau\phi(f)||<||\phi(f)||
$$
and
$$
d\big([\Gamma_{\phi(f)}],[\Gamma_{\phi(w)}]\big)\le \epsilon.
$$
\end{enumerate}
\end{theor}
The definitions of genericity, Whitehead graphs and the uniform
measure $m_A$ are given in the subsequent sections. Informally, if
$\omega\in \partial F$ is an $m_A$-random point, the element
$\omega(n)\in F$ is a ``random'' freely reduced element of length
$n$, which is also close to being cyclically reduced. Normalized
Whitehead graphs of a cyclically reduced word $w$, roughly
speaking, records the frequencies with which the two-letter freely
reduced words occur in $w$.
Thus Theorem~\ref{A} shows that, in terms of the experiments described
above, there will be one ``cluster'' for each $\phi\in \Phi$
consisting of all $\phi_w(w)$ such that $\phi_w=\phi$, $w\in W_2$.
Note that the Whitehead automorphism $\tau$ in the statement of
Theorem~\ref{A} is algorithmically computable in terms of $\phi\in
Aut(F)$, although the complexity of such an algorithm is a priori
exponential in terms of the word length of the outer automorphism
$[\phi]$ in $Out(F)$. The main tool used in the proof of
Theorem~\ref{A} is the machinery of \emph{geodesic currents} on free
groups, discussed in more detail in Section~\ref{sect:curr} below.
Together with a recent result of
S.~Francaviglia~\cite{Fr}, the proof of Theorem~\ref{A} turns out to imply
the existence of the following ``universal" length-reducing factorization for
automorphisms of free groups (when applied to ``random" elements of $F$):
\begin{corol}\label{B}\cite{Fr}
Let $\phi\in Aut(F)$ be an arbitrary automorphism that is not a
composition of a relabelling and an inner automorphisms. Then there
exists a factorization
$$
\phi=\sigma_m\sigma_{m-1}\dots \sigma_1 \alpha,
$$
where $m\ge 1$, the automorphism $\alpha$ is a
composition of a relabelling and an inner automorphisms, where $\sigma_i$
are Whitehead automorphisms of the second kind, and such that the
following holds.
Denote $\psi_0=\alpha$, $\psi_i=\sigma_i\sigma_{i-1}\dots \sigma_1
\alpha$ for $i=1,\dots, m$. Thus $\psi_m=\phi$.
Then for $m_A$-a.e. point $\omega\in \partial F$ as $n\to\infty$ we have
$$
||\psi_i\omega_n||<||\psi_{i+1}\omega_n||, \qquad i=1, \dots, m-1
$$
so that
$$
||\omega_n||=||\psi_0\omega_n||<||\psi_1\omega_n||<\dots <||\psi_m\omega_n||=||\phi\omega_n||.
$$
\end{corol}
\section{Geodesic Currents} \label{sect:curr}
We recall some basic notions related to geodesic currents on free
groups. We refer the reader to~\cite{Ka,Ka1,Ma} for a more
comprehensive discussion.
\begin{conv}
We identify the hyperbolic boundary $\partial F$ with the set of all
geodesic rays from $1$ in $X$ or equivalently, with the set of all
semi-infinite freely reduced words
$$
\omega=a_1a_2\dots a_n\dots, \text{ where } a_i\in A^{\pm 1}.
$$
The boundary $\partial F$ is endowed with the Cantor-set topology and
with the homeomorphic left $F$-action by left translations, as usual.
We also denote
$$
\partial^2 F:=\{(\zeta,\xi): \zeta,\xi\in \partial F \text{ and }
\zeta\ne \xi\}.
$$
Note that $\partial^2 F$ comes equipped with the diagonal left
$F$-action by homeomorphisms.
For a directed geodesic segment $\gamma=[x,y]$ in $X$ with $x,y\in F$,
$x\ne y$ we denote by $Cyl_X(\gamma)$ the set of all $(\zeta,\xi)\in
\partial^2 F$ such that the geodesic $[\zeta,\xi]$ in $X$ passes
through $\gamma$ in the correct direction. Note that
$Cyl_X(\gamma)\subseteq\partial^2 F$ is an open-closed compact subset
of $\partial^2 F$.
We denote by $\mathcal P(X)$ the set of all directed geodesic segments
of positive length in $X$ with endpoints in $VX=F$. Also, denote
$F_\ast:=F-\{1\}$.
\end{conv}
\begin{defn}[Uniform measure]
For $v\in F_\ast$ denote by $Cyl_A(v)$ the set of all geodesic rays
$\omega\in \partial F$ that begin with $v$.
The \emph{uniform measure} $m_A$ on $\partial F$ is the Borel
probability measure on $\partial F$ defined by
$$
m_A(Cyl_A(v))=\frac{1}{2k(2k-1)^{|v|-1}}\quad \text{ for every } v\in
F_\ast.
$$
\end{defn}
\begin{defn}[Geodesic currents]
A \emph{geodesic current} on $F$ is a locally finite (that is finite
on compact subsets) positive Radon measure $\nu$ on $\partial^2 F$
such that $\nu$ is $F$-invariant. The set of all geodesic currents on
$F$ is denoted by $Curr(F)$. The space $Curr(F)$ comes equipped with
the natural weak topology which can be described as follows.
For $\nu_n,\nu\in Curr(F)$ we have
$$
\lim \nu_n =\nu
$$
if and only if
$$
\lim_{n\to\infty}\nu_n(Cyl_X(\gamma))=\nu(Cyl_X(\gamma)) \text{ for
every } \gamma\in \mathcal P(X).
$$
\end{defn}
\begin{defn}[The coordinates on $Curr(F)$]
If $\nu\in Curr(F)$ and $\gamma=[x,y]\in \mathcal P(X)$ then by
$F$-invariance of $\nu$ the value $\nu(Cyl_X(\gamma))$ only depends
on $\nu$ and the \emph{label} $v:=x^{-1}y\in F$ of $\gamma$. For a
nontrivial $v\in F$ we denote
$$
\langle v,\nu\rangle:=\nu(Cyl_X(\gamma))
$$
where $\gamma\in \mathcal P(X)$ is any geodesic segment labelled by
$v$.
We call $\langle v,\nu\rangle$ the \emph{number of occurrences of $v$
in $\nu$}.
\end{defn}
The following lemma~\cite{Ka1} summarizes some basic invariance
properties satisfied by the coordinates of a geodesic current:
\begin{lem}\label{inv}
Let $\nu\in Curr(F)$. Then for every $v\in F_\ast$
$$
\langle v,\nu\rangle=\sum_{a\in A^{\pm 1}, |va|=|v|+1} \langle
va,\nu\rangle =\sum_{a\in A^{\pm 1}, |av|=1+|v|} \langle
av,\nu\rangle.
$$
\end{lem}
A current $\nu\in Curr(F)$ is uniquely determined by the family
$(\langle v, \nu\rangle)_{v\in F_\ast}$. Moreover, as shown
in~\cite{Ka,Ka1}, every nonnegative family $(\langle v,
\nu\rangle)_{v\in F_\ast}$, satisfying the invariance conditions from
Lemma~\ref{inv}, defines a current $\nu\in Curr(F)$.
\begin{defn}[Uniform current]\label{defn:uc}
The \emph{uniform current $n_A\in Curr(F)$} corresponding to the
free basis $A$ of $F$ is the geodesic current defined by:
$$
n_A(Cyl_X(\gamma))=\frac{1}{2k(2k-1)^{|\gamma|-1}} \text{ for every }
\gamma\in \mathcal P(X).
$$
Thus $\langle v,n_A\rangle=\frac{1}{2k(2k-1)^{|v|-1}}$ for every $v\in F_\ast$
\end{defn}
\begin{defn}[Rational currents]
Let $g\in F_\ast$. If $g$ is not a proper power, define
$$
\eta_g:=\sum_{h\in [g]} \delta_{(h^{-\infty},h^{\infty})}.
$$
where $[g]$ is the conjugacy class of $g$ in $F$.
If $g=g_0^s$ where $s\ge 2$ and $g_0\in F_\ast$ is not a proper power,
define
$$
\eta_g:=s \eta_{g_0}.
$$
It is easy to see that $\eta_g$ depends only on the conjugacy class
$[g]$ of $g$ in $F$.
Nonnegative multiples of the currents $\eta_g, g\in F_\ast$, are
called \emph{rational currents}.
\end{defn}
An important basic fact (see~\cite{Ka1}) is:
\begin{prop}
The set of rational currents is dense in $Curr(F)$.
\end{prop}
\begin{conv}[Cyclic words]
We will often think about conjugacy classes of nontrivial elements
of $F$ as \emph{cyclic words}. A \emph{cyclic word} $w$ over $A$ is
a nontrivial cyclically reduced word in $F(A)$ written clockwise on
a circle without specifying an initial point. The length of that
cyclically reduced word is called the \emph{cyclic length} of $w$
and is denoted by $||w||$. The circle is thought of as a labelled
graph subdivided into $||w||$ directed edges, each labelled by a
letter of $A$.
If $v\in F$, we call a vertex on this circle an \emph{occurrence of
$v$ in $w$} if $v$ can be read in the circle starting at that
vertex and going clockwise (we are allowed to stop at a different
vertex from the one where we started). The number of occurrences of
$v$ in $w$ is denoted by $\langle v, w\rangle$.
Also, if $v,g\in F$ are nontrivial elements, we put $\langle v,
g\rangle:=\langle v, w\rangle$ where $w$ is the cyclic word
representing the conjugacy class of $g$.
\end{conv}
The following basic fact gives a useful alternative description of
rational currents:
\begin{lem}\label{lem:rational}
Let $g\in F_\ast$ and let $w$ be the cyclic word determined by the
conjugacy class of $g$. Then for every $v\in F_\ast$ we have
$$
\langle v,w\rangle=\langle v, \eta_g\rangle.
$$
\end{lem}
There is a natural continuous left action of $Aut(F)$ on $Curr(F)$
which factors to the action of $Out(F)$ on $Curr(F)$. If $\phi\in
Aut(F)$ then $\phi$ is a quasi-isometry of the Cayley graph $X$ of
$F$. Therefore $\phi$ induces a canonical boundary homeomorphism
$\partial \phi:\partial F\to\partial F$ which diagonally extends to a
homeomorphism $\partial^2\phi :\partial^2 F\to\partial^2 F$. If
$\nu\in Curr(F)$ and $\phi\in Aut(F)$, the current $\phi\nu\in
Curr(F)$ is defined by setting
$$
\phi\nu(S):=\nu((\partial^2\phi)^{-1}(S))
$$
for every Borel subset $S\subseteq \partial^2 F$. It is not hard to
show (see~\cite{Ka1}) that for every $g\in F_\ast$ and every $\phi\in
Aut(F)$ we have $\phi\eta_g=\eta_{\phi(g)}$.
The following useful statement, established in~\cite{Ka1}, gives a
``coordinate'' description of the action of $Aut(F)$ on $Curr(F)$.
\begin{prop}\label{action}
Let $\phi\in Aut(F)$. there exists an integer $K=K(\phi)>0$ with the
following property.
For every $v\in F_\ast$ there exists a collection of nonnegative
integers $\{c(u,v,\phi): u\in F, |u|=K|v|\}$ such that for every
$\nu\in Curr(F)$
$$
\langle v,\phi \nu\rangle=\sum_{u\in F, |u|=K|v|} c(u,v,\phi)\langle
u,\nu\rangle.
$$
\end{prop}
If $a_n,a\in \mathbb R$ and $\lim_{n\to\infty} =a$, we say that
the convergence in this limit is \emph{exponentially fast} if
there exist $0<\sigma<1$, $b>0$ such that $|a_n-a|\le b \sigma^n$
for all $n\ge 1$.
\begin{defn}[Generic sets]
Let $S\subseteq F$ be an infinite subset. Let $T\subseteq S$. We
say that $T$ is \emph{generic} in $S$, or \emph{$S$-generic} if
$$
\lim_{n\to\infty} \frac{\#\{g\in T: |g|\le n\}}{\#\{g\in S: |g|\le
n\}}=1.
$$
If, in addition, the convergence in this limit is exponentially
fast, we say that $T$ is \emph{exponentially $S$-generic}.
\end{defn}
In practice we will only be interested in the cases where $S=F$ or
$S=C$ (recall that $C$ is the set of all nontrivial cyclically reduced
words in $F$). We refer the reader to~\cite{KMSS,KSS} for more details
regarding genericity and generic-case complexity.
\section{The length functional}
It turns out that the notion of ``cyclic length'' with respect to the
free basis $A$ extends naturally to a continuous linear function on
$Curr(F)$.
\begin{defn}[Length of a current]
Let $\nu\in Curr(F)$. We define the \emph{length $L(\nu)$ of $\nu$
with respect to $A$} as:
$$
L(\nu):=\sum_{a\in A^{\pm 1}} \langle a,\nu\rangle.
$$
\end{defn}
In the language of~\cite{Ka1} we have $L(\nu)=I(\ell_A,\nu)$ where $I$
is the ``intersection form'' and where $\ell_A:F\to\mathbb R$ is the
length function defined as $\ell_A(w)=||w||$ for $w\in F$. Note that
for any automorphism $\phi\in Aut(F)$ the number $L(\phi n_A)$ is
exactly what in~\cite{KKS} is called the \emph{generic stretching
factor} $\lambda_A(\phi)$ of $\phi$ with respect to $A$.
The following basic properties of length follow directly from the
results about the intersection form established in~\cite{Ka1}.
\begin{prop}\label{prop:length}
The following hold:
\begin{enumerate}
\item The function $L: Curr(F)\to\mathbb R$ is continuous and linear.
\item For any integer $m\ge 1$ and for every $\nu\in Curr(F)$ we have
$$
L(\nu)=\sum_{v\in F, |v|=m} \langle v,\nu\rangle.
$$
\item For every $w\in F_\ast$ we have
$$
||w||=L(\eta_w).
$$
\item We have $L(n_A)=1$.
\end{enumerate}
\end{prop}
In view of Proposition~\ref{action} and Proposition~\ref{prop:length}
we obtain:
\begin{prop}\label{prop:le}
Let $\phi\in Aut(F)$.
\begin{enumerate}
\item There is $m\ge 2$ and a collection of integers $\{d(u): u\in F,
|u|=m\}$ such that for every $\nu\in Curr(F)$ we have
$$
L(\phi\nu)=\sum_{|u|=m} d(u)\langle u,\nu\rangle.
$$
\item Suppose $m\ge 1$ is an integer and $\{d(u)\in \mathbb Z: u\in
F,|u|=m\}$ are such that for every cyclic word $w$ we have
$$
||\phi(w)||=\sum_{|u|=m} d(u)\langle u,w\rangle.
$$
Then for every $\nu\in Curr(F)$ we have
$$
L(\phi\nu)=\sum_{|u|=m} d(u)\langle u,\nu\rangle.
$$
\end{enumerate}
\end{prop}
\begin{proof}
Part (1) follows directly from Proposition~\ref{action} and
Proposition~\ref{prop:length}. Suppose the assumptions of part (2)
hold. Then the conclusion of part (2) holds for every current of the
form $\eta_g$, $g\in F_\ast$. Therefore, in view of
Lemma~\ref{lem:rational}, the conclusion of part (2) holds for every
$\nu\in Curr(F)$ since rational currents are dense in $Curr(F)$.
\end{proof}
\section{Whitehead automorphisms}
Recall that $\Sigma=A\cup A^{-1}=\{a_1,\dots, a_k, a_1^{-1}, \dots,
a_k^{-1}\}$. We follow Lyndon and Schupp, Chapter~I~\cite{LS} in our
discussion of Whitehead automorphisms. We recall the basic
definitions and results.
\begin{defn}[Whitehead automorphisms]\label{defn:moves}
A \emph{Whitehead automorphism} of $F$ is an automorphism $\tau$ of
$F$ of one of the following two types:
(1) There is a permutation $t$ of $\Sigma$ such that
$\tau|_{\Sigma}=t$. In this case $\tau$ is called a \emph{relabeling
automorphism} or a \emph{Whitehead automorphism of the first
kind}.
(2) There is an element $a\in \Sigma$, the \emph{multiplier}, such
that for any $x\in \Sigma$
$$
\tau(x)\in \{x, xa, a^{-1}x, a^{-1}xa\}.
$$
In this case we say that $\tau$ is a \emph{Whitehead automorphism of
the second kind}. (Note that since $\tau$ is an automorphism of $F$,
we always have $\tau(a)=a$ in this case). To every such $\tau$ we
associate a pair $(T,a)$ where $a$ is as above and $T$ consists of all
those elements of $\Sigma$, including $a$ but excluding $a^{-1}$, such
that $\tau(x)\in\{xa, a^{-1}xa\}$. We will say that $(T,a)$ is the
\emph{characteristic pair} of $\tau$.
\end{defn}
Note that for any $a\in \Sigma$ the inner automorphism corresponding to
the conjugation by $a$ is a
Whitehead automorphism of the second kind.
\begin{defn}[Minimal elements]
An element $w\in F$ is said to be \emph{automorphically minimal} or
just \emph{minimal} if for \emph{every} $\alpha\in Aut(F)$ we have
$|w|\le |\alpha(w)|$.
\end{defn}
\begin{prop}\label{wh}[Whitehead's Algorithm]
\begin{enumerate}
\item If $u \in F$ is cyclically reduced and not minimal, then there
is a Whitehead automorphism $\tau$ such that $||\tau(u)||<||u||$.
\item Let $u,v \in F$ be minimal (and hence cyclically reduced)
elements with $|u|=|v|=n>0$. Then $Aut(F)u=Aut(F)v$ if and only if
there exists a finite sequence of Whitehead automorphisms
$\tau_s,\dots, \tau_1$ such that $\tau_s\dots \tau_1(u)=v$ and such
that for each $i=1,\dots, s$ we have
$$
||\tau_i\dots \tau_1(u)||=n.
$$
\end{enumerate}
\end{prop}
\begin{defn}[Strict Minimality]
A nontrivial cyclically reduced word $w$ in $F$ is \emph{strictly
minimal} if for every non-inner Whitehead automorphism $\tau$ of
$F$ of the second kind we have
$$
||\tau(w)||>||w||.
$$
\end{defn}
\begin{defn}[Simple automorphisms]
An automorphism $\phi\in Aut(F)$ is called \emph{simple} if it is
the composition of an inner and a relabelling automorphisms.
\end{defn}
Clearly if $\phi$ is simple, then for every $w\in F_\ast$ we have
$||\phi(w)||=||w||$. Proposition~\ref{wh} immediately implies that
every strictly minimal element is minimal and, moreover, if $u$ is
strictly minimal and $\phi\in Aut(F)$ is such that $||u||=||\phi(u)||$
then $\phi$ is simple.
\begin{defn}[Weighted Whitehead graph]
Let $w$ be a nontrivial cyclic word in $F(A)$. The \emph{weighted
Whitehead graph $\Gamma_w$ of $w$} is defined as follows. The
vertex set of $\Gamma_w$ is $\Sigma$. For every $x,y\in \Sigma$ such
that $x\ne y^{-1}$ there is an undirected edge in $\Gamma_w$ from
$x^{-1}$ to $y$ labelled by the sum
$$
\langle xy,w\rangle+\langle y^{-1}x^{-1},w\rangle,
$$
the number of occurrences of the words $xy$ and $y^{-1}x^{-1}$ in $w$.
The \emph{normalized Whitehead graph} $[\Gamma_w]$ of $w$ is the
labelled graph obtained from $\Gamma_w$ by dividing every edge-label
by $||w||$.
\end{defn}
\begin{defn}
An \emph{abstract Whitehead graph} is a labelled graph $\Gamma$
whose vertex and edge sets are the same as those for a weighted
Whitehead graph of a cyclic word and such that each edge $e$ of
$\Gamma$ is labelled by a real number $r(e)$. If $\Gamma,\Gamma'$
are two abstract Whitehead graphs, we define
$$d(\Gamma,\Gamma')=\max_{e\in E\Gamma} |r(e)-r(e')|.$$ This turns
the set of all abstract Whitehead graphs into a metric space
homeomorphic to $\mathbb R^{k(2k-1)}$.
\end{defn}
Note that if $w$ is a cyclic word, then both $\Gamma_w$ and
$[\Gamma_w]$ are abstract Whitehead graphs. Note also that for
$[\Gamma_w]$ the sum of all edge-labels is equal to $1$.
\begin{conv}
Let $w$ be a fixed nontrivial cyclic word. For two subsets
$P,Q\subseteq \Sigma$ we denote by $P\underset{w}{.}Q$ the sum of
all edge-labels in the weighted Whitehead graph $\Gamma_w$ of $w$ of
edges from elements of $P$ to elements of $Q$. Thus for $x\in
\Sigma$ the number $x\underset{w}{.}\Sigma$ is equal to the total
number of occurrences of $x^{\pm 1}$ in $w$.
\end{conv}
The next lemma, which is Proposition~4.16 of Ch.~I in \cite{LS}, gives
an explicit formula for the difference of the lengths of $w$ and
$\tau(w)$, where $\tau$ is a Whitehead automorphism.
\begin{lem}\label{lem:LS}
Let $w$ be a nontrivial cyclically reduced word and let $\tau$ be a
Whitehead automorphism of the second kind with the characteristic
pair $(T,a)$. Let $T'=\Sigma-T$. Then
$$
||\tau(w)||-||w||=T\underset{w}{.}T'-a\underset{w}{.}\Sigma.
$$
\end{lem}
Now Lemma~\ref{lem:LS} and Proposition~\ref{prop:le} immediately
imply:
\begin{cor}\label{cor:wh}
Let $\tau$ be a Whitehead automorphism of the second kind. Then
there exists a collection of integers $\{b(z): z\in F, |z|=2\}$ such
that for every $\nu\in Curr(F)$ we have
$$
L(\tau\nu)=\sum_{|z|=2} b(z)\langle z,\nu\rangle.
$$
\end{cor}
\begin{rem}
Note that in view of Lemma~\ref{lem:LS} and Corollary~\ref{cor:wh}, if
$w$ is a cyclic word and $\tau$ is a Whitehead automorphism of the
second kind, then the quantity
$$
\frac{||\tau(w)||}{||w||}
$$
is completely determined by $\tau$ and the normalized Whitehead graph
$[\Gamma_w]$ of $w$.
\end{rem}
\section{Proof of the main result}
\begin{prop}\label{euler}
For every integer $m\ge 2$ there exists a cyclic word $w$ such that
$$
\langle v,w\rangle =1 \text{ for every } v\in F \text{ with } |v|=m
$$
and that $||w||=2k(2k-1)^{m-1}$.
\end{prop}
\begin{proof}
This follows from a more general result in \cite{Ka}. We present an
argument here for completeness.
If $v\in F$ is a freely reduced word with $|v|\ge 2$, we denote by
$v_{-}$ the initial segment of $v$ of length $|v|-1$ and we denote
by $v_+$ the terminal segment of $v$ of length $|v|-1$.
Let $n\ge 2$. Form a finite directed labelled graph $\Gamma$ as
follows. The vertex set of $\Gamma$ is
$$
V\Gamma:=\{u\in F: |u|=m-1\}.
$$
The set of directed edges of $\Gamma$ is
$$
E\Gamma:=\{v\in F: |v|=m\}.
$$
For each $v\in E\Gamma$ the initial vertex of $v$ in $\Gamma$ is
$v_{-}$ and the terminal vertex of $v$ in $\Gamma$ is $v_+$. Also, the
edge $v\in E\Gamma$ is labelled by the \emph{label} $a(v)\in A^{\pm
1}$ which is the last letter of the word $v$.
Note that for every vertex $u\in V\Gamma$ both the out-degree of $u$
and the in-degree of $u$ in $\Gamma$ are equal to $2k-1$. Thus
$\Gamma$ is a strongly connected directed graph where for each vertex
the in-degree is equal to the out-degree. Therefore there exists an
Euler circuit $c$ is $\Gamma$, that is, a cyclic path passing through
each directed edge of $\Gamma$ exactly once. Let $c$ be represented
by the edge-path
$$
v_1v_2\dots v_{t}, \text{ where } t=|E\Gamma|=2k(2k-1)^{m-1}.
$$
Let $w$ be the cyclic word defined by the word
$$
a(v_1)a(v_2)\dots a(v_t).
$$
Then it is not hard to see that $||w||=t=2k(2k-1)^{m-1}$ and that for
every $v\in F$ with $|v|=m$ we have
$$
\langle v,w\rangle =1,
$$
as required.
\end{proof}
Recall that $n_A$ is the uniform current on $F$ defined in Definition~\ref{defn:uc}.
\begin{prop}\label{prop:wm}[Ideal Whitehead Algorithm]
Let $\phi\in Aut(F)$ be an automorphism such that $\phi$ is not
simple. Then there exists a Whitehead automorphism $\tau$ of the
second kind such that
$$
1=L(n_A)\le L(\tau\phi n_A)<L(\phi n_A).
$$
\end{prop}
\begin{proof}
By~Proposition~\ref{action} there exist an integer $m\ge 2$ and a
collection of nonnegative integers
$$
\{c(v,z): v,z\in F, |v|=m, |z|=2\}
$$
such that for every $\nu\in Curr(F)$ we have
$$
\langle z,\phi\nu\rangle =\sum_{|v|=m}c(v,z) \langle v,\nu\rangle.
$$
Let $w$ be a cyclic word provided by Proposition~\ref{euler}.
Recall that we have $||w||=2k(2k-1)^{m-1}$. Let
$\theta=\frac{\eta_w}{2k(2k-1)^{m-1}}$. Thus for every $v\in F$
with $|v|=m$ we have
$$
\langle v,\theta\rangle=\frac{1}{2k(2k-1)^{m-1}}.
$$
Then for every $z\in F$ with $|z|=2$ we have
$$
\langle z,\phi\theta\rangle=\langle z,\phi n_A\rangle.
$$
Moreover, we have
$$
L(\phi\theta)=\sum_{|z|=2}\langle z,\phi\theta\rangle=
\sum_{|z|=2}\langle z,\phi n_A\rangle=L(\phi n_A).
$$
By Lemma~4.8 of~\cite{KSS} the word $w$ is strictly minimal which
implies, in particular, that $||w||<||\phi(w)||$, since $\phi$ is not
simple. Therefore, by Whitehead's theorem, part (1) of
Proposition~\ref{wh}, there exists a Whitehead automorphism $\tau$ of
the second kind such that
$$
||w||\le ||\tau\phi(w)||<||\phi(w)||.
$$
Therefore
$$
1=L(\theta)\le L(\tau \phi\theta)< L(\phi\theta).
$$
then by the above formulas and Corollary~\ref{cor:wh} we see that
$$
1=L(n_A)\le L(\tau\phi n_A)=L(\tau \theta)<L(\theta)=L(\phi n_A),
$$
as required.
\end{proof}
Note that Proposition~\ref{prop:wm} means that $n_A\in Curr(F)$ is
``minimal" and even ``strictly minimal" in the sense that for
every $\phi\in Aut(F)$
$$
L(n_A)\le L(\phi n_A)
$$
with the equality achieved if and only if $\phi$ is simple.
\begin{cor}\label{cor:ideal}
Let $\phi\in Aut(F)$ be a non-simple automorphism. Then there
exists a factorization
$$
\phi=\sigma_m\sigma_{m-1}\dots \sigma_1 \alpha,
$$
where $m\ge 1$, the automorphism $\alpha$ is simple, $\sigma_i$
are Whitehead automorphisms of the second kind and
$$
L(\sigma_{i-1}\dots \sigma_1\alpha n_A)< L(\sigma_i\sigma_{i-1}\dots \sigma_1
\alpha n_A), \qquad i=1,\dots, m-1.
$$
\end{cor}
\begin{proof}
Put
$$
\Lambda:=\{L(\psi n_A): \psi\in Aut(F)\}.
$$
A recent theorem of S.~Francaviglia~\cite{Fr} shows that $\Lambda$ is
a discrete subset of $\mathbb R$. Also, as proved in~\cite{KSS},
for every $\psi\in Aut(F)$ we have $L(\psi n_A)\ge 1$ and,
moreover, $L(\psi n_A)= 1$ if and only if $\psi$ is simple.
Let $\phi\in Aut(F)$ be a non-simple automorphism. Thus $L(\phi
n_A)>1$. Repeatedly applying Proposition~\ref{prop:wm} we conclude that
there exists a sequence of Whitehead automorphisms
$\tau_1,\tau_2,\dots $ such that $L(\phi n_A)> L(\tau_1\phi
n_A)>L(\tau_2\tau_1\phi n_A)>\dots$. Since $\Lambda$ is a discrete
subset of $[1,\infty)$, the sequence $\tau_1,\tau_2,\dots $ must
terminate in a finite number of steps with some $\tau_m$. Hence
the automorphism $\alpha:=\tau_m\dots \tau_2\tau_1\phi$ must be
simple since otherwise by Proposition~\ref{prop:wm} the sequence of $\tau_i$
could be extended. Then the factorization
$$
\phi=\tau_1^{-1}\dots \tau_m^{-1}\alpha
$$
has the required properties and the corollary is proved.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{A}]
Let $\phi\in Aut(F)$ be an automorphism such that $\phi$ is not simple. By
Proposition~\ref{prop:wm} there exists a Whitehead automorphism
$\tau$ such that
$$
L(\tau\phi n_A)<L(\phi n_A).
$$
Also, as in the proof of Proposition~\ref{prop:wm}, let $w$ be the
cyclic word provided by Proposition~\ref{euler}.
Recall that by Proposition~\ref{action} there exist an integer $m\ge
2$ and a collection of nonnegative integers
$$
\{c(v,z): v,z\in F, |v|=m, |z|=2\}
$$
such that for every $\nu\in Curr(F)$ we have
$$
\langle z,\phi\nu\rangle =\sum_{|v|=m}c(v,z) \langle v,\nu\rangle.
$$
Let $\omega\in \partial F$ be an $m_A$-random point. Then, as
observed in~\cite{Ka1}
$$
\lim_{n\to\infty} \frac{\eta_{\omega_n}}{n}=\lim_{n\to\infty}
\frac{\eta_{\omega_n}}{||\omega_n||}=n_A \text{ in } Curr(F).
$$
Hence
$$
\lim_{n\to\infty} \phi \frac{\eta_{\omega_n}}{n}=\phi n_A
$$
and
$$
\lim_{n\to\infty} \tau\phi \frac{\eta_{\omega_n}}{n}=\tau\phi n_A
$$
Since $L:Curr(F)\to \mathbb R$ is continuous, and $L(\tau\phi
n_A)<L(\phi n_A)$, it follows that for $n\to\infty$
$$
L(\tau\phi \frac{\eta_{\omega_n}}{n})<L(\phi
\frac{\eta_{\omega_n}}{n}),
$$
Then for $n\to\infty$
$$
\frac{||\tau\phi(\omega_n)||}{n}<\frac{||\phi(\omega_n)||}{n}
$$
and therefore
$$
||\tau\phi(\omega_n)||<||\phi(\omega_n)||,
$$
as required.
We have seen in (\ddag) that
$$
\langle z,\phi\theta\rangle=\langle z,\phi n_A\rangle \text{ for each
} z\in F \text{ with } |z|=2
$$
where $\theta=\frac{\eta_w}{2k(2k-1)^{m-1}}$ and
$||w||=2k(2k-1)^{m-1}$. Since \[\lim_{n\to\infty} \phi
\frac{\eta_{\omega_n}}{n}=\phi n_A,\] this implies that for each $z\in
F$ with $|z|=2$ we have
$$
\lim_{n\to\infty} \frac{\langle z, \phi(\omega_n)\rangle}{n}=\langle
z,\phi n_A\rangle=\langle z,\phi\theta\rangle=\frac{\langle
z,\phi(w)\rangle}{2k(2k-1)^{m-1}}.
$$
We also have
$$
||\phi(w)||=\sum_{|z|=2} \langle z,\phi(w)\rangle
=\sum_{|z|=2}\sum_{|v|=m} c(v,z)\langle v,
w\rangle=\sum_{|z|=2}\sum_{|v|=m} c(v,z)
$$
and
$$
||\phi(\omega_n)||=\sum_{|z|=2} \langle z,\phi(\omega_n)\rangle
=\sum_{|z|=2}\sum_{|v|=m} c(v,z)\langle v, \omega_n\rangle.
$$
Therefore for any $z'\in F$ with $|z'|=2$ we have
$$
\frac{\langle z', \phi(w)\rangle}{||\phi(w)||}=\frac{\sum_{|v|=m}
c(v,z')}{\sum_{|z|=2}\sum_{|v|=m} c(v,z)}
$$
and
$$
\frac{\langle z',
\phi(\omega_n)\rangle}{||\phi(\omega_n)||}=\frac{\sum_{|v|=m}
c(v,z')\langle v, \omega_n\rangle}{\sum_{|z|=2}\sum_{|v|=m}
c(v,z)\langle v, \omega_n\rangle}=\frac{\sum_{|v|=m}
c(v,z')\frac{\langle v,
\omega_n\rangle}{n}}{\sum_{|z|=2}\sum_{|v|=m} c(v,z)\frac{\langle
v, \omega_n\rangle}{n}}.
$$
Since $\lim_{n\to\infty} \frac{\eta_{\omega_n}}{n}=n_A$, it follows
that $\lim_{n\to\infty}\frac{\langle v,
\omega_n\rangle}{n}=\frac{1}{2k(2k-1)^{m-1}}$ for every $v\in F$
with $|v|=m$. Therefore for every $z'\in F$ with $|z'|=2$ we have
$$
\lim_{n\to\infty} \frac{\langle z',
\phi(\omega_n)\rangle}{||\phi(\omega_n)||}=\frac{\sum_{|v|=m}
c(v,z')}{\sum_{|z|=2}\sum_{|v|=m} c(v,z)}=\frac{\langle z',
\phi(w)\rangle}{||\phi(w)||}.
$$
It follows that $\lim_{n\to\infty}
[\Gamma_{\phi(\omega_n)}]=[\Gamma_{\phi(w)}]$ as required.
This establishes part (1) of Theorem~\ref{A}.
Recall that by Proposition~6.2 of~\cite{KSS} if $U\subseteq C$ is an
exponentially $C$-generic subset, then the set $W$ consisting of all
$w\in F$ whose cyclically reduced forms are in $U$, is exponentially
$F$-generic. Therefore part (2) of Theorem~\ref{A} implies part (3).
Thus it remains to prove part (2) of Theorem~\ref{A}.
For any $\epsilon'>0$ define
$$
U(\epsilon')=\{u\in C: \left|\frac{\langle
v,u\rangle}{||u||}-\frac{1}{2k(2k-1)^{m-1}}\right|\le \epsilon'
\text{ for every } v\in F, |v|=m\}.
$$
Recall also that there exists a collection of integers $\{d(z): z\in
F, |z|=2\}$ such that
$$
L(\tau\nu)-L(\nu)=\sum_{|z|=2}d(z)\langle z,\nu\rangle\quad \text{ for
every }\nu\in Curr(F).
$$
Since $L(\tau\phi n_A)<L(\phi n_A)$, there is $\epsilon''>0$ such that
for every $\nu\in Curr(F)$ satisfying
$$
|\langle z,\nu\rangle-\langle z,\phi n_A\rangle|\le \epsilon''
$$
for every $z\in F$ with $|z|=2$ we have $L(\tau\nu)-L(\nu)<0$.
The properties of $c(v,z)$ listed above imply that there is
$\epsilon'>0$ such that for every $u\in U(\epsilon')$ and for every
$z\in F,|z|=2$ we have
$$
\left|\langle z,\phi\frac{\eta_u}{||u||}\rangle-\langle z,\phi
n_A\rangle\right|\le \epsilon''.
$$
Hence for every $u\in U(\epsilon')$
$$
L(\tau\phi\frac{\eta_u}{||u||})-L(\phi\frac{\eta_u}{||u||})<0,
$$
that is
$$
\frac{||\tau\phi(u)||}{||u||}<\frac{||\phi(u)||}{||u||}\quad\Rightarrow\quad ||\tau\phi(u)||<||\phi(u)||.
$$
The set $U(\epsilon')\subseteq C$ is exponentially $C$-generic, as was
observed in~\cite{KKS}. The proof of the Whitehead graph assertion of
part (2) of Theorem~\ref{A} is similar to that used in part (1). One
shows that if $\epsilon>0$ is arbitrary then for $\epsilon'>0$ small
enough
$$
d([\Gamma_{\phi(u)}],[\Gamma_{\phi{w}}])\le \epsilon \text{ for all }
u\in U(\epsilon').
$$
We leave the details to the reader.
This completes the proof of Theorem~\ref{A}.
\end{proof}
Corollary~\ref{cor:ideal} implies Corollary~\ref{B} from the
Introduction in a way similar to the proof of part (1) of
Theorem~\ref{A} and we leave the details to the reader.
\section{Acknowledgements}
The author is grateful to Alexey G. Myasnikov for suggesting to
consider the question addressed by Theorem~\ref{A}. The author
especially thanks Paul Schupp for useful conversations and for
help with computer experimentation. The author was supported by
the NSF grant DMS-0404991.
| 150,889
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OK, sometimes it just happens. You write a blog entry one day, about the monkeys, and the very next day a dozen monkeys come by while you’re hanging laundry. They are actually Bonnet Macaques and are common in this part of India. I used photos from our visit to Mysore, but I described anecdotes about them breaking into apartments here at IISc. Today, I happened to be on the roof hanging laundry, and a band of monkeys strolled across the roof. My new personal rule– never hang laundry without your camera at hand – paid off handsomely.
This group, with at least a dozen monkeys, traveled through the trees and landed first on my neighbors’ roof. They strolled across the connecting stairwell, around my laundry, and down into the narrow courtyard that separates us from another neighbor. In that courtyard, they scrambled across the clotheslines (and clothes), explored the windows in hopes of finding one open, and entered a foyer in hopes of finding a door open. In the foyer they found what appeared to be apple peels wrapped in newspaper. These they nibbled as they climbed back up and headed on their way across the next roof.
I took over 200 photos, but selected a few good ones.
This post was transferred from MobileMe to WordPress in 2020, with an effort to retain the content as close to the original as possible; I recognize that some comments may now seem dated or some links may now be broken.
| 134,047
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TITLE: Are there polyominoes that can't tile the plane, but scaled copies can?
QUESTION [13 upvotes]: I'm wondering where there is a finite set $\mathcal{T}$ of polyominoes that are pairwise similar that can tile the plane, but a single element from the set cannot. (All orientations are allowed.)
To show what I mean, here is a tiling by two similar T-tetrominoes. This example is not interesting because T-tetrominoes of the same size already tile the plane.
The reason polyominoes cannot tile the plane is usually because of reasons that seem unlikely that the inclusion of scaled copies could solve, but showing this is the case generally seems difficult.
There are tilings of rectangles that are not possible with a single size, but can be done with multiple sizes, as this example shows. (This is also not an example of what I am looking for, since a single piece can in fact tile the plane).
Here are the small polyominoes that don't tile the plane; each of these (together with scaled copies), is a candidate set, although the ones I tried did not seem very promising.
REPLY [11 votes]: We can achieve this using two polyominoes with one having double the dimensions of the other and the second copy rotated $90^\circ$ and reflected.
The basic element showing the two polyominoes is below. Its bounding box is $165\times98$ and the key to the tiling is that the longer vertical edges are both $82$ units long while the shorter vertical edges are both $16$ units long. It's clear that one polyomino can't tile the plane on its own; the second polyomino is needed for them to hook together.
And here's the tiling (click to enlarge):
The general approach is to hook two polyominoes together in the pattern below. Imagine this initially as a series of rectangles from bottom-left to top-right: $2\times1$ at bottom-left, then $2\times4$ just above that, then $8\times4$ and finally $8\times16$ in the large rectangle at top-right, with the other dimensions being fully determined from this.
A Python program was used to vary the dimensions of the smallest rectangle and redraw the resulting polyominoes. The data from the program was used to find the correct size of rectangle to generate the required matching edges. In the final polyomino the small rectangle is $2\times16$.
| 12,671
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\begin{document}
\title{M\"obius structures and timed causal spaces on the circle}
\author{Sergei Buyalo\footnote{This work is supported by RFBR Grant
17-01-00128a}}
\date{}
\maketitle
\begin{abstract} We discuss a conjectural duality between
hyperbolic spaces on one hand and spacetimes on the other hand,
living on the opposite sides of the common absolute. This
duality goes via M\"obius structures on the absolute,
and it is easily recognized in the classical case of symmetric
rank one spaces. In a general case, no trace of such duality
is known. As a first step in this direction, we show how M\"obius
structures on the circle from a large class including those which
stem from hyperbolic spaces give rise to 2-dimensional spacetimes,
which are axiomatic versions of de Sitter 2-space, and vice versa.
The paper has two Appendices, one of which is written by V.~Schroeder.
\end{abstract}
\noindent{\small{\bf Keywords:} M\"obius structures, cross-ratio, harmonic 4-tuples,
hyperbolic spaces, spacetimes, de Sitter space}
\medskip
\noindent{\small{\bf Mathematics Subject Classification:} 51B10, 53C50}
\section{Introduction}
It is classical that the quadratic form
$$g(v)=x^2+y^2-z^2$$
on
$\R^3$, $v=(x,y,z)\in\R^3$,
induces on any connected component of the set
$g(v)=-1$
a Riemannian metric of the hyperbolic plane
$\hyp^2$,
while on the set
$g(v)=1$
a Lorentz metric of de Sitter 2-space
$\ds^2$.
The (set of lines in the) cone
$g(v)=0$
serves as the common absolute
$S^1$
of the both
$\hyp^2$
and
$\ds^2$.
A similar picture takes place in any dimension and even for all rank one
symmetric spaces of noncompact type.
In other words, we observe a life on the other side of the absolute
$S^1$
of
$\hyp^2$
that is de Sitter space
$\ds^2$.
For mathematical aspects of the duality between hyperbolic spaces
$\hyp^{n+1}$
and de Sitter spacetimes
$\ds^{n+1}$
see e.g. \cite{Ge}, \cite{Yu}. Interplay between geometry of hyperbolic surfaces and
Lorentz (2+1)-spaces is exploited in the famous paper \cite{Mes}, see
also \cite{A-S}. Duality for quadratic forms of arbitrary signature
is discussed in \cite{Ro}. For physical aspects of de Sitter spaces
see e.g. \cite{SSV} and references therein.
In sect.~\ref{sect:other_side}, we describe this duality
in intrinsic terms. The basic feature is the canonical M\"obius structure
$M_0$
on the absolute
$S^1$,
which governs its both sides
$\hyp^2$
and
$\ds^2$.
In particular, the isometry groups of
$\hyp^2$
and
$\ds^2$
coincide with the group of M\"obius automorphisms of
$M_0$.
We show how to recover the hyperbolic plane
$\hyp^2$
and de Sitter 2-space
$\ds^2$
purely out of
$M_0$.
Moreover, we explain a mechanism of the passage from
$\hyp^2$
to
$\ds^2$
and back. Shortly,
$\hyp^2$
is the homogeneous space of the
$M_0$-automorphism
group
$\PSL_2(\R)$
over a compact elliptic subgroup isomorphic to
$S^1$,
while
$\ds^2$
is the homogeneous space of
$\PSL_2(\R)$
over a (closed) hyperbolic subgroup isomorphic to
$\R$.
This rises a bold question: Is there any life (a spacetime) on the other side
of the absolute, i.e., the boundary at infinity, of any Gromov hyperbolic space
with the same symmetry group? The main result of the paper is the answer
``yes'' for a large class of hyperbolic spaces with the absolute
$S^1$,
see Theorem~\ref{thm:main}.
A M\"obius structure on a set
$X$
is a class of semi-metrics having one and the same cross-ratio
on any given ordered 4-tuple of distinct points in
$X$,
see sect.~\ref{sect:moeb}. Every hyperbolic space
$Y$
induces on its boundary at infinity
$X=\di Y$
a M\"obius structure which encodes most essential properties of
$Y$
and in a number of cases allows to recover
$Y$
completely, e.g., in the case
$Y$
is a rank one symmetric space of noncompact type, see
\cite{BS2}, \cite{BS3}. In sect.~\ref{subsect:boundary_continuous},
we explain this for the class of {\em boundary continuous} hyperbolic
spaces. In Appendix~1, sect.~\ref{sect:appendix_1}, we show that every proper
Gromov hyperbolic
$\CAT(0)$
space is boundary continuous.
We axiomatically describe a class
$\cM$
of {\em monotone} M\"obius structures on the circle
$S^1$,
see sect.~\ref{sect:monotone}. The class
$\cM$
includes every M\"obius structure
$M$
which is induced on
$S^1$
by a hyperbolic
$\CAT(0)$
surface
$Y$
without singular points, see Theorem~\ref{thm:without_singular}.
In particular, the isometry group of
$Y$
is included in the group of M\"obius automorphisms of
$M$.
Furthermore, the canonical
M\"obius structure
$M_0$
is the most symmetric representative from
$\cM$.
On the other hand, the set
$\ay$
of unordered pairs of distinct points on the circle
$X=S^1$
has a natural causal structure, which is independent of anything else, see
sect.~\ref{sect:other_side}. Points of
$\ay$
are called {\em events}. There is a large class
$\cT$
of 2-dimensional spacetimes compatible with that causal structure,
and we characterize it axiomatically in sect.~\ref{sect:timed_causal}.
Any spacetime
$T\in\cT$
is a triple
$T=(\ay,\cH,t)$,
where
$\cH$
is a class of timelike curves in
$\ay$
called {\em timelike lines}, which are actually timelike geodesics,
and
$t$
is the time between events in the causal relation. The spacetime
$T\in\cT$
is called {\em timed causal space}.
We prove
\begin{thm}\label{thm:main} There are natural mutually inverse maps
$\wh T:\cM\to\cT$
and
$\wh M:\cT\to\cM$
such that the groups of automorphisms of any
$M\in\cM$
and of the respective
$T=\wh T(M)\in\cT$
are canonically isomorphic.
\end{thm}
It follows from constructions of sect.~\ref{sect:other_side} that
the canonical M\"obius structure
$M_0$
on
$S^1$
determines de Sitter space
$\ds^2$,
that is,
$\wh T(M_0)=\ds^2$
and
$\wh M(\ds^2)=M_0$.
In other words, Theorem~\ref{thm:main} says that a monotone M\"obius structure on
$S^1$
on one hand, and the respective timed causal space with the absolute
$S^1$
on the other hand, are different sides of one and the same phenomenon
also in a general case.
The fundamental feature of spacetimes is the {\em time inequality.}
In section~\ref{sect:time_inequality}, we discuss a hierarchy
of time conditions, in particular, we introduce the {\em weak time inequality,}
and show that every timed causal space
$T\in\cT$
satisfies the weak time inequality, see Theorem~\ref{thm:wti}.
In sect.~\ref{subsect:monotone_vp}, we introduce Increment Axiom~(I)
which implies the time inequality, and show that the subset
$\cI\sub\cM$
of M\"obius structures satisfying (I) contains the canonical structure
$M_0$, $M_0\in\cI$,
(Proposition~\ref{pro:canonical_vp}) with a neighborhood of
$M_0$
in the fine topology (Proposition~\ref{pro:pertubed_canonical_vp}).
In sect.~\ref{subsect:convex_moeb}, we introduce Convexity Axiom~(C)
for monotone M\"obius structures
$M\in\cM$,
which implies convexity of a functional
$F_{ab}$
playing an important role in the hierarchy of time conditions and
show that the subset
$\cC\sub\cM$
of convex M\"obius structures contains
$M_0$
(Proposition~\ref{pro:canonical_convex}).
The spacetimes of the class
$\cT$
are related to de Sitter 2-space
$\ds^2$
in a sense at least as hyperbolic
$\CAT(0)$
surfaces without singular points with the absolute
$S^1$
are related to the hyperbolic plane
$\hyp^2$.
If one would extend results of this paper to more general hyperbolic
spaces even with 1-dimension boundary at infinity, then this potentially
could produce new interesting classes of spacetimes e.g. having
a branching time (timelike lines).
\bigskip
{\em Acknowledgments.} The author is very much thankful to
Prof.Dr. Viktor Schroeder for attention to this paper and valuable remarks.
Especially, for pointing me out that Axiom~(t6) from Sect.~\ref{subsect:time}
follows from the other axioms of timed causal spaces. This is explained
in details in Appendix~2 written by Viktor Schroeder.
\tableofcontents
\section{On the other side of the absolute}
\label{sect:other_side}
This section serves as a motivation, contains no new result, and its constructions
are widely known. Here, we show how the both sides
$\hyp^2$
and
$\ds^2$
of the common absolute
$S^1$
can be recovered out of the canonical M\"obius structure
$M_0$
on
$S^1$.
It is common to define
$\hyp^2$
and
$\ds^2$
in the quotient of
$\R^3$
by the antipodal map
$x\mapsto -x$.
This does not affect
$\hyp^2$,
while
$\ds^2$
becomes a nontrivial line bundle over
$S^1$,
that is, the open M\"obius band.
\subsection{Recovering the hyperbolic plane $\hyp^2$}
\label{subsect:recovering_hyp}
The canonical M\"obius structure
$M_0$
on the circle
$S^1$
is determined by the condition that any its representative with
infinitely remote point is a standard metric (up to a positive factor) on
$\wh\R=\R\cup\{\infty\}$
extended in the sense that the distance between any
$x\in\R$
and
$\infty$
is infinite.
To recover
$\hyp^2$
from
$M_0$,
we consider the space
$Y$
of all M\"obius involutions
$s:S^1\to S^1$
with respect to
$M_0$
without fixed points. The space
$Y$
serves as the underlying space for
$\hyp^2$,
and what remains to do is to introduce a respective metric on
$Y$.
A line in
$Y$
is determined by a pair
$x$, $x'\in S^1$
of distinct points and consists of all involutions
$s\in Y$
which permute
$x$, $x'$, $sx=x'$.
Given two distinct points
$s$, $s'\in Y$,
the compositions
$s's$, $ss':S^1\to S^1$
have one and the same fixed point set consisting of
two distinct points
$x$, $x'\in S^1$.
Thus there is a uniquely determined line in
$Y$
through
$s$, $s'$.
We say that an ordered 4-tuple
$q=(x,x',y,z)\in(S^1)^4$
of pairwise distinct points is {\em harmonic} if
\begin{equation}\label{eq:harmonic}
|xy|\cdot|x'z|=|xz|\cdot|x'y|
\end{equation}
for some and hence any metric on
$S^1$
from
$M_0$.
For the canonical M\"obius structure
$M_0$,
harmonicity of
$(x,x',y,z)$
is equivalent to that the geodesic lines
$xx'$, $yz\sub\hyp^2$
are mutually orthogonal.
A {\em sphere}
$S$
between
$x$, $x'\in S^1$
is of a pair
$(y,z)\sub S^1$
such that the 4-tuple
$(x,x',y,z)$
is harmonic. We take spheres
$S$, $S'\sub S^1$
between
$x$, $x'$
such that
$S$
is invariant under
$s$, $s(S)=S$,
and
$S'$
is invariant under
$s'$, $s'(S')=S'$.
The spheres
$S$, $S'$
with these properties exist and are uniquely determined. Now, we
take
$y\in S$, $y'\in S'$
and put
\begin{equation}\label{eq:distance_log}
|ss'|=|\ln\langle x,y,y',x'\rangle|,
\end{equation}
where
$\langle x,y,y',x'\rangle=\frac{|xy'|\cdot|yx'|}{|xy|\cdot|y'x'|}$
is the cross-ratio of the 4-tuple
$(x,y,y',x')$.
This is well defined and independent of the choice
$y\in S$, $y'\in S'$.
It is easy to show that
$|ss'|$
is the distance in the geometry of
$\hyp^2$,
see sect.~\ref{subsect:auto}.
\begin{rem}\label{rem:rank_one_extension} This construction is easily
extended to any rank one symmetric space of noncompact type,
see \cite{BS2}.
\end{rem}
\subsection{Recovering de Sitter space $\ds^2$}
\label{subsect:recoveting_desitter}
Let
$\ay$
be the space of unordered pairs
$(x,y)\sim(y,x)$
of distinct points on
$S^1$
with the induced from
$S^1$
topology, that is,
$\ay=S^1\times S^1\sm\De/\sim$,
where
$\De=\set{(x,x)}{$x\in S^1$}$
is the diagonal. Then
$\ay$
is a nontrivial
$\R$-bundle
over
$\rp^1\approx S^1$,
i.e.,
$\ay$
is the open M\"obius band. In this case,
$S^1$
is the boundary of
$\ay$
at infinity,
$\di\ay=S^1$.
Points of
$\ay$
are called {\em events.}
We say that two events
$e$, $e'\in\ay$
are in the {\em causal relation} if and only if
$e$, $e'$
do not separate each other as pairs of points in
$S^1$.
This defines the {\em canonical causality structure} on
$\ay$.
A {\em light line} in
$\ay$
is determined by any
$x\in S^1$
and consists of all events
$a=(x,x')\in\ay$, $x'\in S^1\sm x$.
For this light line
$p_x$,
$x$
is the unique point at infinity. Two distinct light lines
$p_x$, $p_y$
have a unique common event
$(x,y)\in\ay$,
and any two events on a light line are in the causal relation.
The canonical causality structure as well as light lines are inherent in
$\ay$,
and they do not depend of anything else.
\begin{rem}\label{rem:high_dim} In higher dimensional case, a
causality structure can be defined similarly, but then it
depends on the M\"obius structure because events are codimension one
spheres in
$S^n$.
\end{rem}
A {\em timelike line} in
$\ay$
is determined by any event
$e\in\ay$
and consists of all
$a\in\ay$
such that the 4-tuple
$(e,a)$
is harmonic. For the timelike line
$h_e\sub\ay$
determined by
$e=(x,x')$,
the points
$x$, $x'\in S^1$
are the ends of
$h_e$
at infinity. It follows from definitions that
$a\in h_e$
if and only if
$e\in h_a$.
Any two events on a timelike line are in the causal relation.
Conversely, for any two events
$a$, $a'\in\ay$
which are in the causal relation and not on a light line there is
a unique timelike line (the common perpendicular)
$h_e$
with
$a$, $a'\in h_e$
(this amounts to existence and uniqueness of a common perpendicular
to divergent geodesics in
$\hyp^2$.
For a (de Sitter) proof see Corollary~\ref{cor:common_perpendicular_prescribed}
and Lemma~\ref{lem:unique_common_perpendicular}).
Let
$a=(y,z)$, $a'=(y',z')$, $e=(x,x')$
in this case. Then the {\em time}
$t=t(a,a')$
between the events
$a$, $a'$
is defined by formula (\ref{eq:distance_log})
$$t=|\ln\langle x,y,y',x'\rangle|$$
(note that
$a$, $a'$
are sphere between
$x$, $x'$).
It follows that two timelike lines
$h_e$, $h_{e'}$
intersect each other if and only if the events
$e$, $e'\in\ay$
are in the causal relation and not on a light line. In this case, the intersection
$h_e\cap h_{e'}$
is a unique event.
An {\em elliptic line} in
$\ay$
is determined by any M\"obius involution without fixed points
$s\in Y$
and consists of all events
$a\in\ay$
such that
$sa=a$.
No two distinct events on an elliptic line are in the causal
relation.
\begin{rem}\label{rem:existence_elliptic} The last definition make sense
only for the canonical M\"obius structure
$M_0$
because in a general case a M\"obius structure may not admit any M\"obius
involution without fixed points.
\end{rem}
\subsection{Automorphisms of $M_0$}
\label{subsect:auto}
To introduce a metric structure on
$\ay$
we consider the Lie algebra
$\mathfrak g$
of the Lie group
$G=\SL_2(\R)$.
Given
$\al$, $\be\in\mathfrak g$
we have the Killing form
\begin{equation}\label{eq:killing_form}
\langle\al,\be\rangle=\frac{1}{2}\tr(\al\be)
\end{equation}
as a scalar product. Note that the matrices
$\si_1,\si_2,\si_3\in\mathfrak g$,
$$\si_1=
\begin{bmatrix}
1&0\\
0&-1
\end{bmatrix},\quad
\si_2=
\begin{bmatrix}
0&1\\
1&0
\end{bmatrix},\quad
\si_3=
\begin{bmatrix}
0&1\\
-1&0
\end{bmatrix},
$$
are mutually orthogonal and
$\|\si_1\|^2=\langle\si_1,\si_1\rangle=1=\|\si_2\|^2$,
$\|\si_3\|^2=-1$.
The group
$G$
acts on
$\wh\R=\R\cup\{\infty\}$
by linear-fractional transformations
\[x\mapsto\frac{ax+b}{cx+d}\quad\text{for}\quad\begin{bmatrix}
a&b\\
c&d
\end{bmatrix}
\in G\]
which are M\"obius with respect to the canonical M\"obius structure
$M_0$.
The action is not effective with the kernel
$\Z_2=\{\pm\id\}$, $G/\Z_2=\PSL_2(\R)$.
The group
$\PSL_2(\R)$
with the left invariant Lorentz metric (\ref{eq:killing_form})
is anti de Sitter 3-space
$\ads^3$.
We denote by
$K_i=\set{\exp(t\si_i)}{$t\in\R$}$, $i=1,2,3$,
a 1-parametric subgroup in
$G$,
and by
$\wh K_i$
its image in
$\PSL_2(\R)$.
Note that
$\wh K_i=K_i$
for
$i=1,2$
and
that
$\wh K_3=K_3/\Z_2$.
The space
$Y$
of M\"obius involutions
$s:S^1\to S^1$
without fixed points can be identified with the homogeneous space
$G/K_3=\PSL_2(\R)/\wh K_3$
because the group
$$K_3=\set{g_3(t)=\exp(t\si_3)=\begin{bmatrix}
\cos t&\sin t\\
-\sin t&\cos t
\end{bmatrix}}{$t\in\R$}$$
stabilizes
$s=g_3(\frac{\pi}{2})=\begin{bmatrix}
0&1\\
-1&0
\end{bmatrix}$
which acts on
$\wh\R$
as the M\"obius involution
$s(x)=-\frac{1}{x}$
without fixed points. The space
$G/K_3$
carries a left-invariant Riemannian metric
$h_3$
originated from the subspace
$L_3\sub\mathfrak g$
spanned by
$\si_1$, $\si_2$,
and
$(G/K_3,h_3)$
is isometric to
$\hyp^2$.
To see that we compute the respective Riemannian distance
between two involutions
$s_1$, $s_2\in Y$.
By conjugation we can assume that
$s_1=s$, $s_2=s'$,
where
$s'=g_1(t)\cdot s\cdot g_1^{-1}(t)$
for some
$t\in\R$,
$$g_1(t)=\exp(t\si_1)=\begin{bmatrix}
e^t&0\\
0&e^{-t}.
\end{bmatrix}
$$
Then
$s'=\begin{bmatrix}
0&e^{2t}\\
-e^{-2t}&0
\end{bmatrix}$
and
$s'(x)=-\frac{e^{4t}}{x}$.
The curve
$t\mapsto g_1(t)$
is a unit speed geodesic in
$G$.
While projected to
$\PSL_2(\R)$
the speed is doubled because of linear-fractional action of
$\PSL_2(\R)$,
so we have
$|ss'|=2t$.
In the upper half-plane model of
$\hyp^2$
the involution
$s$
fixes
$i=(0,1)$
with Euclidean distance
$|0i|_e=1$
and
$s'$
fixes
$ie^{2t}=(0,e^{2t})$
with Euclidean distance
$|0ie^{2t}|_e=e^{2t}$,
hence
$|ss'|=2t$
equals the
$\hyp^2$-distance
$|(0,1)(0,e^{2t})|=\ln\frac{|0ie^{2t}|_e}{|0i|_e}=2t$.
The action of
$\PSL_2(\R)$
on
$\wh\R\approx S^1$
induces the {\em standard} action of
$\PSL_2(\R)$
on
$\ay$.
Note that
$a=\{-1,1\}\in\ay$
is a fixed point for
$K_2$
because
$$\exp(t\si_2)=\begin{bmatrix}
\cosh t&\sinh t\\
\sinh t&\cosh t
\end{bmatrix},$$
and
$$\exp(t\si_2)a=\left\{\frac{\cosh t(-1)+\sinh t}{\sinh t(-1)+\cosh t},
\frac{\cosh t+\sinh t}{\sinh t+\cosh t}\right\}=\{-1,1\}=a.$$
It follows that
$\ay$
can be identified with the homogeneous space
$\PSL_2(\R)/K_2$,
or similarly with
$\PSL_2(\R)/K_1$.
The space
$\ay=\PSL_2(\R)/K_2$
carries a left-invariant Lorentz metric
$h_2$
originated from the subspace
$L_2\sub\mathfrak g$
spanned by
$\si_1$, $\si_3$,
and
$\ds^2=(\ay,h_2)$.
In the rest of the paper, we explain how a M\"obius structure
$M$
from a large class of structures on the circle gives rise to a spacetime,
and vice versa, without any assumption on symmetries of
$M$.
\section{Timed causal spaces on the circle}
\label{sect:timed_causal}
In this section we list axioms for timed causal spaces
on the circle.
\subsection{The canonical causality structure}
\label{subsect:causality_structure}
Recall that on the space
$\ay$
of unordered pairs of distinct point in
$X=S^1$,
which is homeomorphic to the open M\"obius band, we have the canonical causal
structure. That is, events
$e$, $e'\in\ay$
are in the causal relation if and only if they do not separate
each other as pairs of points in
$X$.
In the opposite case, we also say that events
$e$, $e'\in\ay$
separate each other. In the case
$e$, $e'\in\ay$
are in the causal relation and not on a light line, we say
that the events
$e$, $e'$
are in the {\em strong} causal relation.
The canonical causal structure and light lines inherent to
$\ay$,
see sect.~\ref{subsect:recoveting_desitter}.
For a fixed event
$e\in\ay$
the set
$C_e$
of all
$e'\in\ay$
in the causal relation with
$e$
is called the {\em causal cone}. The pair
$e\sub X$
decomposes
$X$
into two closed arcs, which we denote by
$e^+$, $e^-$,
with
$e^+\cap e^-=e$.
Every
$a\in\ay$
with
$a\sub e^\pm$
is in the causal relation with
$e$.
We let
$$C_e^\pm=\set{a\in\ay}{$a\sub e^\pm$}.$$
Therefore, a choice of
$e^+$, $e^-$
induces the decomposition
$C_e=C_e^+\cup C_e^-$
of the causal cone
$C_e$
into the {\em future} cone
$C_e^+$
and the {\em past} cone
$C_e^-$
with
$C_e^+\cap C_e^-=e$,
and moreover introduces a partial order on
$\ay$
in the following way. Every
$a\in C_e^\pm$
decomposes
$X$
into two closed arcs, and if
$a\neq e$,
we canonically define
$a_e^\pm$
as one of them that does not contain
$e$,
otherwise
$a_e^\pm=e^\pm$.
Now we define:
$a\le_e a'$
if and only if one of the following holds
\begin{itemize}
\item $a\in C_e^-$, $a'\in C_e^+$
\item $a'\sub a_e^+$
if
$a,a'\in C_e^+$
and
$a\sub (a')_e^-$
if
$a,a'\in C_e^-$.
\end{itemize}
As usual, we say that
$a<_e a'$,
if
$a\le_e a'$
and
$a\neq a'$.
Note that there is no global partial order on
$\ay$
compatible with the canonical causal structure, and the order above
only appears if an event
$e\in\ay$,
future
$e^+$
and past
$e^-$
arcs are chosen.
\subsection{Timelike lines and a causal space}
\label{subsect:hyplines}
The notion of a {\em timelike line} is not inherent in
$\ay$,
and we define this notion axiomatically.
\medskip\noindent
{\bf Axioms for timelike lines}
\begin{itemize}
\item[(h1)] every event
$e\in\ay$
uniquely determines a timelike line
$h_e\sub\ay$,
and every timelike line in
$\ay$
is of form
$h_e$
for some
$e\in\ay$;
\item[(h2)] any event
$a\in h_e$
separates
$e$;
\item[(h3)] any two events on a timelike line are in the causal
relation;
\item[(h4)] for any point
$x\in X\sm e$
there is a unique event
$x_e=(x,y)\in h_e$;
\item[(h5)] if an event
$a\in\ay$
is on a timelike line
$h_e$,
then
$e\in h_a$;
\item[(h6)] for any two distinct events
$a$, $a'\in\ay$
there is at most one timelike line
$h_e$
with
$a$, $a'\in h_e$.
\end{itemize}
The space
$\ay$
with a fixed collection
$\cH$
of subsets satisfying the axioms of
timelike lines is called a {\em causal space}. We use
notation
$(\ay,\cH)$
for a causal space. In view of Axiom~(h1), we say that an event
$e\in\ay$
and the timelike line
$h_e\sub\ay$
are {\em dual} to each other.
It follows from (h4) that for every event
$e=(z,u)\in\ay$
we have a well defined map
$\rho_e:X\to X$
given by
$\rho_e(z)=z$, $\rho_e(u)=u$,
and
$(x,\rho_e(x))=x_e$
for every
$x\in X\sm e$.
The map
$\rho_e$
is called the {\em reflection with respect to}
$e$.
\begin{lem}\label{lem:reflection} For every
$e=(z,u)\in\ay$,
the map
$\rho_e:X\to X$
is an involutive homeomorphism.
\end{lem}
\begin{proof} By definition,
$\rho_e^2(z)=z$, $\rho_e^2(u)=u$.
Let
$y=\rho_e(x)$
for
$x\in X\sm e$.
Then events
$(x,y),(\rho_e(y),y)\in h_e$,
hence
$\rho_e(y)=x$
by (h4). Thus
$\rho_e^{-1}=\rho_e$,
and
$\rho_e$
is a bijection.
The event
$e$
decomposes
$X$
into two closed arcs
$e^+$, $e^-$
with
$X=e^+\cup e^-$, $e^+\cap e^-=e$.
An orientation of
$X$
determines linear orders on
$e^+$, $e^-$.
It follows from (h2) and (h3) that
$\rho_e:e^+\to e^-$
reverses the orders. Hence,
$\rho_e$
is continuous and, therefore, a homeomorphism.
\end{proof}
\begin{pro}\label{pro:timelike_lines} Let
$(\ay,\cH)$
be a causal space. Then
\begin{itemize}
\item[(a)] for any
$e\in\ay$
the line
$h_e$
is homeomorphic (in the induced from
$\ay$
topology) to
$\R$,
and the boundary of the closure
$\ov h_e\sub\ay\cap\di\ay$
is
$e$, $\d\ov h_e=e$;
\item[(b)] for any two distinct events
$a$, $a'\in\ay$
there is a timelike line
$h$
including
$a$, $a'$
if and only if
$a$
and
$a'$
are in the strong causal relation. In this case,
$h$
is unique with this property;
\item[(c)] any two distinct timelike lines
$h_e$, $h_{e'}\in\cH$
have a common event
$a$
if and only if
$e$, $e'$
are in the strong causal relation.
In this case,
$a$
is unique;
\item[(d)] a light line
$p_x\sub\ay$
intersects a timelike line
$h_e\sub\ay$
if and only if
$x\not\in e$.
In this case, the common event
$a\in p_x\cap h_e$
is unique.
\end{itemize}
\end{pro}
\begin{proof} (a) Let
$\rho_e:X\to X$
be the reflection with respect to
$e$, $e^+\sub X$
one of the two closed arcs, in which
$e$
decomposes
$X$.
Then the map
$\intr e^+\to h_e$, $x\mapsto(x,\rho_e(x))$
is an order preserving bijection. Extended to
$e^+$,
it gives an order preserving bijection to
$\ov h_e=h_e\cup e$.
Thus
$h_e$
is homeomorphic to
$\intr e^+\approx\R$,
and
$\d\ov h_e=e$.
(b) Any distinct
$a$, $a'\in\ay$
on a timelike line
$h$
are in the causal relation by (h3), and by (h4)
they are not on a light line. Hence,
$a$, $a'$
are in the strong causal relation. Conversely, assume that events
$a$, $a'\in\ay$
are in the strong causal relation. Let
$\rho=\rho_a\circ\rho_{a'}$
be the composition of respective reflections,
$a^+\sub X$
the closed arc determined by
$a$
that does not contain
$a'$.
Then
$\rho(a^+)\sub\intr a^+$,
and thus there is a fixed point
$x\in\intr a^+$
of
$\rho$, $\rho(x)=x$.
It follows that the both reflections
$\rho_a$, $\rho_{a'}$
preserve the event
$e=(x,y)$,
where
$y=\rho_{a'}(x)$.
Hence,
$e\in h_a\cap h_{a'}$,
and by (h5),
$a,a'\in h_e$.
By (h6),
$h_e$
is unique with this property.
(c) By duality (h5), this is a reformulation of (b).
(d) This immediately follows from (h2) and (h4).
\end{proof}
\begin{rem}\label{rem:axiom_h6} Axiom~(h6) is not used in
Lemma~\ref{lem:reflection}, and it is only used in
Proposition~\ref{pro:timelike_lines} to prove the uniqueness in (b) and (c).
Thus all the conclusions of Proposition~\ref{pro:timelike_lines}
except for the uniqueness in (b) and (c) hold true without Axiom~(h6).
We use this remark in sect.~\ref{subsect:timelike_lines}.
\end{rem}
\subsection{Timed causal space}
\label{subsect:time}
The notion of a {\em time} is also defined axiomatically.
\medskip\noindent
{\bf The time axioms}
\begin{itemize}
\item[(t1)]
A time
$t(e,e')\ge 0$
between two events
$e$, $e'\in\ay$
is determined if and only if
$e$, $e'$
are in the causal relation;
\item[(t2)] $t(e,e')=0$
if and only if
$e$, $e'$
are events on a light line;
\item[(t3)] $t(e,e')=t(e',e)$
whenever it is defined;
\item[(t4)] timelike lines are
$t$-geodesics:
(a) if
$e,e',e''\in h_a$
are events on a timelike line such that
$e\le e'\le e''$,
then
$t(e,e')+t(e',e'')=t(e,e'')$;
(b) for every
$e\in h_a$
and every
$s>0$
there are
$e_\pm\in h_a\cap C_e^\pm$
with
$t(e,e_\pm)=s$;
\item[(t5)] for any events
$e=(x,y)$, $d=(z,u)$
in the strong causal relation, we have
$t(z_e,u_e)=t(x_d,y_d)$;
\item[(t6)] for any
$e=(x,y), d=(z,u)\in h_e$
the 4-tuple
$(d,e)$
is {\em harmonic} in the sense that
$t(y_a,u_a)=t(y_b,z_b)$,
where
$a=(x,z)$, $b=(x,u)$.
\end{itemize}
A {\em timed causal space} is defined as
$T=(\ay,\cH,t)$,
where
$t$
is a time on the causal space
$(\ay,\cH)$.
This is a version of Busemann (locally) timelike spaces,
see \cite{Bus}, and also an axiomatic version of the de Sitter space
$\ds^2$.
Since
$\ds^2$
is recovered from the canonical M\"obius structure
$M_0$
on the circle, see sect.~\ref{subsect:recoveting_desitter},
it follows from results of sect.~\ref{sect:monotone} (see
Proposition~\ref{pro:safisfies_hypline_axiom},
Lemma~\ref{lem:unique_common_perpendicular} and
Proposition~\ref{pro:satisfies_time_axioms}), that
$\ds^2$
is a timed causal space.
We denote by
$\cT$
the set of all timed causal spaces
$(\ay,\cH,t)$,
where the collection
$\cH$
of timelike line satisfies Axioms~(h1)--(h6), and the time
$t$
satisfies Axioms~(t1)--(t6). A
$T$-{\em automorphism}, $T=(\ay,\cH,t)\in\cT$,
is a bijection
$g:\ay\to\ay$
that preserves the timelike lines
$\cH$
and the time
$t$, $t(g(e),g(e'))=t(e,e')$
whenever
$t(e,e')$
is defined (we do not require to preserve the causality structure because this is
automatic).
\rem\label{rem:terminology} I am not satisfied with a terminology from
e.g. \cite{Bus}, \cite{PY}, where the term ``timelike (metric) space''
is used, because a respective object is never a metric space and its basic
feature is a causality relation. On the other hand to say ``timelike causal space''
sounds a little bit tautologically. Thus I use the term ``timed causal space'' instead.
\rem\label{rem:axioms_t5_t6} Strange looking Axioms~(t5), (t6) are automatically
satisfied for timed causal spaces induced by monotone M\"obius structures
on the circle, see sect.~\ref{sect:monotone}. However, they value and importance
are justified by the fact that (t5), (t6) are indispensable while one recovers
a M\"obius structure on the circle from a timed causal space, see
sect.~\ref{sect:timed_causal_moeb_structures}, especially
Lemma~\ref{lem:monb} and Lemma~\ref{lem:pro:timed_monotone:harmonic}.
In fact, Axiom~(t6) follows from the other axioms, see Appendix~2.
\section{M\"obius structures and hyperbolic spaces}
\label{sect:moeb}
On the boundary at infinity of any boundary continuous
Gromov hyperbolic space there is an induced M\"obius
structure. In this section, we recall details of this fact.
\subsection{Semi-metrics and topology}
\label{subsect:semi-metrics_topology}
Let
$X$
be a set. A function
$d:X^2\to\wh\R=\R\cup\{\infty\}$
is called a {\em semi-metric}, if it is symmetric,
$d(x,y)=d(y,x)$
for each
$x$, $y\in X$,
positive outside the diagonal, vanishes on the diagonal
and there is at most one infinitely remote point
$\om\in X$
for
$d$,
i.e. such that
$d(x,\om)=\infty$
for some
$x\in X\sm\{\om\}$.
Moreover, if
$\om\in X$
is such a point, then
$d(x,\om)=\infty$
for all
$x\in X$, $x\neq\om$.
A metric is a semi-metric that satisfies the triangle inequality.
A 4-tuple
$q=(x_1,x_2,x_3,x_4)\in X^4$
is said to be {\em nondegenerate} if all its entries are pairwise
distinct. We denote by
$\reg\cP_4=\reg\cP_4(X)$
the set of ordered nondegenerate 4-tuples.
A {\em M\"obius structure}
$M$
on
$X$
is a class of M\"obius equivalent semi-metrics on
$X$,
where two semi-metrics are equivalent if and only if they have
the same cross-ratios on every
$q\in\reg\cP_4$.
An
$M$-{\em automorphism}
is a bijection
$f:X\to X$
that preserves cross-ratios.
Given
$\om\in X$,
there is a semi-metric
$d_\om\in M$
with infinitely remote point
$\om$.
It can be obtained from any semi-metric
$d\in M$
for which
$\om$
is not infinitely remote by a {\em metric inversion},
$$d_\om(x,y)=\frac{d(x,y)}{d(x,\om)d(y,\om)}.$$
Such a semi-metric is unique up to a homothety, see \cite{FS},
and we use notation
$|xy|_\om=d_\om(x,y)$
for the distance between
$x$, $y\in X$
in that semi-metric. We also use notation
$X_\om=X\sm\{\om\}$.
Every M\"obius structure
$M$
on
$X$
determines the
$M$-{\em topology}
whose subbase is given by all open balls centered at finite points
of all semi-metrics from
$M$
having infinitely remote points.
For the following fact see \cite[Corollary~4.3]{Bu1} in a more
general context of sub-M\"obius structures. We give here its proof for
convenience of the reader.
\begin{lem}\label{lem:continuity_semimetrics} For every
$\om\in X$,
for a semi-metric
$d\in M$
with infinitely remote point
$\om\in X$
and for every
$x\in X_\om$
the function
$f_x:X\to\wh\R$, $f_x(y)=d(x,y)$,
is continuous in the
$M$-topology.
\end{lem}
\begin{proof} The function
$f_x$
takes values in
$[0,\infty]$.
For
$s,t\in[0,\infty]$
let
$B_s(x)=\set{y\in X}{$f_x(y)<s$}$
be the open
$d$-ball
of radius
$s$
centered at
$x$, $C_t(x)=\set{y\in X}{$f_x(y)>t$}$
the complement of the closed
$d$-ball.
The inverse image
$f_x^{-1}(I)$
of any open interval
$I\sub[0,\infty]$
is either an open ball
$B_s(x)$,
or a complement
$C_t(x)$,
or an intersection
$B_s(x)\cap C_t(x)$
for some
$s>t$.
Let
$d_x\in M$
be the metric inversion of
$d$.
Then
$d_x(y,\om)=1/d(x,y)$,
hence
$C_t(x)=\set{y\in X}{$d_x(y,\om)<1/t$}$
is the open
$d_x$-ball of radius
$1/t$
centered at
$\om$.
It follows that
$f_x^{-1}(I)$
is open in the
$M$-topology.
\end{proof}
\subsection{Boundary continuous hyperbolic spaces}
\label{subsect:boundary_continuous}
Let
$Y$
be a metric space. Recall that the Gromov product
$(x|y)_o$
of
$x$, $y\in Y$
with respect to
$o\in Y$
is defined by
$$(x|y)_o=\frac{1}{2}(|xo|+|yo|-|xy|),$$
where
$|xy|$
is the distance in
$Y$
between
$x$, $y$.
We use the following definition of a hyperbolic space
adapted to the case of geodesic metric spaces.
\begin{definition}\label{def:gromov_hyperbolic}
A geodesic metric space
$Y$
is {\em Gromov hyperbolic}, if for some
$\de\ge 0$
and any triangle
$xyz\sub Y$
the following holds: If
$y'\in xy$, $z'\in xz$
are points with
$|xy'|=|xz'|\le(y|z)_x$,
then
$|y'z'|\le\de$.
In this case, we also say that
$Y$
is
$\de$-hyperbolic,
and
$\de$
is a {\em hyperbolicity constant} of
$Y$.
\end{definition}
A Gromov hyperbolic space
$Y$
is {\em boundary continuous} if the Gromov product extends
continuously onto the boundary at infinity
$\di Y=X$
in the following
way: given
$\xi$, $\eta\in X$,
for any sequences
$\{x_i\}\in\xi$, $\{y_i\}\in\eta$
there is a limit
$(\xi|\eta)_o=\lim_i(x_i|y_i)_o$
for every
$o\in Y$,
for more details, see \cite[sect.~3.4.2]{BS1}. Note that in this case
$(\xi|\eta)_o$
is independent of the choice
$\{x_i\}\in\xi$, $\{y_i\}\in\eta$.
This allows one to define for every
$o\in Y$
a function
$(\xi,\eta)\mapsto d_o(\xi,\eta)=e^{-(\xi|\eta)_o}$,
which is a semi-metric on
$X$.
\begin{lem}\label{lem:semi-metrics_moeb} Let
$Y$
be a boundary continuous hyperbolic space. Then for any
$o$, $o'\in Y$,
the semi-metrics
$d_o$, $d_{o'}$
on
$X=\di Y$
are M\"obius equivalent.
\end{lem}
\begin{proof} Given 4-tuple
$(x,y,z,u)\sub Y$,
we put
$$\cd_o(x,y,z,u)=(x|u)_o+(y|z)_o-(x|z)_o-(y|u)_o$$
for a fixed
$o\in Y$.
Then
$\cd_o(x,y,z,u)=\cd(x,y,z,u)$
is independent of the choice of
$o$
because all entries containing
$o$
enter
$\cd_o(x,y,z,u)$
twice with the opposite signs.
Now, given a nondegenerate 4-tuple
$q=(\al,\be,\de,\ga)\in\reg\cP_4(X)$,
for any
$\{x_i\}\in\al$, $\{y_i\}\in\be$, $\{z_i\}\in\ga$, $\{u_i\}\in\de$,
by the boundary continuity of
$Y$,
there is a limit
$$\cd(\al,\be,\ga,\de)=\lim_i\cd(x_i,y_i,z_i,u_i),$$
which coincides with
$(\al|\de)_o+(\be|\ga)_o-(\al|\ga_o)-(\be|\de)_o$.
Thus the cross-ratio
$$\frac{d_o(\al,\ga)d_o(\be,\de)}{d_o(\al,\de)d_o(\be,\ga)}=\exp(-\cd(\al,\be,\ga,\de))$$
is independent of
$o$.
Hence, semi-metrics
$d_o$, $d_{o'}$
are M\"obius equivalent for any
$o$, $o'\in Y$.
\end{proof}
The M\"obius structure
$M$
on the boundary at infinity
$X=\di Y$
of a boundary continuous hyperbolic space
$Y$
generated by any semi-metric
$d_o(\xi,\eta)=\exp(-(\xi|\eta)_o)$, $o\in Y$,
is said to be {\em induced} (from
$Y$).
For any
$\om\in X$, $o\in Y$,
the metric inversion
$d_\om$
of
$d_o$
with respect to
$\om$
is a semi-metric on
$X$
from
$M$
with the infinitely remote point
$\om$.
Recall that any two semi-metrics in
$M$
with a common infinitely remote point are proportional to each other.
Thus metric inversions with respect to
$\om$
of semi-metrics
$d_o$, $d_{o'}$
are proportional to each other for any
$o$, $o'\in Y$.
In {\bf Appendix}, we show that every proper Gromov hyperbolic
$\CAT(0)$
space is boundary continuous, see Theorem~\ref{thm:cat0_boundary_contiuous}.
\section{Monotone M\"obius structures on the circle}
\label{sect:monotone}
\subsection{Axioms for monotone M\"obius structures on the circle}
\label{subsect:axioms_monotone}
We say that a M\"obius structure
$M$
on
$X=S^1$
is {\em monotone}, if it satisfies the following Axioms
\begin{itemize}
\item [(T)] Topology: $M$-topology
on
$X$
is that of
$S^1$;
\item[(M)] Monotonicity: given a 4-tuple
$q=(x,y,z,u)\in X^4$
such that the pairs
$(x,y)$, $(z,u)$
separate each other,
we have
$$|xy|\cdot|zu|>\max\{|xz|\cdot|yu|,|xu|\cdot|yz|\}$$
for some and hence any semi-metric from
$M$.
\end{itemize}
\begin{rem}\label{rem:ref_schroeder} These Axioms have arisen in
a discussion with V.~Schroeder while working on \cite{BS4}.
\end{rem}
A choice of
$\om\in X$
uniquely determines the interval
$xy\sub X_\om$
for any distinct
$x$, $y\in X$
different from
$\om$
as the arc in
$X$
with the end points
$x$, $y$
that does not contain
$\om$.
As an useful reformulation of Axiom~(M) we have
\begin{cor}\label{cor:interval_monotone} Assume for a nondegenerate
4-tuple
$q=(x,y,z,u)\in\reg\cP_4$
the interval
$xz\sub X_u$
is contained in
$xy$, $xz\sub xy\sub X_u$.
Then
$|xz|_u<|xy|_u$.
\end{cor}
\begin{proof} By the assumption, the pairs
$(x,y)$, $(z,u)$
separate each other. Hence, by Axiom~(M) we have
$|xz||yu|<|xy||zu|$
for any semi-metric from
$M$.
In particular,
$|xz|_u<|xy|_u$.
\end{proof}
We denote by
$\cM$
the class of monotone M\"obius structures on
$S^1$.
\subsection{Examples of monotone M\"obius structures on the circle}
\label{subsect:examples_monotone}
By Theorem~\ref{thm:cat0_boundary_contiuous}, every proper
Gromov hyperbolic
$\CAT(0)$
space
$Y$
is boundary continuous, and thus
$\di Y$
possesses an induced M\"obius
structure.
Recall that in any
$\CAT(0)$
space
$Y$,
the angle
$\angle_o(x,x')$
between geodesic segments
$ox$, $ox'$
with a common vertex
$o$
is well defined and by definition it is at most
$\pi$, $\angle_o(x,x')\le\pi$.
A point
$o$
in a
$\CAT(0)$
space
$Y$
with
$\di Y$
homeomorphic to the circle
$S^1$
is said to be {\em singular,}
if there are two geodesics
$\xi\xi'$, $\eta\eta'\sub Y$
through
$o$
such that the pairs of points
$(\xi,\xi')$
and
$(\eta,\eta')$
in
$\di Y$
separate each other, and
$\angle_o(\xi,\eta)+\angle_o(\xi',\eta')\ge 2\pi$.
\begin{thm}\label{thm:without_singular} Let
$Y$
be a Gromov hyperbolic
$\CAT(0)$
surface with
$\di Y=S^1$
and without singular points. Then the induced M\"obius structure
$M$
on
$X=\di Y$
is monotone.
\end{thm}
\begin{proof} For the induced M\"obius structure, the
$M$-topology
on
$X$
coincides with the standard Gromov topology, see \cite[Sect.~2.2.3]{BS1} or
\cite[Lemma~5.1]{Bu1}.
Thus
$M$
satisfies Axiom~(T) by the assumption. To check Axiom~(M), consider a 4-tuple
$q=(\xi,\xi',\eta,\eta')\in X^4$
such that the pairs
$(\xi,\xi')$
and
$(\eta,\eta')$
separate each other. Since
$Y$
is Gromov hyperbolic, there are geodesics
$\xi\xi'$, $\eta\eta'\sub Y$
with the end points at infinity
$\xi,\xi'$
and
$\eta,\eta'$
respectively. The assumption on separation and the fact that
$Y$
is a
$\CAT(0)$
surface imply that these geodesics intersect at some point
$o$.
We have
$(\xi|\xi')_o=0=(\eta|\eta')_o$.
Thus
$|\xi\xi'|=1=|\eta\eta'|$
for the semi-metric
$|xy|=\exp(-(x|y)_o)$
on
$X$.
Recall that this semi-metric is a semi-metric of
$M$.
The angles at
$o$
between the rays
$ox$, $x=\xi,\eta,\xi',\eta'$
in this cyclic order, form two opposite pairs.
Since
$o$
is not singular in
$Y$,
at least one of angles
$\angle_o(x,z)<\pi$
for each opposite pair.
By Corollary~\ref{cor:zero_gromov_product},
$(x|z)_o>0$,
thus
$|xz|<1$.
It follows that
$$|\xi\xi'|\cdot|\eta\eta|>\max\{|\xi\eta|\cdot|\xi'\eta'|,
|\xi'\eta|\cdot|\xi\eta'|\},$$
i.e.,
$M$
satisfies Axiom~(M).
\end{proof}
\begin{exas}\label{exas:monotone_nonmonotone} 1. The canonical M\"obius
structure
$M_0$
on the circle is monotone.
2. Let
$S$
be a closed surface of negative Euler characteristic with an Euclidean metric
having cone type singularities with
complete angles
$>2\pi$
about every singular point,
$Y$
the universal covering of
$S$
with the lifted metric. Then
$Y$
is a Gromov hyperbolic
$\CAT(0)$
surface with
$\di Y=S^1$.
It follows from Theorem~\ref{thm:cat0_boundary_contiuous} that
$Y$
induces a M\"obius structure
$M$
on
$\di Y$.
However,
$M$
is not monotone.
3. Absence of singular points on a Gromov hyperbolic
$\CAT(0)$
surface
$Y$
does not mean that
$Y$
has no metric singularities. Remove an open equidistant neighborhood
of a geodesic line in
$\hyp^2$
and glue remaining pieces by an isometry between their boundaries.
Then the obtained
$Y$
is a proper Gromov hyperbolic
$\CAT(-1)$
surface with
$\di Y=S^1$
without singular points, and Theorem~\ref{thm:without_singular}
can be applied to
$Y$.
At the same time,
$Y$
has metric singularities along the gluing line.
4. Every (topological) embedding
$f:S^1\to\wh\R^2$
induces on
$S^1$
some metric and therefore a M\"obius structure
$M_f$.
For
$f=\id$,
the M\"obius structure
$M_f$
coincides with the canonical one, and thus it is monotone. However, if
$f(S^1)\sub\R^2$
is an ellipse with principal semi-axes
$a$, $b$
such that
$4ab\le a^2+b^2$,
then
$M_f$
is not monotone. That is, the M\"obius structure induced on a convex
curve in
$\wh\R^2$
in general is not monotone.
\end{exas}
In what follows, we assume that a monotone M\"obius structure
$M$
on
$X=S^1$
is fixed.
\subsection{Harmonic pairs}
\label{subsect:harmonic_events}
A pair
$a=(x,y)$, $b=(z,u)\in\ay$
of events is
{\em $M$-harmonic} or form an
$M$-harmonic 4-tuple, if
\begin{equation}\label{eq:harmonic_events}
|xz|\cdot|yu|=|xu|\cdot|yz|
\end{equation}
for some and hence any semi-metric of the M\"obius structure
(this is the same as (\ref{eq:harmonic}). However, for a general M\"obius
structure
$M$
on
$S^1$,
the interpretation of (\ref{eq:harmonic}) for the canonical
$M_0$
as orthogonality of respective geodesic lines in
$\hyp^2$
makes no sense). Nevertheless, in this case we also say that
$a$, $b$
are mutually {\em orthogonal,}
$a\perp b$.
Note that any harmonic 4-tuple
$q=(a,b)$
is nondegenerate.
\begin{lem}\label{lem:harm_separate} If events
$a=(x,y)$, $b=(z,u)\in\ay$
are mutually orthogonal,
$a\perp b$,
then they separate each other and
$z\in xy\sub X_u$
is a unique midpoint.
\end{lem}
\begin{proof} We have
$|xz|_u=|zy|_u$.
By Corollary~\ref{cor:interval_monotone},
$z\in xy\sub X_u$
is a unique midpoint, and thus
$a$, $b$
separate each other.
\end{proof}
\begin{lem}\label{lem:harm_exist} For every
$e\in\ay$
and every
$x\in X\sm e$
there is a unique
$y\in X$
such that the pair
$a=(x,y)$, $e\in\ay$
of events is harmonic.
\end{lem}
\begin{proof} Let
$e=(z,u)$.
By Axiom~(T) and Lemma~\ref{lem:continuity_semimetrics}, the functions
$f_z,f_u:X\to\wh\R=\R\cup\{\infty\}$,
$f_z(t)=|zt|_x$, $f_u(t)=|ut|_x$
are continuous on
$X=S^1$,
and they take values between
$0=f_z(z)=f_u(u)$
and
$f_z(u)=f_u(z)>0$
on the segment
$zu\sub X_x$.
Thus there is a midpoint
$y\in zu\sub X_x$
between
$z$
and
$u$, $|zy|_x=|yu|_x$.
By Corollary~\ref{cor:interval_monotone}, such a point
$y$
is unique.
\end{proof}
\begin{pro}\label{pro:monotone_moeb_to_timed_space} For every
monotone M\"obius structure
$M\in\cM$
there is a uniquely determined timed causal space
$T=\wh T(M)\in\cT$
such that the automorphism group
$G_M$
of
$M$
injects into the automorphism group
$G_T$
of
$T$:
If
$g:X\to X$
is an
$M$-M\"obius
automorphism, then
the induced
$\wh g:\ay\to\ay$
is an automorphism of
$T$.
\end{pro}
We prove Proposition~\ref{pro:monotone_moeb_to_timed_space}
in sections~\ref{subsect:timelike_lines} and \ref{subsect:time_events}.
\subsection{Timelike lines}
\label{subsect:timelike_lines}
Any timelike line
$h$
in
$\ay$
is associated with an event
$e\in\ay$, $h=h_e$,
and is defined as the set of events
$a\in\ay$
such that the pair
$(a,e)$
is harmonic,
$h_e=\set{a\in\ay}{$a\perp e$}$.
We denote by
$\cH=\cH_M$
the collection of timelike lines in
$\ay$.
\begin{pro}\label{pro:safisfies_hypline_axiom} The collection
$\cH$
satisfies Axioms~(h1)--(h5).
\end{pro}
\begin{proof} Axioms~(h1), (h5) hold by definition, (h2)
follows from Lemma~\ref{lem:harm_separate}, (h4) from
Lemma~\ref{lem:harm_exist}.
To check (h3), assume
$e=(z,u)\in\ay$, $a=(x,y)$, $a'=(x',y')\in h_e$.
Then
$z$
is the midpoint of the segments
$xy$, $x'y'\sub X_u$.
Thus by Axiom~(M), the pairs
$(x,y)$, $(x',y')$
do not separate each other, that is, the events
$a$
and
$a'$
are in the causal relation. Hence, (h3).
\end{proof}
We say that an event
$a\in\ay$
is a {\em common perpendicular} to events
$e$, $e'\in\ay$,
if
$e$, $e'\in h_a$.
\begin{cor}\label{cor:common_perpendicular_prescribed} Given
$e$, $e'\in\ay$
in the strong causal relation, there is a common perpendicular
$a\in\ay$
to
$e$, $e'$.
\end{cor}
\begin{proof} By Proposition~\ref{pro:safisfies_hypline_axiom},
the collection
$\cH=\cH_M$
of timelike lines in
$\ay$
determined by the M\"obius structure
$M$
satisfies Axioms~(h1)--(h5). Thus the assertion follows from
Proposition~\ref{pro:timelike_lines}(b), see Remark~\ref{rem:axiom_h6}.
\end{proof}
The proof of (h6) we postpone to sect.~\ref{subsect:time_events},
see Lemma~\ref{lem:unique_common_perpendicular}.
\subsection{Time between events}
\label{subsect:time_events}
The time between events
$a$, $a'\in\ay$
is defined if and only they are in the causal relation. We do this
essentially as in sect.~\ref{subsect:recoveting_desitter} using
formula (\ref{eq:distance_log}). First of all, the time between
events on a light line by definition is zero,
$t(a,a')=0$
for
$a$, $a'\in p_x$, $x\in X$.
Next, assume that
$a$, $a'\in\ay$
are in the strong causal relation. Then by Corollary~\ref{cor:common_perpendicular_prescribed}
there is a common perpendicular
$e\in\ay$
to
$a$, $a'$,
that is,
$a$, $a'\in h_e$.
We let
$e=(x,y)$, $a=(z,u)$, $a'=(z',u')$.
Then by definition
\begin{equation}\label{eq:time}
t_e(a,a')=\left|\ln\frac{|xz'|\cdot|yz|}{|xz|\cdot|yz'|}\right|
\end{equation}
for some and hence any semi-metric on
$X$
from
$M$.
It follows from harmonicity of
$(a,e)$
and
$(a',e)$
that
\begin{equation}\label{eq:time_different}
t_e(a,a')=\left|\ln\frac{|xu'|\cdot|yu|}{|xu|\cdot|yu'|}\right|=
\left|\ln\frac{|xu'|\cdot|yz|}{|xz|\cdot|yu'|}\right|=
\left|\ln\frac{|xz'|\cdot|yu|}{|xu|\cdot|yz'|}\right|,
\end{equation}
and we often use these different representations of
$t_e(a,a')$.
\begin{lem}\label{lem:unique_common_perpendicular} Given distinct
$a$, $a'\in\ay$
in the causal relation, there is at most one common perpendicular
$b\in\ay$
to
$a$, $a'$.
In particular, the time
$t(a,a')=t_e(a,a')$
is well defined, and Axiom~(h6) is fulfilled for the collection
$\cH$
of timelike lines.
\end{lem}
\begin{proof} The idea is taken from \cite{BS4}. If events
$a$, $a'$
are on a light line, then they cannot lie on a timelike line,
say
$h_e$,
because by Axiom~(h1) they both must separate
$e$,
which would contradict (h4). Thus we assume that
$a$, $a'$
are not on a light line.
Assume there are common perpendiculars
$b=(z,u)$, $b'=(z',u')\in\ay$
to
$a$, $a'$,
that is
$b\perp a,a'$
and
$b'\perp a,a'$,
or which is the same,
$b$, $b'\in h_a\cap h_{a'}$.
By already established Axiom~(h3), see Proposition~\ref{pro:safisfies_hypline_axiom},
$b$
and
$b'$
do not separate each other.
Let
$a=(x,y)$, $a'=(x',y')$.
Without loss of generality, we assume that on
$X_x$
we have the following order of points
$zz'yy'u'ux'$.
By Axiom~(h5),
$a$, $a'\in h_b\cap h_{b'}$.
The times
$t=t_b(a,a')$, $t'=t_{b'}(a,a')$
are already defined by (\ref{eq:time}). Computing them in
a semi-metric of the M\"obius structure with infinitely remote point
$x$,
we obtain
$$e^t=\frac{|zx'|}{|x'u|},\quad
e^{t'}=\frac{|z'x'|}{|x'u'|}.$$
Using the order of points
$zz'yy'u'ux'$
on
$X_x$,
we have, in particular, that the interval
$z'x'$
is contained in the interval
$zx'$.
By Corollary~\ref{cor:interval_monotone},
$|zx'|\ge|z'x'|$.
Similarly,
$x'u\sub x'u'$
and hence
$|x'u|\le|x'u'|$.
Thus
$t\ge t'$
and if
$b'\neq b$,
the inequality is strong. Applying this argument with infinitely remote point
$y$,
we obtain
$t\le t'$.
Therefore
$t=t'$
and
$b=b'$.
\end{proof}
\begin{pro}\label{pro:satisfies_time_axioms} The time
between events in
$\ay$
defined above satisfies Axioms~(t1)--(t6).
\end{pro}
\begin{proof} Axiom~(t1) is satisfied by the definition of the time
$t$.
Axiom~(t2): If events
$a$, $a'$
are on a light line, then
$t(a,a')=0$
by definition. Conversely, assume
$t(a,a')=0$
for events
$a$, $a'\in\ay$
in the causal relation, which are not on a light line, in particular,
$a\neq a'$.
Then by Lemmas~\ref{cor:common_perpendicular_prescribed}
and \ref{lem:unique_common_perpendicular}, there is a unique
$e\in\ay$
with
$a$, $a'\in h_e$.
We let
$e=(x,y)$, $a=(z,u)$, $a'=(z',u')$.
Since
$a\neq a'$,
we have
$z'\neq z,u$.
On the other hand, it follows from (\ref{eq:time}) that
$|xz'|\cdot|yz|=|xz|\cdot|yz'|$
for any semi-metric on
$X$
from
$M$.
In particular,
$|xz'|_y=|xz|_y$,
and by monotonicity (M),
$x$
is the midpoint between
$z$, $z'$
in
$X_y$.
Hence,
$\rho_e(z)=z'=u$,
a contradiction.
Axiom~(t3) follows from the definition of the time
$t$
and (\ref{eq:time}).
Axiom~(t4a): Let
$e$, $e'$, $e''\in h_a$
with
$e\le e'\le e''$.
If
$e'$
coincides with
$e$
or
$e''$,
then the required equality is trivial. Thus we assume that
$e<e'<e''$.
Without loss of generality, we can assume that for
$a=(x,y)$, $e=(z,u)$, $e'=(z',u')$, $e''=(z'',u'')$,
the points
$z$, $z'$, $z''$
lie on one and the same arc determined by
$a$
in the order
$xzz'z''y$.
Then
$$\exp(t(e,e'))=\frac{|xz'|\cdot|yz|}{|xz|\cdot|yz'|},\quad
\exp(t(e',e''))=\frac{|xz''|\cdot|yz'|}{|xz'|\cdot|yz''|},$$
and we obtain
$$\exp(t(e,e')+t(e',e''))=\frac{|xz''|\cdot|yz|}{|xz|\cdot|yz''|}
=\exp(t(e,e'')).$$
Axiom~(t4b): Given an event
$e=(z,u)$
on a timelike line
$h_a\sub\ay$
with
$a=(x,y)$,
and
$s>0$,
we take a semi-metric from
$M$
with the infinitely remote point
$y$.
Then
$|xz|_y=|xu|_y:=t>0$,
and without loss of generality, we can assume that
$t=1$.
Note that the function
$f_x:X_y\to\R$, $f_x(x')=|xx'|_y$,
is continuous, see Lemma~\ref{lem:continuity_semimetrics},
and monotone, see Corollary~\ref{cor:interval_monotone}.
It varies from
$0=f_x(x)$
to
$\infty=f_x(y)$.
Thus there are
$z_-\in xz$, $z_+\in zy$
with
$f_x(z_\pm)=e^{\pm s}$.
For the events
$e_\pm=(z_\pm,u_\pm)\in h_a$,
where
$u_\pm=\rho_a(z_\pm)$,
we have
$$t(e,e_\pm)=\left|\ln\frac{|xz_\pm|_y}{|xz|_y}\right|=|\ln|xz_\pm|_y|=s.$$
Choosing a decomposition
$X=e^+\cup e^-$
determined by
$e$
so that
$y\in e^+$,
we have
$e_\pm\in h_a\cap C_e^\pm$
and
$t(e,e_\pm)=s$.
Axiom~(t5): Let
$e=(x,y)$, $d=(z,u)$
be events in the strong causal relation. Then we have by (\ref{eq:time}),
(\ref{eq:time_different})
$$t(z_e,u_e)=\left|\ln\frac{|xu|\cdot|yz|}{|xz|\cdot|yu|}\right|=t(x_d,y_d).$$
Axiom~(t6): Given
$e=(x,y)\in\ay$, $d=(z,u)\in h_e$,
we put
$a=(x,z)$, $b=(x,u)$.
Then we have by (\ref{eq:time}), (\ref{eq:time_different})
$$t(y_a,u_a)=\left|\ln\frac{|xy|\cdot|zu|}{|xu|\cdot|zy|}\right|,\quad
t(y_b,z_b)=\left|\ln\frac{|xy|\cdot|zu|}{|xz|\cdot|yu|}\right|.$$
On the other hand, the pair
$(a,e)$
is harmonic, thus
$|xu|\cdot|zy|=|xz|\cdot|yu|$.
Hence,
$t(y_a,u_a)=t(y_b,z_b)$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pro:monotone_moeb_to_timed_space}]
Given a monotone M\"obius structure
$M\in\cM$,
we have defined above a class
$\cH=\cH_M$
of timelike lines in
$\ay$
that satisfies Axioms~(h1)--(h6), and a time
$t$
on
$(\ay,\cH)$
that satisfies Axioms~(t1)--(t6). Therefore, a timed causal space
$T=(\ay,\cH,t)$, $T=\wh T(M)$,
is defined.
Let
$g:X\to X$
be an
$M$-M\"obius
automorphism. Then the induced
$\wh g:\ay\to\ay$
preserves the causality structure, the class of timelike lines
$\cH$
and the time
$t$
because the last two are defined via cross-ratios. Thus
$\wh g$
is an automorphism of
$T$.
If
$\wh g=\id_T$,
then, in particular, it preserves every light line.
Hence,
$g=\id$,
and the group
$G_M$
of the
$M$-automorphisms
injects into the group
$G_T$
of the
$T$-automorphisms.
\end{proof}
\section{Timed causal spaces and M\"obius structures}
\label{sect:timed_causal_moeb_structures}
In this section we adopt the following more advanced point of view to
M\"obius structures, see \cite{Bu1}.
\subsection{M\"obius and sub-M\"obius structures}
\label{subsect:moeb_sub-moeb}
Let
$X$
be a set,
$\reg\cP_4=\reg\cP_4(X)$,
see sect.~\ref{subsect:semi-metrics_topology}. For any semi-metric
$d$
on
$X$
we have three cross-ratios
$$q\mapsto \crr_1(q)=\frac{|x_1x_3||x_2x_4|}{|x_1x_4||x_2x_3|};
\crr_2(q)=\frac{|x_1x_4||x_2x_3|}{|x_1x_2||x_3x_4|};
\crr_3(q)=\frac{|x_1x_2||x_3x_4|}{|x_2x_4||x_1x_3|}$$
for
$q=(x_1,x_2,x_3,x_4)\in\reg\cP_4$,
whose product equals 1, where
$|x_ix_j|=d(x_i,x_j)$.
We associate with
$d$
a map
$M_d:\reg\cP_4\to L_4$
defined by
\begin{equation}\label{eq:moeb_map}
M_d(q)=(\ln\crr_1(q),\ln\crr_2(q),\ln\crr_3(q)),
\end{equation}
where
$L_4\sub\R^3$
is the 2-plane given by the equation
$a+b+c=0$.
Two semi-metrics
$d$, $d'$
on
$X$
are M\"obius equivalent if and only
$M_d=M_{d'}$.
Thus a M\"obius structure on
$X$
is completely determined by a map
$M=M_d$
for any semi-metric
$d$
of the M\"obius structure, and we often identify a M\"obius structure
with the respective map
$M$.
A bijection
$f:X\to X$
is an
$M$-automorphism
if and only if
$M\circ\ov f(q)=M(q)$
for every ordered 4-tuple
$q\in\reg\cP_4$
for the induced
$\ov f:\reg\cP_4\to\reg\cP_4$.
Let
$S_n$
be the symmetry group of
$n$
elements. The group
$S_4$
acts on
$\reg\cP_4$
by entries permutations of any
$q\in\reg\cP_4$.
The group
$S_3$
acts on
$L_4$
by signed permutations of coordinates, where a permutation
$\si:L_4\to L_4$
has the sign
``$-1$''
if and only if
$\si$
is odd.
The {\em cross-ratio} homomorphism
$\phi:S_4\to S_3$
can be described as follows: a permutation of a tetrahedron
ordered vertices
$(1,2,3,4)$
gives rise to a permutation of pairs of opposite edges
$((12)(34),(13)(24),(14)(23))$.
Thus the kernel
$K$
of
$\phi$
consists of four elements 1234, 2143, 4321, 3412,
and is isomorphic to the dihedral group
$D_4$
of a square automorphisms. We denote by
$\sign:S_4\to\{\pm 1\}$
the homomorphism that associates to every odd permutation the sign
``$-1$''.
One easily check that any M\"obius structure
$M:\reg\cP_4\to L_4$
is equivariant with respect to the signed cross-ratio homomorphism,
\begin{equation}\label{eq:signed_cross-ratio_homomorphism}
M(\pi(q))=\sign(\pi)\phi(\pi)M(q)
\end{equation}
for every
$q\in\reg\cP_4$, $\pi\in S_4$,
where
$\phi:S_4\to S_3$
is the cross-ratio homomorphism.
A {\em sub-M\"obius structure} on
$X$
is a map
$M:\cP_4\to L_4$
with the basic property (\ref{eq:signed_cross-ratio_homomorphism})
(we drop details related to degenerated 4-tuples, which can be found in
\cite{Bu1}).
Now, we describe a criterion for a sub-M\"obius structure to be
M\"obius. Given an ordered tuple
$q=(x_1,\dots,x_k)\in X^k$,
we use notation
$$q_i=(x_1,\dots,x_{i-1},x_{i+1},\dots,x_k)$$
for
$i=1,\dots,k$.
For a sub-M\"obius structure
$M$
on
$X$
we define its {\em codifferential}
$\de M:\reg\cP_5\to L_5=L_4^5$
by
$$(\de M(q))_i=M(q_i),\ i=1,\dots,5.$$
Furthermore, we use notation
$M(q_i)=(a(q_i),b(q_i),c(q_i))$, $i=1,\dots,5$, $q\in\reg\cP_5$.
The following theorem has been proved in \cite[Theorem~3.4]{Bu1}.
\begin{thm}\label{thm:submoeb_moeb} A sub-M\"obius structure
$M$
on
$X$
is a M\"obius structure if and only if for every nondegenerate 5-tuple
$q\in X^5$
the following conditions (A), (B) are satisfied
\begin{itemize}
\item [(A)] $b(q_1)+b(q_4)=b(q_3)-a(q_1)$;
\item[(B)] $b(q_2)=-a(q_4)+b(q_1)$.
\end{itemize}
\end{thm}
\begin{rem}\label{rem:equivalent_conditions} Conditions (A) and (B)
are in fact equivalent to each other. This follows from
$S_5$-symmetry
of the codifferential
$\de M$
and is explained in details
in \cite{Bu2}.
\end{rem}
\subsection{Timed causal space and a sub-M\"obius structure}
\label{subsect:timed_sub-moeb}
We begin with
\begin{rem}\label{rem:equivalent_monotone}
Axiom~(M) for monotone M\"obius structures on the circle is equivalent to that
$a(q)<0$
and
$b(q)>0$,
where
$M(q)=(a(q),b(q),c(q))\in L_4$
for any
$q=(x,y,z,u)\in\reg\cP_4$
such that the pairs
$(x,u)$
and
$(y,z)$
separate each other. This follows from Eq.~\ref{eq:moeb_map}.
\end{rem}
With every timed causal space
$T=(\ay,\cH,t)$
we associate a sub-M\"obius structure
$M$
on
$X=S^1$
as follows.
We fix an orientation of
$S^1$.
Then for any 4-tuple
$q\in\reg\cP_4$
we have a well defined {\em cyclic order}
$\co(q)$.
Let
$A\sub\cP_4$
be the set
$\set{\pi q}{$\pi\in S_4$}$, $A=A(q)$,
for a given
$q\in\cP_4$.
Note that the cyclic order
$\co(\pi q)=\co(q)$
is independent of
$\pi\in S_4$,
and we denote it by
$\co(A)$.
We label
$\co(A)=1234$.
With any pair
$i,i+1$
for consecutive points in
$\co(A)$,
we associate the timelike line
$h_{i,i+1}$
dual to the event
$(i,i+1)$.
Two other points of
$\co(A)$
determine the events
$a=(i+2)_{(i,i+1)}$, $a'=(i+3)_{(i,i+1)}\in h_{i,i+1}$,
and we associate with
$(i,i+1)$
the time
$t_{i(i+1)}>0$
between
$a$, $a'$.
In that way, every pair
$(i,i+1)$
of consecutive points in
$\co(A)$
is labeled by a positive time
$t_{i(i+1)}$.
Adjacent pairs are labeled in general by distinct numbers
$t_{i(i+1)}$, $t_{(i+1)(i+2)}$, however, by Axiom~(t5), the opposite pairs
are labeled by one and the same number, that is,
$t_{i(i+1)}=t_{(i+2)(i+3)}$,
where indexes are taken modulo 4.
Assume that a 4-tuple
$q=(x,y,z,u)\in\reg\cP_4$
is obtained from
$\co(A)$
by fixing the initial point and by the transposition of two last
entries of
$\co(A)$,
that is,
$x$
is the chosen initial point and
$\co(A)=xyuz$.
Then we put
\begin{equation}\label{eq:def_moeb_timed}
M(q)=(-t_{xy},t_{yu},t_{xy}-t_{yu})\in L_4.
\end{equation}
We denote by
$B\sub A$
the subset consisting of all 4-tuples obtained from
$\co(A)$
by fixing an initial entry and transposing
two last entries of
$\co(A)$.
The set
$B$
consists of 4 elements,
$|B|=4$,
and it is an orbit of a cyclic subgroup
$\Ga_\pi\sub S_4$,
generated by the permutation
$\pi=2413\in S_4$,
that is,
$\Ga_\pi=\{\id,\pi,\pi^2,\pi^3\}$
and
$B=\set{\si q}{$\si\in\Gamma_\pi$}$
for any
$q\in B$.
For example, if
$\co(A)=xyuz$,
then
$q=(x,y,z,u)$, $\pi q=(y,u,x,z)$,
$\pi^2q=(u,z,y,x)$, $\pi^3q=(z,x,u,y)\in B$.
\begin{lem}\label{lem:monb} For any
$q$, $q'=\si q\in B$
with
$\si\in\Ga_\pi$
we have
$$M(q')=\sign(\si)\phi(\si)M(q).$$
\end{lem}
\begin{proof} It suffices to prove the equality for the generator
$\si=2413$
of
$\Ga_\pi$.
Assume without loss of generality that
$q=(x,y,z,u)$
and, hence,
$\co(A)=xyuz$.
We also write
$\co(A)=1234$.
Then
$q'=\si q=(y,u,x,z)$.
By definition,
$M(q)=(-t_{12},t_{23},t_{12}-t_{23})$, $M(q')=(-t_{23},t_{34},t_{23}-t_{34})$.
On the other hand,
$\sign(\si)=-1$
because
$\si$
is odd, and
$\phi(\si)=213$.
Therefore,
$$\sign(\si)\phi(\si)M(q)=-(t_{23},-t_{12},t_{12}-t_{23})=(-t_{23},t_{12},t_{23}-t_{34})
=M(q'),$$
because
$t_{12}=t_{34}$
by Axiom~(t5).
\end{proof}
Furthermore, for every
$p\in A$
there are
$\si\in S_4$
and
$q\in B$
such that
$p=\si q$.
We put
\begin{equation}\label{eq:equivar}
M(p)=\sign(\si)\phi(\si)M(q).
\end{equation}
\begin{pro}\label{pro:submoebius_welldefined} The equation (\ref{eq:equivar})
defines unambiguously a map
$M:\reg\cP_4\to L_4$
which is a sub-M\"obius structure on
$X$.
\end{pro}
\begin{proof} We show that for a different representation
$p=\si'q'$
with
$\si'\in S_4$, $q'\in B$,
Eq.~(\ref{eq:equivar}) gives the same value
$M(p)$.
We have
$\si'q'=\si q$,
thus
$q'=\rho q$
with
$\si'\rho=\si$.
Since
$q$, $q'\in B$
and the group
$S_4$
acts on
$A$
effectively, we have
$\rho\in\Gamma_\pi$.
Then by Lemma~\ref{lem:monb}
$$M(q')=\sign(\rho)\phi(\rho)M(q),$$
and we obtain
$$\sign(\si')\phi(\si')M(q')=\sign(\si'\rho)\phi(\si'\rho)M(q)=M(p).$$
Thus Eq.~(\ref{eq:equivar}) defines unambiguously a map
$M:\reg\cP_4\to L_4$,
which now satisfies (\ref{eq:equivar}) for any
$p=\si q$
with
$q\in\reg\cP_4$, $\si\in S_4$.
Hence,
$M$
is a sub-M\"obius structure on
$X$.
\end{proof}
Note that to define the sub-M\"obius structure
$M$
we do not use Axiom~(t6).
\subsection{The sub-M\"obius structure $M$ is a M\"obius one}
\label{subsect:sub-moeb_moeb}
Given a nondegenerate 5-tuple
$q\in\reg\cP_5$,
we label its cyclic order by
$\co(q)=12345$.
Assuming that the order of
$q=xyzuv$
is cyclic, we have
$\co(q_i)=\co(q)_i$
for
$i=1,\dots,5$.
We consider with every
$i\in\co(q)$
three variables
$t_{(i+1)(i+2)}^i$, $t_{(i+2)(i+3)}^i$, $t_{(i+3)(i+4)}^i$,
associated with the 4-tuple
$\co(q_i)=\co(q)_i$
as in sect.~\ref{subsect:timed_sub-moeb}, where indexes are
taken modulo 5. These 15 variables satisfy 10 equations
\begin{equation}\label{eq:time_(t4)}
t_{(i+1)(i+2)}^i=t_{(i+3)(i+4)}^i
\end{equation}
\begin{equation}\label{eq:time_(t6)}
t_{(i+2)(i+3)}^i=t_{(i+2)(i+3)}^{i+1}+t_{(i+2))i+3)}^{i+4},
\end{equation}
which follow from Axioms~(t5) and (t4a) respectively. We compute
$\de M(q)=v$
for
$q\in\reg\cP_5$
with
$\co(q)=12345$
as follows. The time-labeling of
$\co(q_i)=\co(q)_i$
is given by
$t_{(i+1)(i+2)}^i$, $t_{(i+2)(i+3)}^i$, $t_{(i+3)(i+4)}^i$, $t_{(i+4)(i+6)}^i$.
Thus according to our definition of the sub-M\"obius structure
$M$
we have
$$(-t_{(i+1)(i+2)}^i,t_{(i+2)(i+3)}^i,t_{(i+1)(i+2)}^i-t_{(i+2)(i+3)}^i)=M(\pi \si^{i-1}q_i),$$
where
$\pi=1243$, $\si=4123$.
Therefore,
$$M(\si^{i-1}q_i)=\sign(\pi)\phi(\pi)M(\pi\si^{i-1}q_i)
=(t_{(i+1)(i+2)}^i,t_{(i+2)(i+3)}^i-t_{(i+1)(i+2)}^i,-t_{(i+2)(i+3)}^i),$$
and we obtain
$$\de M(q)=\begin{pmatrix}
a_1&b_1&c_1\\
a_2&b_2&c_2\\
a_3&b_3&c_3\\
a_4&b_4&c_4\\
a_5&b_5&c_5
\end{pmatrix}
=\begin{pmatrix}
t_{23}^1 &t_{34}^1-t_{23}^1 &-t_{34}^1\\
t_{45}^2 &t_{34}^2-t_{45}^2 &-t_{34}^2\\
t_{12}^3 &t_{15}^3-t_{12}^3 &-t_{15}^3\\
t_{12}^4 &t_{23}^4-t_{12}^4 &-t_{23}^4\\
t_{12}^5 &t_{23}^5-t_{12}^5 &-t_{23}^5
\end{pmatrix}.$$
\begin{thm}\label{thm:submoeb_is_moeb} The sub-M\"obius structure
$M$
associated with any timed causal space
$(\ay,\cH,t)$
is M\"obius.
\end{thm}
\begin{proof} We show that
$M$
satisfies equations (A) and (B) of Theorem~\ref{thm:submoeb_moeb}.
It suffices to check that for every unordered 5-tuple
$x,y,z,u,v\sub X$
of pairwise distinct points, equations (A) and (B) are satisfied
for some ordering
$q\in\reg\cP_5$
of the 5-tuple, because in this case
$\de M(q)$
lies in an irreducible invariant subspace
$R$
of respective representation of
$S_5$,
describing M\"obius structures, see \cite{Bu2}. Hence,
$\de M(q)\in R$
for any ordering of the 5-tuple. Or, applying the procedure above,
to check equations (A) and (B) directly.
Thus we assume that
$q=(x,y,z,u,v)\in\reg\cP_5$
has the cyclic order
$xyzuv$.
Equation~(A) can be rewritten as
$$0=b_1+b_4-b_3+a_1=-c_1-b_3+b_4=:A(v)$$
because
$a_1+b_1+c_1=0$,
and we compute using (\ref{eq:time_(t4)}), (\ref{eq:time_(t6)})
\begin{align*}
A(v)=-c_1-b_3+b_4&=t_{34}^1+t_{12}^3-t_{15}^3+t_{23}^4-t_{12}^4\\
&=t_{34}^1-t_{15}^3+t_{23}^4-t_{12}^5\\
&=t_{34}^5+t_{34}^2-t_{15}^3+t_{23}^4-t_{12}^5\\
&=t_{34}^2-t_{15}^3+t_{23}^4\\
&=t_{34}^2+t_{23}^4-t_{15}^2-t_{15}^4\\
&=t_{34}^2-t_{15}^2=0
\end{align*}
Similarly, Equation~(B) can be rewritten as
$0=b_1-b_2-a_4=:B(v)$
and we compute using (\ref{eq:time_(t4)}), (\ref{eq:time_(t6)})
\begin{align*}
B(v)=b_1-b_2-a_4&=t_{34}^1-t_{23}^1-t_{34}^2+t_{45}^2-t_{12}^4\\
&=t_{34}^5-t_{23}^1+t_{45}^2-t_{12}^4\\
&=t_{34}^5-t_{23}^1+t_{45}^1+t_{45}^3-t_{12}^4\\
&=t_{34}^5+t_{45}^3-t_{12}^4\\
&=t_{34}^5+t_{45}^3-t_{12}^3-t_{12}^5=0.
\end{align*}
Therefore, by Theorem~\ref{thm:submoeb_moeb},
$M$
is a M\"obius structure.
\end{proof}
\begin{pro}\label{pro:timed_monotone} The M\"obius structure
$M=\wh M(T)$
associated with a timed causal space
$T\in\cT$
is monotone,
$M\in\cM$,
and the timed causal space
$\wh T(M)$
associated with
$M$
coincide with
$T$, $\wh T(M)=T$.
\end{pro}
The proof proceeds in three steps, Lemmas~\ref{lem:pro:timed_monotone:axiom_m} --
\ref{lem:semi-metric_balls}.
\begin{lem}\label{lem:pro:timed_monotone:axiom_m} The M\"obius structure
$M=\wh M(T)$
satisfies Axiom~(M), and the time of the timed causal space
$T=(\ay,\cH,t)$
is computed in the usual way via
$M$-cross-ratios.
\end{lem}
\begin{proof} We check Axiom~(M) and simultaneously compute the time
$t(e,e')$
between events
$e$, $e'\in\ay$
assuming without loss of generality that
$e=(y,y')$, $e'=(u,u')\in h_a$
for
$a=(x,z)$
such that
the 4-tuple
$q=(x,y,z,u)\in\reg\cP_4$
is obtained from
$\co(q)=xyuz$
by fixing the initial point
$x$
and by the transposition of two last entries of
$\co(q)$.
Note that the pairs
$(x,u)$
and
$(y,z)$
separate each other. Then by definition,
$M(q)=(a(q),b(q),c(q))=(-t_{xy},t_{yu},t_{xy}-t_{yu})$
with the negative first entry
$a(q)=-t_{xy}$
and the positive second entry
$b(q)=t_{yu}$.
By Theorem~\ref{thm:submoeb_is_moeb}, we have
$M(q)=M_d(q)$
for any semi-metric
$d\in M$.
Thus
$$\crr_1(q)=e^{a(q)}=\frac{d(x,z)d(y,u)}{d(x,u)d(y,z)}<1,\quad
\crr_2(q)=e^{b(q)}=\frac{d(x,u)d(y,z)}{d(x,y)d(z,u)}>1.$$
This shows that
$M$
satisfies Axiom~(M), see Remark~\ref{rem:equivalent_monotone}, and that
$t(e,e')=t_{xz}=t_{yu}=\ln\crr_2(q)$.
\end{proof}
\begin{lem}\label{lem:pro:timed_monotone:harmonic} Let
$h=h_e$
be a timelike line in a timed causal space
$T$.
An event
$d\in h_e$
if and only if the 4-tuple
$(d,e)$
is
$M$-harmonic, that is, harmonic with respect to the M\"obius
structure
$M=\wh M(T)$.
\end{lem}
\begin{proof} Let
$e=(x,y)$
and
$d=(z,u)$.
If
$d\in h_e$,
then by Axiom~(h2),
$d$
separates
$e$,
and by Axiom~(t6) we have
$t(y_a,u_a)=t(y_b,u_b)$,
where
$a=(x,z)$, $b=(x,u)$.
Thus we can assume without loss of generality that
$\co(q)=xzyu$
for the nondegenerate 4-tuple
$q=(e,d)$.
Note that
$t(y_a,u_a)=t_{xz}$
and
$t(y_b,z_b)=t_{xu}$.
Thus
$t_{xz}=t_{xu}$.
The 4-tuple
$\wt q=(u,x,y,z)$
is obtained from
$\co(q)=uxzy$
by fixing the first entry
$u$
and permuting two last entries
$z$, $y$.
Therefore, by definition,
$M(\wt q)=(-t_{xu},t_{xz},0)$.
Using that
$M(\wt q)=M_d(\wt q)$
for any semi-metric
$d\in M$,
we obtain
$$1=\crr_3(\wt q)=\frac{d(x,u)\cdot d(y,z)}{d(y,u)\cdot d(x,z)}.$$
Hence
$(d,e)$
is
$M$-harmonic.
Conversely, if
$(d,e)$
is
$M$-harmonic,
then by Lemma~\ref{lem:harm_separate},
$d$
and
$e$
separate each other. By Axiom~(h3), there is a unique
$u'\in X\sm e$
such that
$d'=(z,u')\in h_e$.
By the first part of the proof, the 4-tuple
$(d',e)$
is
$M$-harmonic.
Taking a semi-metric
$d\in M$
with the infinitely remote point
$z$,
we observe that
\begin{equation}\label{eq:midpoints}
d(x,u)=d(u,y)\quad\textrm{and}\quad d(x,u')=d(u',y),
\end{equation}
because the 4-tuples
$(d,e)$, $(d',e)$
are
$M$-harmonic.
Assume
$u\neq u'$.
Then the 4-tuple
$(x,y,u,u')$
is nondegenerate, and
$u$, $u'$
are on the arc determined by
$e$
that does not contain
$z$.
Without loss of generality, we assume that
$(x,u)$
separate
$(y,u')$.
By Lemma~\ref{lem:pro:timed_monotone:axiom_m},
$M$
satisfies Axiom~(M). Thus
$$d(x,u)\cdot(y,u')>d(x,u')\cdot d(y,u)$$
in contradiction with (\ref{eq:midpoints}). Hence
$u=u'$
and
$d=d'\in h_e$.
\end{proof}
\begin{lem}\label{lem:semi-metric_balls} The set
$\cA$
of open arcs in
$X$
coincides with the set
$\cB$
of open balls with respect to semi-metrics
$d\in M=\wh M(T)$
with infinitely remote points centered at finite points of
$d$, $\cA=\cB$.
\end{lem}
\begin{proof} Let
$\al\in\cA$
be an open arc in
$X$
and let
$x$, $y\in X$
be the end points of
$\al$.
We put
$e=(x,y)\in\ay$
and take
$z\in\al$.
Then for
$u=\rho_e(z)$
the event
$d=(z,u)$
lies on the timelike line
$h_e$.
By Lemma~\ref{lem:pro:timed_monotone:harmonic}, the 4-tuple
$(d,e)$
is
$M$-harmonic.
Thus
$z$
is the midpoint between
$x$, $y$
with respect to any semi-metric
$d\in M$
with infinitely remote point
$u$.
By Axiom~(M),
$v\in\al$
if and only if
$d(z,v)<r:=d(x,z)=d(y,z)$.
Therefore
$\al$
coincides with the open ball
$B_r(z)$
with respect to
$d$
of radius
$r$
centered at
$z$.
It means that
$\cA\sub\cB$.
Let
$\be=B_r(o)\in\cB$
be the open ball with respect to a semi-metric
$\de\in M$
with the infinitely remote point
$\om$
of radius
$r>0$
centered at
$o\in X_\om$.
We show that
$\be\in\cA$.
Let
$d=(o,\om)\in\ay$.
By (h4), for any
$y\in X_\om$, $y\neq o$,
there is a unique event
$e=y_d=(y,y')\in h_d$.
We fix such an
$y$,
denote by
$e^+\sub X$
the closed arc determined by
$e$
that contains
$\om$,
and consider the respective linear order
$<=<_e$
on
$h_d$
with the future arc
$e^+$.
First,
we show that for
$r=\de(y,o)$
the open ball
$B_r(o)$
coincides with the open arc
$\intr e^-\in\cA$
determined by
$e$
that contains
$o$.
We denote by
$d^+\sub X$
the closed arc determined by
$d$
that contains
$y$,
by
$d^-$
the opposite closed arc. Then
$\intr e^-=(d^+\cap\intr e^-)\cup(d^-\cap\intr e^-)$.
We have
$u\in d^+\cap\intr e^-$
if and only if pairs
$(y,o)$, $(u,\om)$
separate each other. By Axiom~(M) this is equivalent to
$\de(u,o)<\de(y,o)=r$.
On the other hand,
$u\in d^-\cap\intr e^-$
if and only if
$u'=\rho_d(u)\in d^+\cap\intr e^-$.
By above, this is equivalent to
$\de(u',o)<r$.
By Lemma~\ref{lem:pro:timed_monotone:harmonic}, the 4-tuple
$(d,u_d)$
is harmonic, where
$u_d=(u,u')$.
Thus
$\de(u,o)=\de(u',o)<r$.
Therefore,
$\intr e^-=B_r(o)$
for
$r=\de(y,o)$.
It remains to show that for any
$r>0$
there is
$y\in X_\om$
with
$\de(y,o)=r$.
We fix some
$y\in X_\om$, $y\neq o$,
and use the notations introduced above.
By (t4b), for any
$s>0$
there is
$e_\pm=(u_\pm,u_\pm')\in h_d\cap C_e^\pm$
with
$t(e,e_\pm)=s$.
By Lemma~\ref{lem:pro:timed_monotone:harmonic}, the 4-tuples
$(d,e)$, $(d,e_\pm)$
are
$M$-harmonic. Hence
$\de(o,y)=\de(o,y')$, $\de(o,u_\pm)=\de(o,u_\pm')$.
As above, Axiom~(M) implies
$$\de(u_-,o)<\de(y,o)<\de(u_+,o).$$
By Lemma~\ref{lem:pro:timed_monotone:axiom_m}, the time
$t(e,e_\pm)$
is computed via
$M$-cross-ratios,
$$t(e,e_\pm)=\left|\ln\frac{\de(\om,y)\de(u_\pm,o)}{\de(\om,u_\pm)\de(y,o)}\right|
=\left|\ln\frac{\de(u_\pm,o)}{\de(y,o)}\right|,$$
hence
$s=\pm\ln\frac{\de(u_\pm,o)}{\de(y,o)}$.
This shows that for any
$\la>0$
there is
$u\in X_\om$, $u\neq o$,
with
$\de(u,o)=\la \de(y,o)$.
Hence, for any
$r>0$
there is
$y\in X_\om$
with
$\de(y,o)=r$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pro:timed_monotone}]
By Lemma~\ref{lem:pro:timed_monotone:axiom_m}, the M\"obius structure
$M=\wh M(T)$
satisfies Axiom~(M) for any timed causal space
$T=(\ay,\cH,t)\in\cT$.
It follows from Lemma~\ref{lem:semi-metric_balls} that
$M$
satisfies Axiom~(T). Thus
$M$
is monotone,
$M\in\cM$.
Let
$T'=(\ay,\cH',t')=\wh T(M)\in\cT$
be the timed causal space determined by
$M$.
By Lemma~\ref{lem:pro:timed_monotone:harmonic},
$\cH'=\cH$,
and by Lemma~\ref{lem:pro:timed_monotone:axiom_m},
$t'=t$.
Thus
$T'=T$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Given
$M\in\cM$,
we show that
$M'=M$,
where
$M'=\wh M\circ\wh T(M)$,
that is,
$M'(q)=M(q)$
for every
$q\in\reg\cP_4$.
Using Eq.~\ref{eq:signed_cross-ratio_homomorphism}, we can assume
without loss of generality that the cyclic order of
$q=(x,y,z,u)$
is
$\co(q)=xyuz$,
and thus
$q$
is obtained from
$\co(q)$
by picking up the first entry
$x$
and permuting the last two entries. In particular,
$(x,u)$
and
$(y,z)$
separate each other. Then by definition~(\ref{eq:def_moeb_timed})
we have
$$M'(q)=(-t_{xy},t_{yu},t_{xy}-t_{yu}),$$
where
$t_{xy}=t(z_a,u_a)$, $t_{yu}=t(x_b,z_b)$
for
$a=(x,y)$, $b=(y,u)\in\ay$,
see Axiom~(h4),
for
$T=\wh T(M)=(\ay,\cH,t)\in\cT$.
By definition~(\ref{eq:time}) of the time
$t$
we have
$t(z_a,u_a)=\left|\ln\frac{|xz|\cdot|yu|}{|xu|\cdot|yz|}\right|=-\ln\crr_1(q)$,
$t(x_b,z_b)=\left|\ln\frac{|xu|\cdot|yz|}{|xy|\cdot|uz|}\right|=\ln\crr_2(q)$
(to choose the signs, we have used that
$(x,u)$, $(y,z)$
separate each other and monotonicity of
$M$).
Therefore,
$M'(q)=M(q)$.
Together with Proposition~\ref{pro:timed_monotone} this shows that
$\wh T:\cM\to\cT$
and
$\wh M:\cT\to\cM$
are mutually inverse maps.
Let
$\wh g:\ay\to\ay$
be an automorphism of some
$T=(\ay,\cH,t)\in\cT$.
Since
$t(e,e')=0$
if and only if the events
$e$, $e'\in\ay$
lie on a light line, and
$\wh g$
preserves the time
$t$,
we see that
$\wh g$
maps every light line to a light line. Thus
$\wh g$
determines a map
$g:X\to X$
with
$\wh g(p_x)=p_{g(x)}$,
see sect.~\ref{subsect:recoveting_desitter}. For any event
$e=(x,y)\in\ay$
we have
$e=p_x\cap p_y$.
Thus
$\wh g(e)=\wh g(p_x)\cap\wh g(p_y)=p_{g(x)}\cap p_{g(y)}=(g(x),g(y))$.
Hence,
$\wh g$
is induced by
$g$.
Since
$T=\wh T(M)$
for some
$M\in\cM$,
the timelike lines and the time of
$T$
are determined by cross-ratios of
$M$,
see Proposition~\ref{pro:monotone_moeb_to_timed_space}. Therefore,
$g$
is an
$M$-automorphism.
If
$g=\id_X$,
then
$\wh g=\id_{\ay}$.
Thus the group
$G_T$
of
$T$-automorphisms
injects into the group
$G_M$
of
$M$-automorphisms.
Together with Proposition~\ref{pro:monotone_moeb_to_timed_space}
this shows that the groups
$G_M$
and
$G_T$
are canonically isomorphic.
\end{proof}
\section{Time inequalities}
\label{sect:time_inequality}
The {\em time inequality} for de Sitter 2-space
$\ds^2$
says that
$$t(a,b)+t(b,c)\le t(a,c)$$
for any events
$a<b<c$
with the equality in the case
$t(a,c)>0$
if and only if
$a,b,c$
are events on a timelike line. We first show in sect.~\ref{subsect:time_inequality_ds}
that this inequality follows from properties of Lambert quadrilaterals.
Then in sect.~\ref{subsect:hierarchy}, we discuss a hierarchy of time conditions,
which includes the time inequality,
and show that every timed causal space
$T\in\cT$
satisfies the {\em weak time inequality,} see Theorem~\ref{thm:wti}.
In sect.~\ref{subsect:monotone_vp} we describe monotone M\"obius structures
which satisfy Variational Principle, (VP), the most strong time condition
from the list, and in sect.~\ref{subsect:convex_moeb} also {\em convex}
M\"obius structures. We show that these two classes contain the canonical M\"obius structure,
and that the first one contains a neighborhood of the canonical structure in a fine
topology.
\subsection{The time inequality for $\ds^2$ via $\hyp^2$}
\label{subsect:time_inequality_ds}
The time inequality for de Sitter 2-space
$\ds^2$
follows from properties of Lambert quadrilaterals in
$\hyp^2$.
This goes of course via the canonical M\"obius structure
$M_0$
on the common absolute
$S^1$.
More precisely, we use the fact that harmonicity
of a 4-tuple
$((x,y),(z,u))\sub S^1$
with respect to
$M_0$
is equivalent to orthogonality of the geodesics
$xy$, $zu\sub\hyp^2$.
Recall that a {\em Lambert quadrilateral}
$\al\be\ga o$
in the hyperbolic plane
$\hyp^2$
has three right angles at
$\al$, $\be$
and
$\ga$.
The fours angle at
$o$
is acute, and
$|\al\be|<|o\ga|$, $|\be\ga|<|\al o|$.
Now, we explain how these properties imply the time inequality for
$\ds^2$.
\begin{figure}[htbp]
\centering
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{c}{$c$}
\psfrag{h}{$d$}
\psfrag{hpr}{$p$}
\psfrag{hprpr}{$q$}
\psfrag{o}{$o$}
\psfrag{al}{$\al$}
\psfrag{alpr}{$\al'$}
\psfrag{be}{$\be$}
\psfrag{bepr}{$\be'$}
\psfrag{ga}{$\ga$}
\psfrag{gapr}{$\ga'$}
\includegraphics[width=0.6\columnwidth]{time.eps}
\caption{The time inequality in $\ds^2$}\label{fi:time}
\end{figure}
Let
$a$, $b$, $c$
be events in
$\ay$
such that
$a<b<c$
for the order
$<:=<_b$.
We consider a generic case when no pair of events
$(a,b)$, $(b,c)$
lies on a light line, and the events are not on a common
timelike line. Then there are events
$d$, $p$, $q\in\ay$
with
$a,b\in h_p$, $a,c\in h_d$, $b,c\in h_q$.
We pass to the
$\hyp^2$-picture,
and draw the respective timelike lines as geodesics in
$\hyp^2$
with the same ends on the absolute
$S^1$.
Since the time in
$\ds^2$
and the distance in
$\hyp^2$
are computed via cross-ratios with respect to
$M_0$,
we have
$t(a,c)=|\al\ga|$, $t(a,b)=|\al'\be|$, $t(b,c)=|\be'\ga'|$,
see Figure~\ref{fi:time}.
But
$|\al\ga|=|\al o|+|o\ga|$,
and quadrilaterals
$\al\al'\be o$, $\ga\ga'\be' o$
have right angles at
$\al$, $\al'$, $\be$
respectively at
$\ga$, $\ga'$, $\be'$,
i.e., they are Lambert quadrilaterals. Thus
$|\al o|>|\al'\be|$, $|o\ga|>|\be'\ga'|$,
and we obtain
$$t(a,c)>t(a,b)+t(b,c).$$
\subsection{Hierarchy of time conditions}
\label{subsect:hierarchy}
We assume that a timed causal space
$T=\{\ay,\cH,t\}\in\cT$
is fixed together with the respective monotone M\"obius structure
$M=\wh M(T)\in\cM$.
We say that an event
$b\in\ay$
is {\em strictly between} events
$a$
and
$c\in\ay$
if
$a$
and
$c$
lie on different open arcs in
$X$
defined by
$b$.
Note that in this case,
$a$, $b$, $c$
are pairwise in the strong causal relation, in particular,
$a<b<c$
for appropriately chosen
$<:=<_b$.
Let
$a=(o,o')$, $b=(\om,\om')\in\ay$
be events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other. Then
$(o,\om),(o',\om')$
are also in the strong causal relation. Let
$d=(x,x')\in\ay$
be an event strictly between
$(o,\om)$
and
$(o',\om')$.
We denote by
$$t_d^+(a,b)=t(o_d,\om_d),\quad t_d^-(a,b)=t(o_d',\om_d').$$
In general,
$t_d^+(a,b)\neq t_d^-(a,b)$.
However, if
$a,b\in h_d$,
then
$t_d^+(a,b)=t_d^-(a,b)=t(a,b)$
by definition of
$t(a,b)$,
see (\ref{eq:time}), (\ref{eq:time_different}) and
Lemma~\ref{lem:unique_common_perpendicular}.
We consider the function
$$F_{ab}(d)=\frac{1}{2}(t_d^+(a,b)+t_d^-(a,b))$$
on the set
$D_{ab}$
of events
$d\in\ay$
that are strictly between
$(o,\om)$
and
$(o',\om')$,
and introduce the following list of time conditions for
$T$
and therefore simultaneously for
$M$.
\begin{itemize}
\item [(VP)] Variational principle: the infimum of
$F_{ab}$
is taken at unique
$d_0\in D_{ab}$
for which
$a,b\in h_{d_0}$;
\item[(LQI)] Lambert quadrilateral inequality:
$$F_{ab}(d)>F_{ab}(d_0)$$
for every
$d\in D_{ab}\sm d_0$
such that
$a\in h_d$;
\item[(TI)] Time inequality:
$$t(a,b)+t(b,c)\le t(a,c)$$
for any
$a<b<c$
with the equality in the case
$t(a,c)>0$
if and only if
$a,b,c$
are events on a timelike line;
\item[(WTI)] Weak time inequality:
$$t(a,b)+t(b,c)<t(a,c)$$
for any
$a<b<c$
such that
$b$
lies on a light line either with
$a$
or with
$c$,
and
$a$, $c$
are not on a light line.
\end{itemize}
We have the following implications
$$\mathrm{(VP)}\Rightarrow\mathrm{(LQI)}\Rightarrow\mathrm{(TI)}\Rightarrow\mathrm{(WTI)}.$$
The first and the last implications are obvious, and we explain the second
implication in Proposition~\ref{pro:lqi_ti}. For the canonical
M\"obius structure
$M_0$,
the geometric meaning of the function
$F_{ab}:D_{ab}\to\R$
is especially clear.
\begin{pro}\label{pro:functional_canonical} Let
$a=(o,o')$, $b=(\om,\om')\in\ay$
be events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other,
$d=(x,x')\in D_{ab}$.
Then for the canonical M\"obius structure
$M_0$,
the value
$F_{ab}(d)$
is the distance in
$\hyp^2$
between the points
$p=oo'\cap xx'$, $q=\om\om'\cap xx'$
at which the geodesic
$xx'\sub\hyp^2$
intersects the geodesics
$oo'$
and
$\om\om'$.
\end{pro}
\begin{proof} Using that the time between events in a timed causal
space and the distance in
$\hyp^2$
are computed via the respective cross-ratios, we see that
$t_d^+(a,b)=t(o_d,\om_d)$
is distance in
$\hyp^2$
between projections
$\wh o$, $\wh\om$
of
$o$, $\om$
to the geodesic
$xx'\sub\hyp^2$,
and similarly,
$t_d^-(a,b)=t(o_d',\om_d')$
is distance between projections
$\wh o'$, $\wh\om'$
of
$o'$, $\om'$
to the same geodesic
$xx'$.
By the angle parallelism formula,
$\wh op=p\wh o'$
and
$\wh\om q=q\wh\om'$.
Therefore,
$F_{ab}(d)=\frac{1}{2}(|\wh o\wh\om|+|\wh o'\wh\om'|)=|pq|$.
\end{proof}
\begin{cor}\label{cor:vp_canonical} The canonical M\"obius structure
satisfies (VP).
\end{cor}
\begin{proof} This immediately follows from properties of the distance in
$\hyp^2$
between points on geodesics.
\end{proof}
\subsection{The weak time inequality}
\label{subsect:wti}
Here, we prove the following
\begin{thm}\label{thm:wti} Any timed causal space
$T=\{\ay,\cH,t\}\in\cT$
satisfies (WTI).
\end{thm}
\begin{lem}\label{lem:order_harmonic} Assume distinct events
$a$, $b$
lie on a common timelike line,
$a,b\in h_c$,
where
$a=(x,y)$, $b=(z,u)$, $c=(v,w)\in\ay$,
and suppose that
$v$
lies on the open arc
$\ga$
between
$x$, $y$
that does not contain
$b$.
Then for every
$s\in\ga$, $s\neq v$,
and for
$d=(s,t)\in h_a$, $d'=(s,t')\in h_b$
we have:
$t'$
lies on the open arc
$\si$
between
$w$, $t$
that does not contain
$s$.
\end{lem}
\begin{proof} Moving
$s$
along
$\ga$,
observe that for
$s=v$
we have
$t=t'$,
while for
$s$
approaching to
$x$
or
$y$
the point
$t$
is not on the arc between
$z$, $u$
that contains
$w$.
Therefore
$t'$
lies on
$\si$
for these extremal cases. By continuity of reflections
$\rho_a$, $\rho_b:X\to X$
and Lemma~\ref{lem:unique_common_perpendicular} we have
$t'\in\si$
for every
$s\in\ga$.
\end{proof}
\begin{lem}\label{lem:monotone_plus_dist} Let
$a=(o,o')$, $b=(\om,\om')\in\ay$
be events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other. Then the function
$F_{ab}^+(d)=t(o_d,\om_d)$
is monotone on the set
$D_{ab}$
of events
$d\in\ay$
that are strictly between
$e=(o,\om)$
and
$e'=(o',\om')$, $F_{ab}^+(d)<F_{ab}^+(d')$
for any
$d$, $d'\in D_{ab}$
with
$d<d'<e$.
\end{lem}
\begin{proof} Let
$d=(x,y)$.
By Axiom~(t5) we have
$t(o_d,\om_d)=t(x_e,y_e)$.
Thus
$F_{ab}^+(d)=t(x_e,y_e)$.
For
$d'=(x',y')$
between
$d$
and
$e$,
the segment
$x_ey_e\sub h_e$
is contained in the segment
$x_e'y_e'\sub h_e$
and does not coincide with it (though, we do not exclude a possibility
that these segments have a common end). Thus
$t(x_e,y_e)<t(x_e',y_e')$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:wti}] Let
$a,b,c\in\ay$
be events in the causal relation,
$a<b<c$,
and we assume without loss of generality that
$b$, $c$
are on a light line. Then
$t(b,c)=0$,
and the required inequality is reduced to
$t(a,b)<t(a,c)$.
We assume furthermore without loss of generality that
$a=(o,o')$, $b=(\om,\om')$
and the pairs
$(o,\om')$, $(o',\om)$
separate each other. Since
$b$, $c$
are on a light line, we can assume that
$c=(\om,\om'')$.
Then the assumption
$a<b<c$
implies that
$\om''$
is on the (open) arc
$\al$
between
$\om$, $\om'$
that does not contain
$a$.
There is
$d=(x,y)\in\ay$
with
$a,b\in h_d$.
We assume that
$x$
is on the arc
$\be$
between
$o$, $o'$
that does not contain
$b$.
Then
$y\in\al$.
Similarly, there is
$d'=(x',y')\in\ay$
with
$a,c\in h_{d'}$.
We also assume that
$x'$
is on the arc
$\be$.
Then
$y'$
is on the arc
$\al'\sub\al$
between
$\om$, $\om''$.
Note that
$d$, $d'\in D_{ab}$
and that
$d\neq d'$
since
$b\neq c$,
therefore
$x'\neq x$
because
$d$, $d'\in h_a$.
We claim that
$x'$
lies on the arc
$\be$
between
$x$
and
$o$.
Indeed, since
$d$, $d'\in h_a$,
we otherwise would have by Lemma~\ref{lem:order_harmonic}, substituting
$o$
for
$v$, $o'$
for
$w$,
$\om$
for
$s$,
$\om'$
for
$t$,
$\om''$
for
$t'$,
that
$\om''\not\in\al$
in contradiction with the previously established
$\om''\in\al$.
It follows that
$d<d'<e=(o,\om)$.
By Lemma~\ref{lem:monotone_plus_dist},
$F_{ab}^+(d)<F_{ab}^+(d')$.
On the other hand,
$F_{ab}^+(d)=t(o_d,\om_d)=t(a,b)$
and
$F_{ab}^+(d')=t(o_{d'},\om_{d'})=t(a,c)$.
\end{proof}
\subsection{Implication (LQI)$\Longrightarrow$(TI)}
\label{subsect:lqi_ti}
Here, we show that the Lambert quadrilateral inequality implies the
time inequality.
\begin{pro}\label{pro:lqi_ti}
$\mathrm{(LQI)}\Rightarrow\mathrm{(TI)}$.
\end{pro}
Assume
$b\in\ay$
is strictly between
$a,c\in\ay$.
Then by Corollary~\ref{cor:common_perpendicular_prescribed},
there are common perpendiculars
$p$
to
$a,b$
and
$q$
to
$b,c$.
\begin{lem}\label{lem:common_perp_separation} Assume
$a$, $c\in h_d$
and
$b\in\ay\sm h_d$
is strictly between
$a$
and
$c$.
Then
$d$
is strictly between the common perpendiculars
$p$
to
$a$, $b$
and
$q$
to
$b$, $c$.
\end{lem}
\begin{proof} By the assumption,
$b$
is not on the timelike line
$h_d\sub\ay$.
Hence, the common perpendicular
$p=(p',p'')\in\ay$
to
$a$, $b$
is not equal to
$d$, $p\neq d$,
and the common perpendicular
$q=(q',q'')\in\ay$
to
$b$, $c$
is not equal to
$d$, $q\neq d$.
Since
$p$, $d\in h_a$,
the events
$p$, $d$
are not on a light line, and some closed arc in
$X$
determined by
$d$
does not include
$p$.
We denote that arc by
$d^+\sub X$.
Hence,
$p<_dd$
for the respective partial order
$<_d$.
We also denote by
$b^+\sub X$
the closed arc determined by
$b$
that includes
$c$.
Without loss of generality, we assume that
$p''$, $w$, $q'\in b^+$,
where
$d=(v,w)$.
Then by Lemma~\ref{lem:order_harmonic}, applied to
$a$, $b\in h_p$
and
$\ga=b^+$
we see that
$v$
lies on the arc
$\si$
determined by
$(p',t)$
that does not contain
$w$,
where
$r=(t,w)$
is orthogonal to
$b$.
Therefore,
$d<_dr$
and
$t\in d^+$.
We denote by
$r_d^+$
the closed arc in
$X$
determined by
$r$
that does not include
$d$,
see sect.~\ref{subsect:causality_structure}. Since
$d$
is also orthogonal to
$c$,
applying again Lemma~\ref{lem:order_harmonic} to
$b$, $c\in h_q$,
we see that
$t$
lies on the arc
$\si'$
determined by
$(v,q'')$
that does not contain
$q'\in d^+$.
Since
$r$, $q\in h_b$,
it means that
$q\sub r_d^+$,
thus
$r<_dq$.
Therefore,
$p<_dd<_dr<_dq$.
Since by construction,
$p$, $r$, $q\in h_b$,
and
$p$, $d$
are not on a light line, we see that
$d$
is strictly between
$p$
and
$q$.
\end{proof}
\begin{cor}\label{cor:common_perp_separation} Assume
$a$, $c\in h_d$
as in Lemma~\ref{lem:common_perp_separation}. Given
$p$, $q\in\ay$
with
$p\perp a$, $q\perp c$
such that
$d$
is strictly between
$p$
and
$q$,
the common perpendicular to
$p$, $q$
is strictly between
$a$
and
$c$.
\end{cor}
\begin{proof} Since
$d$
is strictly between
$p$
and
$q$,
the events
$p$, $q$
are in the strong causal relation. Thus their common perpendicular
$b\in\ay$
exists and is uniquely determined by Corollary~\ref{cor:common_perpendicular_prescribed}
and Lemma~\ref{lem:unique_common_perpendicular}. Since
$a$
is the common perpendicular to
$d$, $p$,
and
$c$
is the common perpendicular to
$d$, $q$,
Lemma~\ref{lem:common_perp_separation} implies that
$b$
is strictly between
$a$
and
$c$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{pro:lqi_ti}] Assume
$a<b<c$
for events in
$\ay$.
If
$t(a,c)=0$
then by Axiom~(t2),
$a$, $c$
are on a light line,
$a,c\in p_x$
for some
$x\in X$.
Then
$b\in p_x$,
and we have
$t(a,b)=t(b,c)=t(a,c)=0$.
Therefore, we can assume that
$t(a,c)>0$
and hence
$a,c\in h_d$
for some timelike line
$h_d\sub\ay$.
Using Theorem~\ref{thm:wti}, we also can assume that
$b$
lies on a light line neither with
$a$
nor with
$b$.
If
$b$
is also on
$h_d$,
then by Axiom~(t4a),
$t(a,b)+t(b,c)=t(a,c)$.
To complete the proof, we show that the assumption
$b\not\in h_d$
implies the strict inequality in the time inequality.
In this case,
$b$
is strictly between
$a$, $c$
by our assumption, and there are
$p$, $q\in\ay$
with
$a,b\in h_p$, $b,c\in h_q$.
By Lemma~\ref{lem:common_perp_separation},
$d$
is strictly between
$p$
and
$q$.
Since
$a\in h_p$
and
$p$, $d\in D_{ab}$,
(LQI) applied to
$a$, $b$
gives
$F_{ab}(d)>F_{ab}(p)$.
Since
$c\in h_q$
and
$q$, $d\in D_{bc}$,
(LQI) applied to
$b$, $c$
gives
$F_{bc}(d)>F_{bc}(q)$.
On the other hand,
$F_{ab}(p)=t(a,b)$, $F_{bc}(q)=t(b,c)$,
and it remains to show that
$F_{ab}(d)+F_{bc}(d)=t(a,c)$.
We fix decomposition
$X=d^+\cup d^-$, $d^+\cap d^-=d$,
induced by
$d$,
and write
$a=(a^+,a^-)$, $b=(b^+,b^-)$, $c=(c^+,c^-)$,
where
$a^\pm,b^\pm,c^\pm\in d^\pm$.
By (t4a), we have
$t(a_d^\pm,b_d^\pm)+t(b_d^\pm,c_d^\pm)=t(a_d^\pm,c_d^\pm)$.
Therefore
$F_{ab}(d)+F_{bc}(d)=\frac{1}{2}(t(a_d^+,c_d^+)+t(a_d^-,c_d^-))=t(a,c)$
because
$a,c\in h_d$.
\end{proof}
\begin{cor}\label{cor:vp_ti} Variational principle implies the time
inequality,
(VP)$\Longrightarrow$(TI),
cp. \cite{PY}.
\end{cor}
\subsection{Monotone M\"obius structures with (VP)}
\label{subsect:monotone_vp}
Some important properties of M\"obius structures
$\cM$
which do not follow from monotonicity Axiom~(M) can be expressed
as an inequality
$\crr(q)>\crr(q')$
between cross-ratios of 4-tuples
$q$, $q'$
with two common entries,
$|q\cap q'|=2$,
under an assumption that a symmetry between
$q$, $q'$
is broken down in a definite way.
We use notation
$\reg\cP_n$
for the set of ordered nondegenerate
$n$-tuples
of points in
$X=S^1$, $n\in\N$.
For
$q\in\reg\cP_n$
and a proper subset
$I\sub\{1,\dots,n\}$
we denote by
$q_I\in\reg\cP_k$, $k=n-|I|$,
the
$k$-tuple
obtained from
$q$
(with the induced order) by crossing out all entries which correspond to elements of
$I$.
We introduce the following Axiom for a M\"obius structure
$M\in\cM$,
which implies the variational principle (VP).
(I) Increment: for any
$q\in\reg\cP_7$
with cyclic order
$\co(q)=1234567$
such that
$q_{247}$
and
$q_{157}$
are harmonic, we have
$$\crr_1(q_{345})>\crr_1(q_{123}).$$
It means the following. Assume we are given two events
$e=(o,\om)$, $e'=(o',\om')\in\ay$
in the strong causal relation such that
$(o,\om')$
and
$(o',\om)$
separate each other. Let
$oo'\sub X$
be the arc between
$o$, $o'$
that does not contain
$\om$, $\om'$,
and let
$(u,v)\in\ay$, $u\in oo'$,
be the common perpendicular to
$a=(o,o')$, $b=(\om,\om')$,
that is
$(u,v)\in h_a\cap h_b$.
Given
$x\in oo'$
such that
$(o,u)$
and
$(o',x)$
separate each other, we put
$g_+(u,x)=\exp t_e(u_e,x_e)$, $g_-(u,x)=\exp(-t_{e'}(u_{e'},x_{e'}))$,
$\De(u,x)=g_+(u,x)g_-(u,x)$.
Then Axiom~(I) tells that
$\De(u,x)>1$.
Indeed, consider
$q=(o,\om,v,\om',o',u,x)\in\reg\cP_7$
written in the cyclic order
$\co (q)=1234567$.
The assumption that 4-tuples
$q_{247}$
and
$q_{157}$
are harmonic means that 4-tuples
$(u,o,v,o')$
and
$(u,\om,v,\om')$
are harmonic with the common axis
$(u,v)$,
i.e.,
$(u,v)\in h_a\cap h_b$.
Since
$q_{345}=(o,\om,u,x)$, $q_{123}=(\om',o',u,x)$,
we have
$g_+(u,x)=\crr_1(q_{345})$, $g_-(u,x)=1/\crr_1(q_{123})$.
Thus the condition
$\crr_1(q_{345})>\crr_1(q_{123})$
means that
$\De(u,x)>1$.
\begin{pro}\label{pro:canonical_vp} The canonical M\"obius structure
$M_0$
on
$X$
satisfies Axiom~(I).
\end{pro}
\begin{proof} Let
$q=(o,\om,v,\om',o',u,x)\in\reg\cP_7$
be as above. In the metric on
$X$
from
$M_0$
with infinitely remote point
$u$,
we have
$|vo|=|vo'|$, $|v\om|=|v\om'|$.
Since
$M_0$
is canonical,
$|vo|=|v\om|+|o\om|$,
thus
$|o\om|=|o'\om'|$.
Furthermore,
$\crr_1(q_{345})=\crr_1(o,\om,u,x)=|x\om|/|ox|$
and
$\crr_1(q_{123})=\crr_1(\om',o',u,x)=|xo'|/|x\om'|$.
Note that
$xo\sub x\om'\sub X_u$.
Thus
$|xo|<|x\om'|$.
Using
$|x\om|=|xo|+|o\om|$
and
$|xo'|=|x\om'|+|o'\om'|=|x\om'|+|o\om|$,
we obtain
$|x\om|/|ox|>|xo'|/|x\om'|$.
Hence,
$\crr_1(q_{345})>\crr_1(q_{123})$,
and
$M_0$
satisfies (I).
\end{proof}
\begin{pro}\label{pro:increment_variational_principle} Increment Axiom~(I)
implies Variational Principle~(VP).
\end{pro}
\begin{proof} Let
$a=(o,o')$, $b=(\om,\om')\in\ay$
be events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other. Then the events
$e=(o,\om)$, $e'=(o',\om')$
are also in the strong causal relation.
Let
$d_0=(u,v)\in D_{ab}$
be the unique event with
$a$, $b\in h_{d_0}$.
We show that
$F_{ab}(d)>F_{ab}(d_0)$
for any
$d=(x,x')\in D_{ab}$, $d\neq d_0$.
Let
$oo'\sub X$
be the arc between
$o$, $o'$
that does not include
$b$.
Without loss of generality we can assume that
$u,x\in oo'$
and
$x\neq u$.
It suffices to show that
$F_{ab}(d)>F_{ab}(d')$
for
$d'=(u,x')$.
Let
$\si\sub h_e$
be the segment between
$u_e$, $x_e'\in h_e$, $\si'\sub h_{e'}$
the segment between
$u_{e'}$, $x_{e'}'\in h_{e'}$.
Since
$x\neq u$,
one of the events
$x_e\in h_e$, $x_{e'}\in h_{e'}$
lies in the respective segment
$\si$, $\si'$,
while the other not. We assume without loss of generality that
$x_{e'}\in\si'$.
Then
$x_e\not\in\si$,
and moreover
$u_e$
separates the events
$x_e$
and
$x_e'$
on the timelike line
$h_e$.
Thus
$t(x_e,x_e')>t(u_e,x_e')$
while
$t(x_{e'},x_{e'}')<t(u_{e'},x_{e'}')$.
By Axiom~(I),
$t(x_e,u_e)>t(x_{e'},u_{e'})$,
thus
$t(x_e,x_e')-t(u_e,x_e')>t(u_{e'},x_{e'}')-t(x_{e'},x_{e'}')$.
Recall that
$$F_{ab}(d)=\frac{1}{2}(t_d^+(a,b)+t_d^-(a,b)),$$
where
$t_d^+(a,b)=t(o_d,\om_d)$, $t_d^-(a,b)=t(o_d',\om_d')$.
By (t5) we have
$t(o_d,\om_d)=t(x_e,x_e')$, $t(o_d',\om_d')=t(x_{e'},x_{e'}')$.
Hence
$$F_{ab}(d)-F_{ab}(d')=\frac{1}{2}(t(x_e,x_e')-t(u_e,x_e')
+t(x_{e'},x_{e'}')-t(u_{e'},x_{e'}'))>0,$$
which completes the proof.
\end{proof}
Using Corollary~\ref{cor:vp_ti}, we immediately obtain
\begin{cor}\label{cor:i_ti} Increment Axiom~(I) implies the time
inequality, (TI).
\end{cor}
\subsection{The fine topology and Axiom (I)}
\label{subsect:fine_topology}
We denote by
$\cI$
the class of monotone M\"obius structures on the circle
which satisfies Axiom~(I). This work does not provide tools,
which allow to answer natural questions like to
characterize hyperbolic spaces
$Y$
with
$\di Y=S^1$
for which the respective M\"obius structure is in the class
$\cI$.
We only show here that a neighborhood of the canonical M\"obius structure
$M_0$
on
$X=S^1$
in an appropriate topology lies in
$\cI$.
Recall that a M\"obius structure
$M$
on a set
$X$
determines the
$M$-topology
on
$X$
(see sect.~\ref{subsect:semi-metrics_topology})
and hence the induced topology on the set
$\reg\cP_n(X)\sub X^n$.
One can consider a M\"obius structure as a map defined on
$\reg\cP_4$
with values in a vector space (see sect.~\ref{subsect:moeb_sub-moeb}).
Thus it not clear how to define a topology on a set of M\"obius
structures on
$X$
because the topology of
$X$
may change together with change of a M\"obius structure.
However, for monotone M\"obius structures on
$X=S^1$
such a problem does not exist in view of Axiom~(T): all M\"obius
structures
$M\in\cM$
induce on
$X$
one and the same topology of the circle. We define a fine topology on
$\cM$
as follows.
Let
$\reg^+\cP_7\sub X^7$
be the subset of
$\reg\cP_7$
which consists of all
$q\in\reg\cP_7$
with the cyclic order. That is, for
$q\in\reg^+\cP_7$
we have
$\co(q)=q$.
We take on
$\reg^+\cP_7$
the topology induced from the standard topology of the 7-torus
$X^7$.
We associate with a M\"obius structure
$M\in\cM$
a section of the trivial bundle
$\reg^+\cP_7\times\R^4\to\reg^+\cP_7$
given by
$$M(q)=(q,\crr_2(q_{247}),\crr_2(q_{157}),\crr_1(q_{345}),\crr_1(q_{123}))$$
for
$q=1234567\in\reg^+\cP_7$.
Taking the product topology on
$\reg^+\cP_7\times\R^4$,
we define the {\em fine} topology on
$\cM$
with base given by sets
$$U_V=\set{M\in\cM}{$M(\reg^+\cP_7)\sub V$},$$
where
$V$
runs over open subsets of
$\reg^+\cP_7\times\R^4$.
We show that the canonical M\"obius structure
$M_0$
on
$X$
possesses a neighborhood
$U_V$
in the fine topology which lies in
$\cI$,
that is, every M\"obius structure
$M\in U_V$
satisfies Axiom~(I). To this end, consider a function
$\ep:\reg^+\cP_7\to\R$
given by
$$\ep(q)=\frac{|o\om|_0^2}{4|x\om'|_0^2}$$
for
$q=(o,\om,v,\om',o',u,x)\in\reg^+\cP_7$,
where
$|\cdot\,\cdot\,|_0$
is a standard metric on
$X_u=\R$
from the canonical M\"obius structure
$M_0$
with infinitely remote point
$u$.
Such a metric is determined up to a homothety, but clearly
$\ep$
does not depend on that.
\begin{lem}\label{lem:ep_continuous} The function
$\ep:\reg^+\cP_7\to\R$
is continuous.
\end{lem}
\begin{proof} Obviously, it suffices to check that
$\ep$
varies continuously in the variable
$u\in q$.
We switch to the notation
$d_u(x,y)=|xy|_0$
for a metric from
$M_0$
with infinitely remote point
$u$.
Applying to
$u'\in X$, $u'\neq u$,
a metric inversion, we have
$$d_{u'}(x,y)=\frac{d_u(x,y)}{d_u(u',x)d_u(u',y)}.$$
The point
$u'\in X$
is infinitely remote for
$d_{u'}$.
Thus for
$q'=(o,\om,v,\om',o',u',x)$, $q=(o,\om,v,\om',o',u,x)$
we obtain
$$\ep(q')=\frac{d_{u'}^2(o,\om)}{4d_{u'}^2(x,\om')}
=\ep(q)\frac{d_u^2(u',\om')d_u^2(u',x)}{d_u^2(u',o)d_u^2(u',\om)}.$$
The factor after
$\ep(q)$
in the right hand side tends to 1 as
$u'\to u$.
Thus
$\ep(q')\to\ep(q)$
as
$u'\to u$,
that is, as
$q'\to q$.
\end{proof}
The set
$$V=\set{(q,r)\in\reg^+\cP_7\times\R^4}{$|r-\pr_2\circ M_0(q)|<\ep(q)$},$$
where
$\pr_2:\reg^+\cP_7\times\R^4\to\R^4$
is the projection to the second factor, is the
$\ep$-neighborhood
of
$M_0(\reg^+\cP_7)$
with variable
$\ep=\ep(q)$
in
$\reg^+\cP_7\times\R^4$.
It follows from Lemma~\ref{lem:ep_continuous} that
$V$
is open in
$\reg^+\cP_7\times\R^4$.
Thus the set
$$U_V=\set{M\in\cM}{$M(\reg^+\cP_7)\sub V$}$$
of M\"obius structures is open in the fine topology.
The following is a pertubed version of Proposition~\ref{pro:canonical_vp}.
\begin{pro}\label{pro:pertubed_canonical_vp} Every M\"obius structure
$M\in U_V$
satisfies Increment Axiom~(I), that is,
$U_V\sub\cI$.
\end{pro}
\begin{proof} Given
$M\in U_V$,
for any
$q\in\reg^+\cP_7$, $q=1234567$,
such that 4-tuples
$q_{247}$, $q_{157}$
are
$M$-harmonic,
that is,
$\crr_2(q_{247})=1=\crr_2(q_{157})$,
we have to show that
$\crr_1(q_{345})>\crr_1(q_{123})$
for
$M$-cross-ratios.
We assume that
$q=(o,\om,v,\om',o',u,x)$,
and for (semi-)metrics
$d_u\in M$, $d_u^0\in M_0$
with infinitely remote point
$u$
we use notations
$d_u(a,b)=|ab|$, $d_u^0(a,b)=|ab|_0$.
The assumption
$M\in U_V$
implies
$|\crr_2^0(q_{247})-1|<\ep$, $|\crr_2^0(q_{157})-1|<\ep$
for
$M_0$-cross-ratios,
where
$\ep=\ep(q)$.
Since
$q_{247}=(o,v,o',u)$, $q_{157}=(\om,v,\om',u)$,
we have
$1=\crr_2(q_{247})=\frac{|vo'|\cdot|ou|}{|ov|\cdot|o'u|}=\frac{|vo'|}{|ov|}$,
$1=\crr_2(q_{157})=\frac{|v\om'|\cdot|\om u|}{|\om v|\cdot|\om'u|}=\frac{|v\om'|}{|\om v|}$.
Hence,
\begin{equation}\label{eq:ep_harm_canon}
\left|\frac{|vo'|_0}{|ov|_0}-1\right|<\ep,\quad \left|\frac{|v\om'|_0}{|\om v|_0}-1\right|<\ep.
\end{equation}
Using that
$|o\om|_0=|ov|_0-|\om v|_0$, $|\om'o'|_0=|vo'|_0-|v\om'|_0$,
because
$M_0$
is canonical, we have
$$|o\om|_0-|\om'o'|_0=|ov|_0-|vo'|_0+|v\om'|_0-|\om v|_0$$
and thus using (\ref{eq:ep_harm_canon}) we obtain
\begin{equation}\label{eq:ep_oom}
-\ep(|ov|_0+|\om v|_0)\le|o\om|_0-|\om'o'|_0\le\ep(|ov|_0+|\om v|_0).
\end{equation}
Similarly, since
$|x\om|_0=|xo|_0+|o\om|_0$, $|xo'|_0=|x\om'|_0+|\om'o'|_0$,
we have
$$\crr_1^0(q_{345})-\crr_1^0(q_{123})=\frac{|x\om|_0}{|xo|_0}-\frac{|xo'|_0}{|x\om'|_0}
=\frac{|o\om|_0}{|xo|_0}-\frac{|\om'o'|_0}{|x\om'|_0}.$$
Using (\ref{eq:ep_oom}) and that
$|x\om'|_0-|xo|_0=|o\om'|_0$
we obtain
\begin{equation}\label{eq:diff_cr1_below_0}
\crr_1^0(q_{345})-\crr_1^0(q_{123})\ge\frac{|o\om|_0\cdot|o\om'|_0}{|xo|_0\cdot|x\om'|_0}
-\ep\frac{|ov|_0+|\om v|_0}{|x\om'|_0}.
\end{equation}
By the assumption
$M\in U_V$,
we have
$|\crr_1(p)-\crr_1^0(p)|<\ep$
for
$p=q_{345}$
and
$p=q_{123}$.
Hence
$\crr_1(q_{345})-\crr_1(q_{123})\ge\crr_1^0(q_{345})-\crr_1^0(q_{123})-2\ep$.
Thus using (\ref{eq:diff_cr1_below_0}) we obtain
\begin{equation}\label{eq:diff_cr1_below}
\crr_1(q_{345})-\crr_1(q_{123})\ge\frac{|o\om|_0\cdot|o\om'|_0}{|xo|_0\cdot|x\om'|_0}
-\ep\left(2+\frac{|ov|_0+|\om v|_0}{|x\om'|_0}\right).
\end{equation}
We have
$o\om\sub o\om'$, $xo\sub x\om'$, $\om v\sub ov\sub x\om'$
in
$X_u$.
Thus
$|o\om|_0<|o\om'|_0$, $|xo|_0<|x\om'|_0$, $|\om v|_0<|ov|_0<|x\om'|_0$,
and hence
$$\frac{|o\om|_0\cdot|o\om'|_0}{|xo|_0\cdot|x\om'|_0}>\frac{|o\om|_0^2}{|x\om'|_0^2},\quad
\frac{|ov|_0+|\om v|_0}{|x\om'|_0}<\frac{2|ov|_0}{|x\om'|_0}<2.$$
Therefore
$\crr_1(q_{345})-\crr_1(q_{123})>\frac{|o\om|_0^2}{|x\om'|_0^2}-4\ep=0$.
\end{proof}
\subsection{Convex M\"obius structures}
\label{subsect:convex_moeb}
We introduce the following Axiom for a M\"obius structure
$M\in\cM$,
which implies convexity of the function
$F_{ab}$.
(C) Convexity: for any
$q\in\reg\cP_6$
with cyclic order
$\co(q)=123456$
such that
$\crr_3(q_{46})=\crr_3(q_{26})$
we have
$$\crr_1(q_{12})>\crr_1(q_{14}).$$
A M\"obius structure
$M\in\cM$
is {\em convex}, if it satisfies Axiom~(C).
Axiom~(C) can be rewritten in the following way. Assume we have
$q=(o',x,y,z,o,\om)\in\reg\cP_6$
written in the cyclic order,
$\co(q)=123456$.
Then
$q_{46}=(o',x,y,o)$, $q_{26}=(o',y,z,o)$,
and the assumption
$\crr_3(q_{46})=\crr_3(q_{26})$
is equivalent to
$\de_{x,y,z}(o)=\de_{x,y,z}(o')$,
where
$$\de_{x,y,z}(o)=\frac{|yo|^2}{|xo|\cdot|zo|}.$$
Further, we have
$q_{12}=(y,z,o,\om)$, $q_{14}=(x,y,o,\om)$.
Thus the condition
$\crr_1(q_{12})>\crr_1(q_{14})$
is equivalent to
$\de_{x,y,z}(o)>\de_{x,y,z}(\om)$.
\begin{pro}\label{pro:canonical_convex} The canonical M\"obius structure
$M_0$
on
$X$
is convex.
\end{pro}
\begin{proof} In the metric from
$M_0$
with infinitely remote point
$o'$,
we have
$\de_{x,y,z}(o')=1$.
Thus we have
$\de_{x,y,z}(o)=1$
and hence
$|yo|^2=|xo|\cdot|zo|$.
Let
$\si=|o\om|$.
Using that
$M_0$
is canonical, we have
$|y\om|=|yo|+\si$, $|x\om|=|xo|+\si$, $|z\om|=|zo|+\si$.
Therefore,
$$\de_{x,y,z}(\om)=\frac{(|yo|+\si)^2}{(|xo|+\si)(|zo|+\si)}=
\frac{1+\al\si+\be\si^2}{1+\ga\si+\be'\si^2},$$
where
$\al=2/|yo|$, $\be=1/|yo|^2$, $\ga=\frac{|xo|+|zo|}{|xo|\cdot|zo|}$,
$\be'=1/(|xo|\cdot|zo|)$.
Since
$|yo|^2=|xo|\cdot|zo|$,
we have
$\be=\be'$,
and thus the inequality
$\de_{x,y,z}(\om)<1$
is equivalent to
$\sqrt{|xo|/|zo|}+\sqrt{|zo|/|xo|}>2$,
which is always true because
$x\neq z$.
\end{proof}
Let
$a=(o,o')$, $b=(\om,\om')\in\ay$
be events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other. Using the parametrization
$x\leftrightarrow x_a$
of the arc
$oo'$
between
$o$, $o'$
that does not contain
$b$
by the timelike line
$h_a$, $x\in oo'$, $x_a\in h_a$,
and the parametrization
$x'\leftrightarrow x_b'$
of the arc
$\om\om'$
between
$\om$, $\om'$
that does not contain
$a$
by the timelike line
$h_b$,
we consider the function
$F_{ab}:D_{ab}\to\R$,
see sect.~\ref{subsect:hierarchy}, as a function defined on
$h_a\times h_b$, $F_{ab}:h_a\times h_b\to\R$.
\begin{pro}\label{pro:convexity_axiom_convex} Convexity Axiom~(C)
implies that the function
$F_{ab}:h_a\times h_b\to\R$
is strictly convex for any events
$a$, $b\in\ay$
in the strong causal relation.
\end{pro}
\begin{rem}\label{rem:convexity} 1. The convexity of the function
$F_{ab}$
is a precise analog of the convexity of the distance function in
$\CAT(-1)$ spaces, cp. Proposition~\ref{pro:functional_canonical}.
2. The convexity property depends on a parametrization up to an affine
equivalence. Here, the parametrization of
$D_{ab}$
by
$h_a\times h_b$
is chosen because
$h_a\times h_b$
is an affine space isomorphic to
$\R\times\R$.
\end{rem}
\begin{proof}[Proof of Proposition~\ref{pro:convexity_axiom_convex}]
As usual, we assume that
$a=(o,o')$, $b=(\om,\om')\in\ay$
are events in the strong causal relation such that the pairs
$(o,\om')$
and
$(o',\om)$
separate each other, and
$e=(o,\om)$, $e'=(o',\om')$.
We show that the increment of the function
$F_{ab}$
strictly increases along any line in
$h_a\times h_b=\R^2$.
To this end, it suffices to show that for any
$x_a$, $y_a$, $z_a\in h_a$, $x_a<y_a<z_a$,
such that
$t(x_a,y_a)=t(y_a,z_a)$
we have
$\De F_{a,b}(z_a,y_a)>\De F_{a,b}(y_a,x_a)$,
where
$$\De F_{a,b}(y_a,x_a)=\frac{1}{2}(t(y_e,x_e')+t(y_{e'},x_{e'}')
-t(x_e,x_e')-t(x_{e'},x_{e'}'))
$$
for some
$x_b'\in h_b$
which is independent of
$x_b'$
(recall that we use here parametrizations
$x\leftrightarrow x_a$
and
$x'\leftrightarrow x_b'$).
Indeed, without loss of generality, we assume that
$q=(o',x,y,z,o,\om)\in\reg\cP_6$
is written in the cyclic order. Then
$t(y_e,x_e')-t(x_e,x_e')=t(y_e,x_e)$,
$t(y_{e'},x_{e'}')-t(x_{e'},x_{e'}')=-t(y_{e'},x_{e'})$,
and thus
$$\De F_{a,b}(y_a,x_a)=\frac{1}{2}(t(y_e,x_e)-t(y_{e'},x_{e'})).$$
The condition
$t(x_a,y_a)=t(y_a,z_a)$
is equivalent to
$\frac{|yo'|\cdot|xo|}{|yo|\cdot|xo'|}=\frac{|zo'|\cdot|yo|}{|zo|\cdot|yo'|}$
for any semi-metric from
$M$,
or which is the same to
$\de_{x,y,z}(o)=\de_{x,y,z}(o')$.
Axiom~(C) implies
$\de_{x,y,z}(o)>\de_{x,y,z}(\om)$.
Since
$$t(z_e,y_e)=\frac{|yo|\cdot|z\om|}{|zo|\cdot|y\om|}\quad\textrm{and}\quad
t(y_e,x_e)=\frac{|xo|\cdot|y\om|}{|yo|\cdot|x\om|},$$
this is equivalent to
$t(z_e,y_e)>t(y_e,x_e)$.
Applying the same argument to
$q'=(o,z,y,x,o',\om')\in\reg\cP_6$,
we obtain that Axiom~(C) implies
$\de_{x,y,z}(o')>\de_{x,y,z}(\om')$,
which is equivalent to
$t(z_{e'},y_{e'})<t(y_{e'},x_{e'})$.
Therefore,
$\De F_{a,b}(z_a,y_a)>\De F_{a,b}(y_a,x_a)$,
and the strict convexity of the function
$F_{ab}$
follows.
\end{proof}
\begin{rem}\label{rem:axioms_I_C} By Proposition~\ref{pro:convexity_axiom_convex},
Axiom~C implies that the function
$F_{ab}:D_{ab}\to\R$
achieves the infimum at a unique point
$d_0'\in D_{ab}$
for any
$a$, $b\in\ay$
in the strong causal relation because
$F_{ab}(d)\to\infty$
as
$d$
approaches to the boundary
$\d D_{ab}$
of
$D_{ab}$.
However, in general there is no reason that
$d_0'=h_a\cap h_b$.
It seems that Axioms~(I) and (C) are independent of each other.
\end{rem}
\section{Appendix 1}
\label{sect:appendix_1}
We show that Gromov hyperbolic spaces from a large class
are boundary continuous, see sect.~\ref{subsect:boundary_continuous}.
\begin{thm}\label{thm:cat0_boundary_contiuous} Every proper Gromov
hyperbolic $\CAT(0)$ space
$Y$
is boundary continuous.
\end{thm}
For
$\CAT(-1)$
spaces this is established in \cite[Proposition~3.4.2]{BS1}.
Here, we extend this result to
$\CAT(0)$
spaces. A distinction between
$\CAT(-1)$
and
$\CAT(0)$
cases relevant to arguments is that
$\dist(\ga,\ga')=\inf\set{d(s,s')}{$s\in\ga,s'\in\ga'$}=0$
for asymptotic geodesic rays
$\ga$, $\ga'$
in the former case, while that distance is only finite in last case.
This distinction is compensated by the following Lemma.
We use the notation
$o_t(1)$
for a quantity with
$o_t(1)\to 0$
as
$t\to\infty$.
\begin{lem}\label{lem:o(1)_difference} Let
$xyz\sub\R^2$
be a triangle with
$|yz|\le d$
for some fixed
$d>0$
and
$|xy|,|xz|\ge t$.
Assume
$\angle_z(x,y)$, $\angle_y(x,z)\ge\pi/2-o_t(1)$.
Then
$||xy|-|xz||=o_t(1)$.
\end{lem}
\begin{proof} The required estimate follows from the convexity of the distance function on
$\R^2$
and the first variation formula. We leave details to the reader.
\end{proof}
Recall that in a geodesic metric space, the Gromov product is {\em monotone}
in the following sense, see e.g. \cite[Lemma~2.1.1]{BS1}.
\begin{lem}\label{lem:gromov_product_monotone} Let
$Y$
be a geodesic metric space,
$xyz\sub Y$
a geodesic triangle. Then for any
$y'\in xy$, $u\in yz$
we have
$$(y'|z)_x\le (y|z)_x\le\min\{(y|u)_x,(u|z)_x\}.$$
\end{lem}
\begin{proof} The left hand side inequality is equivalent to
$|y'x|-|y'z|\le|yx|-|yz|$,
which follows from the triangle inequality
$|yz|\le|yy'|+|y'z|$
because
$|yx|-|y'x|=|yy'|$.
A similar argument using
$|yz|=|yu|+|uz|$
proves the right hand side inequality.
\end{proof}
All necessarily information about
$\CAT(0)$
spaces like definition of angles, the triangle inequality for angles,
the comparison of angles, the first variation formula etc used in the proof below can be
found in \cite{BH}.
\begin{proof}[Proof of Theorem~\ref{thm:cat0_boundary_contiuous}]
Given
$o\in Y$, $\xi$, $\xi'\in\di Y$,
we have to show that for any sequences
$\{x_i\}\in\xi$, $\{x_i'\}\in\xi'$
there is a limit
$\lim_i(x_i|x_i')_o$.
We can assume that
$\xi\neq\xi'$
because otherwise there is nothing to prove.
We use the notation
$\xi=\xi(t)$
for the unit speed parametrization of the geodesic ray
$o\xi$
with
$\xi(0)=o$.
By monotonicity of the Gromov product, see Lemma~\ref{lem:gromov_product_monotone},
there is a limit
$$a=\lim_{t\to\infty}(\xi(t)|\xi'(t))_o.$$
We have
$a<\infty$
because
$Y$
is hyperbolic and
$\xi\neq\xi'$,
which implies that the geodesic segment
$\xi(t)\xi'(t)$
stays at uniformly in
$t$
bounded distance from
$o$.
Since
$Y$
is proper, the segments
$\xi(t)\xi'(t)$
subconverge in the compact-open topology as
$t\to\infty$
to a geodesic
$\ga\sub Y$
with the end points
$\xi$, $\xi'$
at infinity.
(1) We fix
$p\in\ga$
and show that
$|x_ip|+|px_i'|=|x_ix_i'|+o_i(1)$.
The geodesic segments
$px_i$, $px_i'$
converge to subrays
$p\xi$, $p\xi'\sub\ga$
respectively in the compact-open topology as
$i\to\infty$.
It follows that the angle
$\angle_p(x_i,x_i')\ge\pi-o_i(1)$.
Let
$q_i\in x_ix_i'$
be the point closest to
$p$.
By hyperbolicity of
$Y$
we have
$|pq_i|=\dist(p,x_ix_i')\le d$
for some
$d>0$
and all
$i$.
For the triangles
$\De_i=pq_ix_i$, $\De_i'=pq_ix_i'$
we have
$\angle_{q_i}(p,x_i)$, $\angle_{q_i}(p,x_i')\ge\pi/2$,
and
$\angle_p(x_i,q_i)+\angle_p(q_i,x_i')\ge\angle_p(x_i,x_i')\ge\pi-o_i(1)$.
Using the comparison of angles for
$\CAT(0)$
spaces, we see that the comparison triangles
$\wt\De_i=\wt p\wt q_i\wt x_i$, $\wt\De_i'=\wt p\wt q_i\wt x_i'\sub\R^2$
have angles
$\ge\pi/2$
at
$\wt q_i$,
and
$\angle_{\wt p}(\wt x_i,\wt q_i)\ge\angle_p(x_i,q_i)$,
$\angle_{\wt p}(\wt q_i,\wt x_i')\ge\angle_p(q_i,x_i')$.
Thus
$\angle_{\wt p}(\wt x_i,\wt q_i)$, $\angle_{\wt p}(\wt q_i,\wt x_i')<\pi/2$,
and we obtain
$$\pi-o_i(1)\le\angle_{\wt p}(\wt x_i,\wt q_i)+\angle_{\wt p}(\wt q_i,\wt x_i')<\pi.$$
Hence,
$\angle_{\wt p}(\wt x_i,\wt q_i)$, $\angle_{\wt p}(\wt q_i,\wt x_i')\ge\pi/2-o_i(1)$.
Using that
$|\wt p\wt q_i|\le d$,
we can apply Lemma~\ref{lem:o(1)_difference} and conclude that
$|\wt x_i\wt p|=|\wt x_i\wt q_i|+o_i(1)$,
$|\wt p\wt x_i'|=|\wt q_i\wt x_i'|+o_i(1)$.
Therefore
$|x_ip|+|px_i'|=|x_ix_i'|+o_i(1)$.
(2) By hyperbolicity of
$Y$,
there are points
$u\in o\xi$, $u'\in o\xi'$, $v_t\in\xi(t)\xi'(t)$
with mutual distances bounded above independent of
$t$.
Thus
$\angle_{\xi(t)}(o,\xi'(t))=\angle_{\xi(t)}(o,v_t)=o_t(1)$,
$\angle_{\xi'(t)}(o,\xi(t))=\angle_{\xi'(t)}(o,v_t)=o_t(1)$,
that is, the segment
$ov_t$
is observed from
$\xi(t)$
and
$\xi'(t)$
under arbitrarily small angles as
$t\to\infty$.
(3) Let
$\eta(t)$, $\eta'(t)\in\ga$
be points closest to
$\xi(t)$, $\xi'(t)$
respectively. Since the geodesic
$\ga$
is convex as a set in
$Y$,
we have
$|\eta(t)\eta'(t)|\le|\xi(t)\xi'(t)|$.
Our next goal is to show that
$|\xi(t)\xi'(t)|\le|\eta(t)\eta'(t)|+o_t(1)$.
Since the geodesic rays
$o\xi$, $p\xi$
are asymptotic, the distance
$\dist(\xi(t),\ga)$
is uniformly bounded above.
Using convexity of the distance function on
$Y$,
we conclude that
$g(t)=\dist(\xi(t),\ga)$
and similarly
$g'(t)=\dist(\xi'(t),\ga)$
decrease as
$t\to\infty$.
Then for
$t'>t$
we have
$g(t')\le g(t)\le|\xi(t)\eta(t')|$
and similarly
$g'(t')\le g'(t)\le|\xi'(t)\eta'(t')|$.
The first variation formula for
$\CAT(0)$
spaces, see \cite[Corollary~3.6]{BH}, implies that
$\angle_{\xi(t)}(\eta(t),o)$, $\angle_{\xi'(t)}(\eta'(t),o)\ge\pi/2$
for all
$t>0$.
Combining that with the estimates from (2) for the angles
$\angle_{\xi(t)}(o,\xi'(t))$, $\angle_{\xi'(t)}(o,\xi(t))=o_t(1)$,
we conclude that
$\angle_{\xi(t)}(\eta(t),\xi'(t))$, $\angle_{\xi'(t)}(\eta'(t),\xi(t))\ge\pi/2-o_t(1)$.
Therefore, all the angles of the quadrilateral
$\eta(t)\xi(t)\xi'(t)\eta'(t)$
are at least
$\pi/2-o_t(1)$.
We also note that
$g(t)=|\xi(t)\eta(t)|$
and
$g'(t)=|\xi(t)\eta(t)|\le c$
for all
$t\ge 0$
and some
$c>0$
independent of
$t$.
Let
$x(t)y(t)u(t)$, $y(t)z(t)u(t)$
be comparison triangles in
$\R^2$
with vertices
$x(t)$, $z(t)$
separated by the common side
$y(t)u(t)$
for triangles
$\eta(t)\xi(t)\eta'(t)$, $\xi(t)\xi'(t)\eta'(t)$
in
$Y$
respectively. Using the comparison of angles in
$\CAT(0)$
spaces and the triangle inequality for angles, we obtain that all
the angles of the quadrilateral
$x(t)y(t)z(t)u(t)\sub\R^2$
are at least
$\pi/2-o_t(1)$.
Since
$|x(t)y(t)|$, $|z(t)u(t)|\le c$,
we have
$\angle_{y(t)}(z(t),u(t))$, $\angle_{u(t)}(x(t),y(t))=o_t(1)$.
Thus
$\angle_{y(t)}(x(t),u(t))$, $\angle_{u(t)}(z(t),y(t))\ge\pi/2-o_t(1)$.
By Lemma~\ref{lem:o(1)_difference},
$|y(t)z(t)|$, $|x(t)u(t)|=|y(t)u(t)|+o_t(1)$,
hence
$|\xi(t)\xi'(t)|\le|\eta(t)\eta'(t)|+o_t(1)$.
(4) Now, we show that
$\al(t)$, $\al'(t)\ge\pi/2-o_t(1)$,
where
$\al(t)=\angle_{\xi(t)}(\eta(t),\xi)$,
$\al'(t)=\angle_{\xi'(t)}(\eta'(t),\xi')$.
For brevity, we only prove this estimate for the angles
$\al(t)$.
By the first variation formula, we have
$|\xi(t+s)\eta(t)|=|\xi(t)\eta(t)|-s\cos\al(t)+o(s)$
for all sufficiently small
$s\ge 0$.
On the other hand, the function
$g=g(t)$
is convex. Thus it has at every point the right derivative
$d_+ g/dt$,
which is non decreasing. It is nonpositive because
$g(t)$
decreases. Thus
$-d_+ g(t)/dt=o_t(1)$.
Using that
$g(t+s)\le|\xi(t+s)\eta(t)|$
for every
$s\ge 0$,
we obtain
$$g(t)-s\cos\al(t)+o(s)=|\xi(t+s)\eta(t)|\ge g(t+s)\ge g(t)+s\cdot d_+g(t)/dt$$
for all sufficiently small
$s>0$,
hence
$\cos\al(t)\le-d_+g(t)/dt=o_t(1)$,
and therefore
$\al(t)\ge \pi/2-o_t(1)$.
(5) We show that
$|\xi(t)x_i|=|\eta(t)x_i|+o_{t,i}(1)$
for every sufficiently large fixed
$t$,
and similarly
$|\xi'(t)x_i'|=|\eta'(t)x_i'|+o_{t,i}(1)$.
The geodesic segments
$\xi(t)x_i$, $\eta(t)x_i$
converge in the compact-open topology to subrays
$\xi(t)\xi$, $\eta(t)\xi$
respectively as
$i\to\infty$.
Thus
$\angle_{\xi(t)}(\eta(t),x_i)\ge\al(t)-o_i(1)$
and
$\angle_{\eta(t)}(\xi(t),x_i)\ge\be(t)-o_i(1)$,
where
$\be(t)=\angle_{\eta(t)}(\xi(t),\xi)\ge\pi/2$.
Using (4) and the comparison of angles, we obtain that the angles at
$x$, $y$
of the comparison triangle
$xyz\sub\R^2$
for
$\xi(t)\eta(t)x_i$
are
$\ge\pi/2-o_{t,i}(1)$.
By Lemma~\ref{lem:o(1)_difference},
$|\xi(t)x_i|=|\eta(t)x_i|+o_{t,i}(1)$.
(6) Since the geodesic segments
$ox_i$
converge to the ray
$o\xi$,
we have
$|ox_i|=|o\xi(t)|+|\xi(t)x_i|-o_{t,i}(1)$
for every fixed
$t>0$
and all sufficiently large
$i$.
Similarly,
$|px_i|=|p\eta(t)|+|\eta(t)x_i|-o_{t,i}(1)$.
By (5),
$|ox_i|-|px_i|=|o\xi(t)|-|p\eta(t)|+o_{t,i}(1)$.
Using (1), (3) and
$|\eta(t)p|+|p\eta'(t)|=|\eta(t)\eta'(t)|$,
we finally obtain
$(x_i|x_i')_o=(\xi(t)|\xi'(t))_o+o_{t,i}(1)$.
Hence
$\lim_i(x_i|x_i')_o=a$.
\end{proof}
\begin{cor}\label{cor:zero_gromov_product} In a proper
Gromov hyperbolic
$\CAT(0)$
space
$Y$,
we have
$(\xi|\xi')_o=0$
if and only if
$\angle_o(\xi,\xi')=\pi$
for
$o\in Y$, $\xi$, $\xi'\in\di Y$.
\end{cor}
\begin{proof} If
$\angle_o(\xi,\xi')=\pi$,
then
$|xo|+|ox'|=|xx'|$,
and
$(x|x')_o=0$
for every
$x\in o\xi$, $x'\in o\xi'$.
By Theorem~\ref{thm:cat0_boundary_contiuous},
$(\xi|\xi')_o=0$.
Conversely, assume that
$\angle_o(\xi,\xi')<\pi$.
Then for
$x\in o\xi$, $x'\in o\xi'$
sufficiently close to
$o$,
we have
$|xo|+|ox'|>|xx'|$,
and thus
$(x|x')_o>0$.
By monotonicity of the Gromov product and Theorem~\ref{thm:cat0_boundary_contiuous},
$(\xi|\xi')_o\ge (x|x')_o>0$.
\end{proof}
\section{Appendix 2}
\label{sect:appendix_2}
\begin{center} {\Large Viktor Schroeder}
\end{center}
Here, it will be shown that Axiom~(t6) follows from the other axioms of
timed causal spaces. That is, we assume Axioms (h1)--(h6) and (t1)--(t5)
but not (t6) and show that (t6) follows. Given an event
$e=(\al,\be)$
we have a reflection
$\rho=\rho_e:S^1\to S^1$
fixing
$\al$, $\be$.
The M\"obius structure
$M$
is obtained in Theorem~\ref{thm:submoeb_is_moeb} without using (t6).
It gives another timelike line structure
$\cH_M$
and hence for
$e$
another reflection
$\tau=\tau_e:S^1\to S^1$.
Choose
$x$, $y$
in the same component of
$S^1\sm\{\al,\be\}$
in the order
$\al xy\be$.
We use notation
$[\ ,\ ,\ ,\ ]$
for the cross-ratio
$\crr_3$,
$$[x,y,z,u]:=\frac{|xy||zu|}{|xz||yu|}.$$
Then we have
\begin{equation}\label{eq:tau_equality}
[\al,x,\tau(x),\be]=[\al,y,\tau(y),\be]=1.
\end{equation}
This cross-ratio satisfies the {\em cocycle property}
$$[\al,x,y,\be][\al,y,z,\be]=[\al,x,z,\be]$$
for any
$x$, $y$, $z$.
Axiom~(t6) is not used in the proof of Lemma~\ref{lem:pro:timed_monotone:axiom_m}.
By that Lemma, the time of the timed causal space is computed in the usual way via
$M$-cross-ratios. Thus we have
$$\ln[\al,x,y,\be]=-t((x,\rho(x)),(y,\rho(y)))=\ln[\al,\rho(x),\rho(y),\be],$$
and we have by the cocycle property and (\ref{eq:tau_equality})
$[\al,x,y,\be]=[\al,\tau(x),\tau(y),\be]$.
Thus
$$[\al,\rho(x),\tau(x),\be][\al,\tau(x),\rho(y),\be]=[\al,\rho(x),\rho(y),\be]$$
equals
$$[\al,\tau(x),\rho(y),\be][\al,\rho(y),\tau(y),\be]=[\al,\tau(x),\tau(y),\be].$$
Thus
$[\al,\rho(x),\tau(x),\be]$
is constant for
$x$
in a connected component of
$S^1\sm\{\al,\be\}$.
In order to prove the result, we have to show that
$[\al,\rho(x),\tau(x),\be]=1$.
Then be monotonicity
$\rho(x)=\tau(x)$,
and we have (t6).
Now
$[\al,\rho(x),\tau(x),\be]=[\al,\rho(x),x,\be]$
since
$[\al,\tau(x),x,\be]=1$
and, hence, also
\begin{equation}\label{eq:rho_constant}
[\al,x,\rho(x),\be]\quad\textrm{is constant in}\ x.
\end{equation}
\begin{figure}[htbp]
\centering
\psfrag{1}{$1$}
\psfrag{2}{$2$}
\psfrag{3}{$3$}
\psfrag{4}{$4$}
\psfrag{5}{$5$}
\psfrag{6}{$6$}
\psfrag{7}{$7$}
\psfrag{8}{$8$}
\psfrag{9}{$9$}
\psfrag{10}{$10$}
\includegraphics[width=0.6\columnwidth]{fivegone.eps}
\caption{Pentagon $P$}\label{fi:fivegone}
\end{figure}
Now, construct a pentagon
$P=x_1x_2,x_2x_3,\dots,x_9x_{10}$
of consecutively ``orthogonal'' timelike lines, i.e.,
$\rho_{x_i,x_{i+1}}(x_{i+2})=x_{i+3}$
for
$i=1,\dots,9$,
where indexes are taken modulo 10 (existence of
$P$
easily follows from Proposition~\ref{pro:timelike_lines}(b)). Then
(\ref{eq:rho_constant}) implies (we use
$[i,j,k,l]=[x_i,x_j,x_k,x_l])$
\begin{align*}
[1,3,4,2]&=[6,3,4,5]\\
&=[6,8,7,5]\\
&=[9,8,7,10]\\
&=[9,1,2,10]\\
&=[4,1,2,3]=[1,4,3,2]=1/[1,3,4,2],
\end{align*}
hence
$[1,3,4,2]=1$.
| 92,017
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Acquiring not buy large toys for a child if there is not adequate space for them to be used and stored. If you are purchasing a large toy, ensure you have enough room for your child to safely play with it. Storage space for the toy may also be an issue..
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Make sure that all of the toys you purchase have an exchange/return policy. By the time your child gets the gift, their interests may have moved on to something different. It is always a good idea to purchase toys that have an exchange policy or the store gives your money back.. along.
| 237,362
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Hi, everyone. Have anyone of you ever encounter I situation where Android keep asking of password when the device restart. Yes this normal for most android but is abnormal is sometimes you put in the right password but the phone refuses to boot or you may just have forgotten the password.
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Step 1: Enter Incorrect Password many times Till Phone Will Tell You To Restart
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if the problem comes up again, keep doing the same thing as stated above till you get successful result
Thank you as you enjoy your Android.
| 49,397
|
\begin{document}
\title{\bf Cluster structures on quantum coordinate rings}
\author{C. Gei\ss, B. Leclerc, J. Schr\"oer}
\date{}
\maketitle
\begin{abstract}
We show that the quantum coordinate ring of the unipotent subgroup $N(w)$ of a
symmetric Kac-Moody group $G$ associated with a Weyl group element $w$
has the structure of a quantum cluster algebra.
This quantum cluster structure arises naturally from a subcategory
$\CC_w$ of the module category of the corresponding
preprojective algebra.
An important ingredient of the proof is a system of quantum determinantal identities
which can be viewed as a $q$-analogue of a $T$-system.
In case $G$ is a simple algebraic group of type $A, D, E$, we deduce
from these results that the quantum coordinate ring of an open cell of
a partial flag variety attached to $G$ also has a cluster
structure.
\end{abstract}
\bigskip
\setcounter{tocdepth}{1}
\tableofcontents
\section{Introduction}
Let $\g$ be the Kac-Moody algebra associated with a symmetric
Cartan matrix.
Motivated by the theory of integrable systems in statistical mechanics
and quantum field theory, Drinfeld and Jimbo have introduced its
quantum enveloping algebra $U_q(\g)$.
Let $\n$ denote the nilpotent subalgebra arising from
a triangular decomposition of $\g$. In the case when $\g$ is finite-dimensional,
Ringel \cite{Ri} showed that the positive part $U_q(\n)$ of $U_q(\g)$ can be
realized as the (twisted) Hall algebra of the category of representations
over $\fF_{q^2}$ of a quiver $Q$, obtained by orienting the Dynkin diagram of $\g$.
This was a major inspiration for Lusztig's geometric realization of $U_q(\n)$
in terms of Grothendieck groups of categories of perverse sheaves over
varieties of representations of $Q$, which is also valid when $\g$ is infinite-dimensional \cite{Lu0}.
The constructions of Ringel and Lusztig involve the choice of an orientation
of the Dynkin diagram. In an attempt to get rid of this choice, Lusztig
replaced the varieties of representations of $Q$ by the varieties of
nilpotent representations of its preprojective algebra $\L = \L(Q)$,
which depends only on the underlying unoriented graph.
He showed that one can realize the enveloping algebra $U(\n)$
as an algebra of $\C$-valued constructible functions over these nilpotent varieties
\cite{Lu0,Lu4}.
The multiplication of $U(\n)$ is obtained as a convolution-type product
similar to the product of the Ringel-Hall algebra, but using
Euler characteristics of complex varieties instead of
number of points of varieties over finite fields.
Note that this realization of $U(\n)$ is only available when the Cartan
matrix is symmetric.
One of the motivations of this paper was to find a similar construction of
the quantized enveloping algebra $U_q(\n)$, as a kind of Ringel-Hall algebra
attached to a category of representations of $\L$.
Unfortunately, there seems to be no simple way of $q$-deforming Lusztig's
realization of $U(\n)$. In this paper we try to overcome this difficulty by switching
to the dual picture.
The dual of $U(\n)$ as a Hopf algebra can be identified
with the algebra $\C[N]$ of regular functions on the pro-unipotent group $N$
attached to $\n$. Dualizing Lusztig's construction, one can obtain
for each nilpotent representation $X$ of $\L$ a distinguished regular function
$\vphi_X\in\C[N]$, and the product $\vphi_X \vphi_Y$ can be calculated
in terms of varieties of short exact sequences with end-terms $X$ and $Y$
\cite{GLSMult}.
For each element $w$ of the Weyl group $W$ of $\g$,
the group $N$ has a finite-dimensional
subgroup $N(w)$ of dimension equal to the length of $w$.
In particular, when $\g$ is finite-dimensional, we have
$N = N(w_0)$, where $w_0$ is the longest element of $W$.
In \cite{GLS}, we have shown that the coordinate ring $\C[N(w)]$
is spanned by the functions $\vphi_X$ where $X$ goes over
the objects of a certain subcategory $\CC_w$ of $\md(\L)$.
This category was
introduced by Buan, Iyama, Reiten and Scott in \cite{BIRS},
and independently in \cite{GLSUni1} for adaptable $w$.
Moreover, we have proved that $\C[N(w)]$ has a
cluster algebra structure in the sense of Fomin and Zelevinsky \cite{FZ2},
for which the cluster monomials are of the form $\vphi_T$
for rigid objects $T$ of $\CC_w$.
The algebra $\C[N]$ has a quantum deformation, which we denote by
$A_q(\n)$ (see below \S\,\ref{multAqn}), and it is well known that
the algebras $U_q(\n)$ and $A_q(\n)$ are in fact isomorphic.
By works of Lusztig and De Concini-Kac-Procesi, $A_q(\n)$ has a subalgebra $A_q(\n(w))$,
which can be regarded
as a quantum deformation of $\C[N(w)]$.
On the other hand, Berenstein and Zelevinsky \cite{BZ} have introduced the
concept of a quantum cluster algebra. They have conjectured that
the quantum coordinate rings of double Bruhat cells in semisimple algebraic groups should have
a quantum cluster algebra structure.
In this paper, we introduce for every $w$ an explicit quantum cluster algebra
$\AA_{\Q(q)}(\CC_w)$, defined in a natural way in terms of the
category $\CC_w$. In particular, for every reachable rigid object $T$
of $\CC_w$ there is a corresponding quantum cluster monomial $Y_T \in \AA_{\Q(q)}(\CC_w)$.
Our main result is the following quantization of the above theorem of \cite{GLS}.
\begin{Thm}\label{ThIntro}
There is an algebra isomorphism $\k\colon \AA_{\Q(q)}(\CC_w) \stackrel{\sim}{\to} A_q(\n(w))$.
\end{Thm}
Note that quantizations of coordinate rings and quantizations of cluster algebras
are defined in very different ways. For example $A_q(\n) \cong U_q(\n)$ is
given by its Drinfeld-Jimbo presentation, obtained by $q$-deforming the Chevalley-Serre-type
presentation of $U(\n)$. In contrast, quantum cluster algebras are defined
as subalgebras of a skew field of rational functions in $q$-commuting variables,
generated by a usually infinite number of elements given by an inductive procedure.
As a matter of fact, there does not seem to be so many examples
of ``concrete'' quantum cluster algebras in the literature.
Grabowski and Launois \cite{GL} have shown that the quantum coordinate rings of
the Grassmannians $\Gr(2,n) \ (n\ge 2)$, $\Gr(3,6)$, $\Gr(3,7)$, and $\Gr(3,8)$
have a quantum cluster algebra structure.
Lampe \cite{La,La2} has proved two particular instances of Theorem~\ref{ThIntro}, namely
when $\g$ has type $A_n$ or $A_1^{(1)}$ and $w=c^2$ is the square of a Coxeter element.
Recently, the existence of a quantum cluster structure on every algebra $A_q(\n(w))$ was
conjectured by Kimura \cite[Conj.1.1]{Ki}.
Now Theorem~\ref{ThIntro} provides a large class of such examples,
including all algebras $U_q(\n)$ for $\g$ of type $A,D,E$.
By taking $\g = \Sl_n$ and some special permutation $w_k\in S_n$,
one also obtains that the quantum coordinate ring $A_q(\Mat(k,n-k))$
of the space of $k\times(n-k)$-matrices
has a quantum cluster algebra structure for every $1\le k\le n$.
This may be regarded as a cluster structure
on the quantum coordinate ring of an open cell of $\Gr(k,n)$.
More generally, for any simply-laced simple algebraic group~$G$, and any
parabolic subgroup $P$ of $G$, we obtain a cluster structure
on the quantum coordinate ring of the unipotent radical $N_P$ of $P$,
which can be regarded as a cluster structure
on the quantum coordinate ring of an open cell of the partial flag
variety $G/P$.
Note also that, by taking $w$ equal to the square of a Coxeter element,
our result gives a Lie theoretic realization
of all quantum cluster algebras associated with an arbitrary acyclic quiver
(but with a particular choice of coefficients).
Our strategy for proving Theorem~\ref{ThIntro} can be summarized as
follows.
Let $A_q(\g)$ be the quantum analogue of the coordinate ring
constructed by Kashiwara \cite{K}.
We first obtain a general quantum determinantal
identity in $A_q(\g)$ (Proposition~\ref{qmin-id1}). This is
a $q$-analogue (and an extension to the Kac-Moody case) of a determinantal identity of
Fomin and Zelevinsky \cite{FZ}. We then transfer this identity
to $A_q(\n)$ (Proposition~\ref{unip_minor_identity}).
Here a little care must be taken since the restriction
map from $A_q(\g)$ to $A_q(\n)$ is \emph{not} a ring homomorphism.
Appropriate specializations of this identity give rise, for every
$w\in W$, to a system $\Sigma_w$
of equations (Proposition~\ref{Tsystem})
allowing to calculate certain quantum minors depending on $w$ in a recursive
way. This system is a $q$-analogue of a $T$-system
arising in various problems of mathematical physics and
combinatorics (see \cite{KNS}), and we believe it could be of independent
interest. It will turn out that all quantum minors involved in $\Sigma_w$ belong
to the subalgebra $A_q(\n(w))$, and that among them we find a set of
algebra generators.
On the other hand, we show that the generalized determinantal
identities of \cite[Theorem 13.1]{GLS}, which relate a distinguished
subset of cluster variables, take exactly the same form as
the quantum $T$-system $\Sigma_w$
when we lift them to the quantum cluster
algebra $\AA_{\Q(q)}(\CC_w)$.
Therefore, after establishing
that the quantum tori consisting of the initial variables of the
two systems are isomorphic, we can construct an injective algebra homomorphism
$\k$ from $\AA_{\Q(q)}(\CC_w)$ to the skew field of fractions $F_q(\n(w))$
of $A_q(\n(w))$, and show that the image of $\k$ contains a set of
generators of $A_q(\n(w))$. Finally, using an argument of specialization
$q \to 1$ and our result of \cite{GLS},
we conclude that $\k$ is an isomorphism from $\AA_{\Q(q)}(\CC_w)$
to $A_q(\n(w))$.
Another motivation of this paper was the open orbit conjecture of
\cite[\S18.3]{GLS}. This conjecture states that all functions $\vphi_T$
associated with a rigid $\L$-module $T$ belong to the dual canonical
basis of $\C[N]$.
It can be seen as a particular instance of the general principle of Fomin and
Zelevinsky \cite{FZ2} according to which, in every cluster algebra coming from Lie theory,
cluster monomials should belong to the dual canonical basis.
Since the dual canonical basis of $\C[N]$ is obtained by specializing
at $q=1$ the basis $\B^*$ of $A_q(\n)$ dual to Lusztig's canonical basis
of $U_q(\n)$, it is natural to conjecture that, more precisely,
every quantum cluster monomial $Y_T$ of $\AA_{\Q(q)}(\CC_w)$ is mapped by $\k$ to
an element of $\B^*$.
In fact,
it is not too difficult to show that $\k(Y_T)$ always satisfies
one of the two characteristic properties of $\B^*$ (see below \S\ref{canonbasACw}).
But unfortunately, the second property remains elusive, although
Lampe \cite{La,La2} has proved it for all cluster variables in the two special
cases mentioned above.
Finally we note that it is well known that the algebras $A_q(\n(w))$ are
skew polynomial rings. Therefore, by Theorem~\ref{ThIntro},
all quantum cluster algebras of the form $\AA_{\Q(q)}(\CC_w)$ are also
skew polynomial rings, which is far from obvious from their definition.
One may hope that, conversely, the existence of a cluster structure on
many familiar quantum coordinate rings will bring some new insights for studying
their ring-theoretic properties, a very active subject in recent years
(see \eg \cite{BG,GLL,MC,Y} and references therein).
\section{The quantum coordinate ring $A_q(\g)$}
\subsection{The quantum enveloping algebra $U_q(\g)$}
Let $\g$ be a symmetric Kac-Moody algebra
with Cartan subalgebra $\t$.
We follow the notation of \cite[\S1]{K}.
In particular,
we denote by $I$ the indexing set of the simple roots $\a_i\ (i\in I)$
of $\g$, by $P\subset\t^*$ its weight lattice, by $h_i (i\in I)$
the elements of $P^*\subset\t$
such that
$\< h_i, \a_j\>=a_{ij}$ are the entries of
the generalized Cartan matrix of $\g$.
Since $\g$ is assumed to be symmetric, we also have a
symmetric bilinear form $(\cdot,\cdot)$ on $\t^*$ such that
$(\a_i,\a_j) = a_{ij}$.
The Weyl group $W<\GL(\t^*)$ is the Coxeter group generated by the
reflections $s_i$ for $i\in I$, where
\[
s_i(\ga):= \ga -\<h_i,\ga\>\a_i.
\]
The length of $w\in W$ is denoted by $l(w)$.
We will also need the contragradient action of $W$ on $P^*$.
Let $U_q(\g)$ be the corresponding quantum enveloping algebra,
a $\Q(q)$-algebra with generators $e_i, f_i \ (i\in I), q^h\ (h\in P^*)$.
We write $t_i=q^{h_i}$.
We denote by $U_q(\nn)$ (\resp $U_q(\nn_-)$)
the subalgebra of $U_q(\g)$
generated by $e_i\ (i\in I)$ (\resp $f_i\ (i\in I))$.
For $i\in I$, let $U_q(\g_i)$ denote the subalgebra of $U_q(\g)$
generated by $e_i, f_i, q^h\ (h\in P^*)$.
Let $M$ be a (left) $U_q(\g)$-module.
For $\la\in P$,
let
$M_{\la} = \{m\in M \mid q^h m = q^{\<\la,\,h\>} m \mbox{ for every } h\in P^*\}$
be the corresponding weight space of $M$.
We say that $M$ is \emph{integrable} if (i) $M = \oplus_{\la\in P}\ M_\la$,
(ii) for any $i$, $M$ is a direct sum of finite-dimensional $U_q(\g_i)$-modules,
and (iii) for any $m\in M$, there exists $l\ge 0$ such that
$e_{i_1}\cdots e_{i_l} m = 0$ for any $i_1,\ldots, i_l\in I$.
We denote by $\Oint(\g)$ the category of integrable $U_q(\g)$-modules.
This is a semisimple category, with simple objects the irreducible
highest weight modules $V(\la)$ with highest weight $\la\in P_+$, the
monoid of dominant weights.
\subsection{Bimodules}
Let $\vphi$ and $*$ be the $\Q(q)$-linear anti-automorphisms of $U_q(\g)$
defined by
\begin{eqnarray}\label{vphi}
&&\vphi(e_i) = f_i,\quad \vphi(f_i) = e_i, \quad \vphi(q^h) = q^h, \\[2mm]
&&e_i^* = e_i,\quad f_i^* = f_i, \quad (q^h)^* = q^{-h}. \label{star}
\end{eqnarray}
A right $U_q(\g)$-module $N$ gives rise to a left $U_q(\g)$-module $N^\vphi$
by defining
\begin{equation}
x\cdot n = n\cdot\vphi(x), \qquad (n\in N, \ x\in U_q(\g)).
\end{equation}
We say that $N$ is an integrable right module if $N^\vphi$ is an integrable
left module.
In particular, for $\la\in P_+$, we have an irreducible integrable right
module $V^\r(\la)$ such that $V^\r(\la)^\vphi = V(\la)$.
Let $m_\la$ be a highest weight vector in $V(\la)$, \ie $e_i\, m_\la = 0$
for any $i\in I$. Then $m_\la$ can be regarded as a vector $n_\la\in V^\r(\la)$,
which satisfies $n_\la \, f_i = 0$ for any $i\in I$.
Equivalently, $V^\r(\la)$ is isomorphic to the graded dual of $V(\la)$, endowed
with the natural right action of $U_q(\g)$. It follows that we have a natural
pairing $\<\cdot,\cdot\>_\la\colon V^\r(\la) \times V(\la) \to \Q(q)$, which satisfies
$\<n_\la,m_\la\>_\la = 1$, and
\begin{equation}
\<n\,x,\, m\>_\la = \<n,\, x\,m\>_\la, \qquad
(m\in V(\la),\ n\in V^\r(\la),\ x\in U_q(\g)).
\end{equation}
We denote by $\Oint(\g^{\rm op})$ the category of integrable right
$U_q(\g)$-modules.
It is also semisimple, with simple objects $V^\r(\la)\ (\la\in P_+)$.
The tensor product of $\Q(q)$-vector spaces $V^\r(\la)\otimes V(\la)$
has the natural structure of a $U_q(\g)$-bimodule, via
\begin{equation}\label{bimod_matrix_coeff}
x\cdot(n\otimes m)\cdot y = (n\cdot y)\otimes (x\cdot m),\qquad
(x,y \in U_q(\g),\ m\in V(\la),\ n \in V^\r(\la)).
\end{equation}
\subsection{Dual algebra}\label{Uqstar}
Let $U_q(\g)^*=\Hom_{\Q(q)}(U_q(\g),\Q(q))$. This is a $U_q(\g)$-bimodule, via
\begin{equation}\label{bimod_Uqstar}
(x\cdot\psi\cdot y)(z) = \psi(yzx),\qquad
(x,y,z\in U_q(\g),\ \psi\in U_q(\g)^*).
\end{equation}
On the other hand, $U_q(\g)$ is a Hopf algebra, with comultiplication
$\De\colon U_q(\g) \to U_q(\g)\otimes U_q(\g)$
given by
\begin{equation}\label{comult}
\De(e_i) = e_i\otimes 1 + t_i \otimes e_i,\quad
\De(f_i) = f_i\otimes t_i^{-1} + 1 \otimes f_i,\quad
\De(q^h) = q^h\otimes q^h,
\end{equation}
with counit $\eps\colon U_q(\g)\ra\Q(q)$ given by
\begin{equation}
\eps(e_i)=\eps(f_i)=0,\quad\eps(q^h)=1,
\end{equation}
and with antipode $S$ given by
\begin{equation}\label{antipode}
S(e_i)=-t_i^{-1}e_i,\quad S(f_i) = -f_i\,t_i,\quad S(q^h) = q^{-h}.
\end{equation}
Dualizing $\De$, we obtain a multiplication on $U_q(\g)^*$, defined by
\begin{equation} \label{multA}
(\psi\,\theta)(x) = (\psi\otimes\theta)(\De( x)), \qquad
(\psi, \theta \in U_q(\g)^*,\ x\in U_q(\g)).
\end{equation}
Later on it will often be convenient to use Sweedler's notation
$\De(x) = \sum x_{(1)}\otimes x_{(2)}$ for
the comultiplication. Using this notation, (\ref{multA}) reads
$(\psi\,\theta)(x) =\sum \psi(x_{(1)})\theta(x_{(2)})$.
Combining~(\ref{bimod_Uqstar}) and~(\ref{multA}) we obtain
\begin{equation} \label{bimod_alg}
x\cdot(\psi\,\theta)\cdot y =
\sum (x_{(1)}\cdot\psi\cdot y_{(1)})\,(x_{(2)}\cdot\theta\cdot y_{(2)}).
\end{equation}
\subsection{Peter-Weyl theorem}
Following Kashiwara \cite[\S7]{K}, we define $A_q(\g)$ as the subspace of
$U_q(\g)^*$ consisting of the linear forms $\psi$ such that the left
submodule $U_q(\g) \psi$ belongs to $\Oint(\g)$,
and the right submodule $\psi U_q(\g)$ belongs to $\Oint(\g^{\rm op})$.
It follows from the fact that the categories $\Oint(\g)$ and
$\Oint(\g^{\rm op})$
are closed under tensor product that $A_q(\g)$ is a subring of $U_q(\g)^*$.
The next proposition of Kashiwara can be regarded as a $q$-analogue of the Peter-Weyl
theorem for the Kac-Moody group $G$ attached to $\g$ (see \cite{KP}).
We include a proof for the convenience of the reader.
\begin{Prop}[{\cite[Proposition 7.2.2]{K}}] \label{PW-thm}
We have an isomorphism $\Phi$ of $U_q(\g)$-bimodules
\[
\bigoplus_{\la\in P_+} V^\r(\la)\otimes V(\la)
\stackrel{\sim}{\longrightarrow} A_q(\g)
\]
given by
\[
\Phi(n\otimes m)(x) = \<n\,x,\,m\>_\la,\qquad
(m \in V(\la),\ n \in V^\r(\la),\ x\in U_q(\g)).
\]
\end{Prop}
\proof
It follows from (\ref{bimod_matrix_coeff}) and
(\ref{bimod_Uqstar}) that $\Phi$ defines a homomorphism
of $U_q(\g)$-bimodules from $\oplus_{\la\in P_+} V^\r(\la)\otimes V(\la)$ to
$U_q(\g)^*$. Since $V(\la)$ and $V^\r(\la)$ are integrable for all
$\la\in P_+$, we see that $\im\Phi \subseteq A_q(\g)$.
Let us show that $\Phi$ is surjective.
Let $\psi\in A_q(\g)$. We want to show that $\psi\in\im\Phi$.
Since $V:=U_q(\g)\psi$ is integrable, it decomposes as a (finite) direct sum of
irreducible integrable modules. Thus, without loss of generality, we may
assume that $V$ is isomorphic to $V(\la)$ for some $\la\in P_+$.
We may also assume that $\psi$ is a weight vector of $V$ (otherwise
we can decompose it as a sum of weight vectors).
Since $V:=U_q(\g)\psi$ and $W:=\psi U_q(\g)$ are both integrable, we
see that there exist $k$ and $l$ such that
$e_{i_1}\cdots e_{i_k}\cdot\psi = \psi\cdot f_{j_1}\cdots f_{j_l}=0$
for every $i_1, \ldots, i_k, j_1, \ldots, j_l \in I$.
Hence $\psi(xe_{i_1}\cdots e_{i_k})=\psi(f_{j_1}\cdots f_{j_l}x) = 0$
for every $x\in U_q(\g)$.
It follows that the linear form $a\in V^*$ defined by $a(\vphi)=\vphi(1)$
takes nonzero values only on a finite number of weight spaces of $V$.
Hence $a$ is in the graded dual of $V$, which we can identify to
$V^r(\la)$. Moreover,
\[
\Phi(a\otimes \psi)(x) = \<a,\,x\psi\>_\la = a(x\psi) = x\psi(1) = \psi(x)
\]
for every $x\in U_q(\g)$. Therefore $\psi = \Phi(a\otimes \psi)$ belongs
to $\im\Phi$.
Now, $\Phi$ is also injective. Indeed, if for $n\otimes m\in
V^\r(\la)\otimes V(\la)$
we have $\Phi(n\otimes m) = 0$, then for every $x\in U_q(\g)$, $\<nx,\,m\>_\la=0$.
If $n\not = 0$, since $\<\cdot,\,\cdot\>_\la$
is a pairing between $V^\r(\la)$ and $V(\la)$ and $n\,U_q(\g) = V^\r(\la)$,
we get that $m=0$ and $n\otimes m = 0$. Hence the restriction of
$\Phi$ to $V^\r(\la)\otimes V(\la)$
is injective.
Finally, since the bimodules $V^\r(\la)\otimes V(\la)$ are simple and
pairwise non-isomorphic, $\Phi$ is injective. \cqfd
In view of Proposition~\ref{PW-thm}, we can think of $A_q(\g)$ as a $q$-analogue
of the coordinate ring of the Kac-Moody group attached to $\g$ in \cite{KP}.
We therefore call $A_q(\g)$ the \emph{quantum coordinate ring}.
When $\g$ is a simple finite-dimensional Lie algebra,
$A_q(\g)$ is the quantum coordinate ring $\O_q(G)$ of the simply-connected
simple Lie group with Lie algebra $\g$ studied by many authors,
see \eg \cite[\S 9.1.1]{J}.
\subsection{Gradings} Let $Q\subset P$ be the root lattice. It follows from the defining relations
of $U_q(\g)$ that it is a $Q$-graded algebra:
\begin{equation}
U_q(\g)=\bigoplus_{\alpha\in Q} U_q(\g)_\alpha,
\end{equation}
where
\begin{equation}
U_q(\g)_\alpha=\{x\in U_q(\g)\mid q^h x q^{-h} =q^{\< h,\a\>} x\ \mbox{ for all } h\in P^*\}.
\end{equation}
By Proposition~\ref{PW-thm}, we have
\begin{equation}
A_q(\g)=\bigoplus_{\ga,\de\in P} A_q(\g)_{\ga,\de},
\end{equation}
where
\begin{equation}
A_q(\g)_{\ga,\de}=\{\psi\in A_q(\g)\mid
q^l\cdot\psi\cdot q^r=q^{\< r,\ga\>+\< l,\de\>}\psi\ \mbox{ for all } r,l\in P^*\}.
\end{equation}
\begin{Lem}\label{gradings}
\begin{itemize}
\item[(a)]
With the above decomposition $A_q(\g)$ is a $P\times P$-graded algebra.
\item[(b)]
For $x\in U_q(\g)_\a$, $\psi\in A_q(\g)_{\ga,\de}$ and $y\in U_q(\g)_\b$, we have
$x\cdot\psi\cdot y\in A_q(\g)_{\ga-\b,\de+\a}$.
\item[(c)] For $x\in U_q(\g)_\a$, $\psi\in A_q(\g)_{\ga,\de}$, we have $\psi(x)\not = 0$
only if $\a = \ga - \de$.
\end{itemize}
\end{Lem}
\section{Determinantal identities for quantum minors}
\subsection{Quantum minors} \label{ssec:qmin-def}
For our convenience we reproduce \emph{mutatis mutandis}
a part of~\cite[\S9.2]{BZ}.
Using the isomorphism $\Phi$ from Proposition~\ref{PW-thm} we define for
each $\la\in P_+$ the element
\begin{equation} \label{DefPrincMin}
\De^\la:=\Phi(n_\la\otimes m_\la)\in A_q(\g)_{\la,\la}.
\end{equation}
This is a $q$-analogue of a (generalized) principal minor, in the sense
of \cite[\S1.4]{FZ}.
An easy calculation shows that
\begin{equation} \label{PrincMin_Prp}
\De^\la(f\,q^h\,e)=\eps(f)\,q^{\< h,\la\>}\eps(e),\qquad
(f\in U_q(\nn_-),\ h\in P^*,\ e\in U_q(\n)).
\end{equation}
For $(u,v)\in W\times W$, we choose reduced expressions
$\bi=(i_{l(u)},\ldots,i_2,i_1)$ and $\bj=(j_{l(v)},\ldots,j_2,j_1)$ so that
$u = s_{i_{l(u)}}\cdots s_{i_2}s_{i_1}$ and
$v = s_{j_{l(v)}}\cdots s_{j_2}s_{j_1}$.
Next, we introduce positive roots
\begin{equation}\label{defbeta}
\b_k=s_{i_1}s_{i_2}\cdots s_{i_{k-1}}(\a_{i_k}),\quad
\ga_l=s_{j_1}s_{j_2}\cdots s_{j_{l-1}}(\a_{j_l}),\qquad
(1\leq k\leq l(u),\ 1\leq l\leq l(v)).
\end{equation}
Finally, for $\la\in P_+$, we set
\begin{equation}\label{eq-bc}
b_k = (\b_k,\la),\quad c_l = (\ga_l,\la),
\qquad
(1\leq k\leq l(u),\ 1\leq l\leq l(v)),
\end{equation}
and we define the (generalized) \emph{quantum minor}
$\De_{u(\la),v(\la)}\in A_q(\g)$ by
\begin{equation}
\De_{u(\la),v(\la)}= \left(f_{j_{l(v)}}^{(c_{l(v)})}\cdots f_{j_1}^{(c_1)}\right)
\cdot\De^\la\cdot \left(e_{i_1}^{(b_1)}\cdots e_{i_{l(u)}}^{(b_{l(u)})}\right).
\end{equation}
Here, as usual, we denote by $e_i^{(k)}$ (\resp $f_i^{(k)}$) the $q$-divided powers
of the Chevalley generators.
Equivalently, for $x\in U_q(\g)$ we have
\begin{equation}\label{DeltaValue}
\De_{u(\la),v(\la)}(x)=
\De^\la\left(e_{i_1}^{(b_1)}\cdots e_{i_{l(u)}}^{(b_{l(u)})} x\,
f_{j_{l(v)}}^{(c_{l(v)})}\cdots f_{j_1}^{(c_1)}\right).
\end{equation}
It follows from the quantum Verma relations~\cite[Proposition~39.3.7]{Lu}
that $\De_{u(\la),v(\la)}$ depends only on the pair of weights $(u(\la),v(\la))$,
and not on the choice of $u$ and $v$, or of their reduced expressions $\bi$ and $\bj$.
Moreover it is immediate that $\De_{u(\la),v(\la)}\in A_q(\g)_{u(\la),v(\la)}$.
We have the following direct consequence of the definition
of quantum minors.
\begin{Lem}\label{cons_def}
If $l(s_iu)=l(u)+1$ and $l(s_jv)=l(v)+1$ then
\[
\De_{s_iu(\la),s_jv(\la)}=f_j^{(c)}\cdot\De_{u(\la),v(\la)}\cdot e_i^{(b)},
\]
where $b:=( \a_i,u(\la))\geq 0$ (\resp $c:=(\a_j,v(\la))\geq 0$) is the
maximal natural number such that
$\De_{u(\la),v(\la)}\cdot e_i^{(b)}\neq 0$
(\resp $~f_j^{(c)}\cdot\De_{u(\la),v(\la)}\neq 0$).
\end{Lem}
It is convenient to identify $\De_{u(\la),v(\la)}$ with an
extremal vector of weight $(u(\la),v(\la))$ in the
simple $U_q(\g)\otimes U_q(\g)$ highest weight module $V^r(\la)\otimes V(\la)$.
Thus, we have
\begin{align}
f_i\cdot\De_{\ga,\de} &=0 \text{ if }(\a_i,\de)\leq 0, &
e_i\cdot\De_{\ga,\de} &=0 \text{ if }(\a_i,\de)\geq 0, \label{eq:leftz}\\
\De_{\ga,\de}\cdot e_i&=0 \text{ if }(\a_i,\ga)\leq 0, &
\De_{\ga,\de}\cdot f_i&=0 \text{ if }(\a_i,\ga)\geq 0. \label{eq:rightz}
\end{align}
In particular, we have
$f_{j_{l(v)}}\cdot\De_{u(\la),v(\la)}=0=\De_{u(\la),v(\la)}\cdot e_{j_{l(u)}}$.
One may also identify $\De_{u(\la),v(\la)}$ with a matrix coefficient in $V(\la)$.
To do that, let us first denote by
\begin{equation}
m_{v(\la)} := f_{j_{l(v)}}^{(c_{l(v)})}\cdots f_{j_1}^{(c_1)} m_\la
\end{equation}
the extremal weight vector of $V(\la)$ with weight $v(\la)$.
Next, let $(\cdot,\,\cdot)_\la$ be the nondegenerate bilinear form on $V(\la)$ defined
by
\begin{equation}
(xm_\la,\,ym_\la)_\la := \<n_\la\vphi(x),\,ym_\la\>_\la,\qquad (x,y\in U_q(\g)).
\end{equation}
Then, using (\ref{DefPrincMin}) and (\ref{DeltaValue}) we easily get
\begin{equation}\label{matrixCoeff}
\De_{u(\la),v(\la)}(x)= (m_{u(\la)},\,xm_{v(\la)})_\la,\qquad (x\in U_q(\g)).
\end{equation}
\subsection{A family of identities for quantum minors}
Let $(\vpi_i)_{i\in I}$ be the fundamental weights, \ie
we have $\<h_j,\vpi_i\>=(\a_j,\vpi_i)=\de_{ij}$. We note that the fundamental weights
are only determined up to a $W$-invariant element. Moreover, it
is useful to observe that
\begin{equation}\label{sivpi}
s_i(\vpi_j)=\begin{cases}
\vpi_j-\a_j &\text{ if } i=j,\\
\vpi_j &\text{ otherwise.}
\end{cases}
\end{equation}
We can now state the main result of this section. This is a $q$-analogue
of~\cite[Theorem~1.17]{FZ}.
\begin{Prop} \label{qmin-id1}
Suppose that for $u,v\in W$ and $i\in I$ we have $l(us_i)=l(u)+1$ and
$l(vs_i)=l(v)+1$. Then
\begin{equation}\label{eq:2.10}
\De_{us_i(\vpi_i),vs_i(\vpi_i)}\,\De_{u(\vpi_i),v(\vpi_i)}=
q^{-1}\De_{us_i(\vpi_i),v(\vpi_i)}\,\De_{u(\vpi_i),vs_i(\vpi_i)}+
\prod_{j\neq i}\De_{u(\vpi_j),v(\vpi_j)}^{-a_{ji}}
\end{equation}
holds in $A_q(\g)$.
\end{Prop}
The proof of this proposition will be given after some preparation in
Section~\ref{pf:qmin-id1} below, by following essentially
the strategy of~\cite[\S2.3]{FZ}.
We first continue to review material from~\cite[\S9.2]{BZ}.
\begin{Lem} \label{lem:MaxDer}
Let $\psi\in A_q(\g)_{\ga,\de}$ and $\psi'\in A_q(\g)_{\ga',\de'}$. For
$i\in I$, assume that $a=(\a_i,\de)$ and $a'=(\a_i,\de')$ are the maximal
non-negative integers such that $f_i^a\cdot\psi\neq 0$ and
$f_i^{a'}\cdot\psi'\neq 0$. Then
\[
(f_i^{(a)}\cdot \psi)\,(f_i^{(a')}\cdot\psi')= f_i^{(a+a')}\cdot(\psi\,\psi').
\]
\end{Lem}
This follows from the definition of the comultiplication
of $U_q(\g)$ and from~\eqref{bimod_alg}.
Note that we have an analogous result where $f_i$ is replaced by $e_i$ and acts
from the right.
The next lemma is an immediate consequence of Lemma~\ref{cons_def} and Lemma~\ref{lem:MaxDer}.
\begin{Lem} \label{lem:QMinIdRed}
Let $\la',\la''\in P_+$, $u', v', u'', v''\in W$, and $i,j\in I$, be such that
\[
l(s_j v')=l(v')+1,\quad l(s_j v'')=l(v'')+1,\quad l(s_i u')=l(u')+1,\quad l(s_i u'')=l(u'')+1.
\]
Then, putting $a = (\a_j,\, v'(\la')+v''(\la''))$,
and $b = (\a_i,\, u'(\la')+u''(\la''))$, we have
\begin{align*}
f_j^{(a)} \cdot
(\De_{u'(\la'),v'(\la')}\De_{u''(\la''),v''(\la'')}) &=
\De_{u'(\la'),s_jv'(\la')}\De_{u''(\la''),s_jv''(\la'')},\\
(\De_{u'(\la'),v'(\la')}\De_{u''(\la''),v''(\la'')}) \cdot
e_i^{(b)} &=
\De_{s_i u'(\la'),v'(\la')}\De_{s_i u''(\la''), v''(\la'')}.
\end{align*}
\end{Lem}
The next lemma follows easily from Lemma~\ref{lem:QMinIdRed} by induction on the length of $u$ and $v$.
\begin{Lem} \label{rem:MultQMin}
The quantum minors have the following multiplicative property:
\[
\De_{u(\la),v(\la)}\,\De_{u(\mu),v(\mu)}=\De_{u(\la+\mu),v(\la+\mu)},\quad
(u,v\in W,\ \la,\mu\in P_+).
\]
In particular, the factors of the second summand in the right hand side of (\ref{eq:2.10})
pairwise commute.
\end{Lem}
The factors of the first summand in the right hand side of (\ref{eq:2.10})
also commute with each other, as shown by the following lemma.
\begin{Lem}\label{commute}
Suppose that for $u,v\in W$ and $i\in I$ we have $l(us_i)=l(u)+1$ and
$l(vs_i)=l(v)+1$. Then
\[
\De_{us_i(\vpi_i),\,v(\vpi_i)}\,\De_{u(\vpi_i),\,vs_i(\vpi_i)}
=\De_{u(\vpi_i),\,vs_i(\vpi_i)}\,\De_{us_i(\vpi_i),\,v(\vpi_i)}.
\]
\end{Lem}
\proof
For $x\in U_q(\nn)$ we have
\[
x\cdot \De_{s_i(\vpi_i),\,\vpi_i} = x\cdot \De^{\vpi_i}\cdot e_i
= \eps(x) \De^{\vpi_i}\cdot e_i = \eps(x) \De_{s_i(\vpi_i),\,\vpi_i}.
\]
Similarly, for $y\in U_q(\nn_-)$ we have
\[
\De_{\vpi_i,\,s_i(\vpi_i)}\cdot y = \eps(y) \De_{\vpi_i,\,s_i(\vpi_i)}.
\]
By \cite[Lemma~10.2]{BZ} we can then conclude that
\[
\De_{s_i(\vpi_i),\,\vpi_i}\,\De_{\vpi_i,\,s_i(\vpi_i)}=
\De_{\vpi_i,\,s_i(\vpi_i)}\,\De_{s_i(\vpi_i),\,\vpi_i}.
\]
(In \cite{BZ}, $\g$ is generally assumed to be finite-dimensional, but this
assumption plays no role in the proof of \cite[Lemma~10.2]{BZ}.)
This proves the lemma for $u=v=e$. The general case then follows from
successive applications of Lemma~\ref{lem:QMinIdRed}.
\cqfd
\subsection{Proof of Proposition~\ref{qmin-id1}} \label{pf:qmin-id1}
With the help of Lemma~\ref{lem:QMinIdRed}, we see by an easy induction on
the length of $u$ and $v$ that it is sufficient to verify the
special case $u=v=e$, \ie
\begin{equation}
\underbrace{\vphantom{\prod_{j\neq i}}
\De_{s_i(\vpi_i),s_i(\vpi_i)}\,\De_{\vpi_i,\vpi_i}-
q^{-1}\De_{s_i(\vpi_i),\vpi_i}\,\De_{\vpi_i,s_i(\vpi_i)}}_{\psi_1}=
\underbrace{\prod_{j\neq i}\De_{\vpi_j,\vpi_j}^{-a_{ji}}}_{\psi_2}.
\end{equation}
Note that
\[
\ga_i:=-\sum_{j\neq i} a_{ji}\vpi_j = 2\vpi_i-\a_i=\vpi_i+s_i(\vpi_i)\in P_+.
\]
Thus, by Lemma~\ref{rem:MultQMin} we have
\begin{equation} \label{eq:gamma_i}
\psi_2=\De^{\ga_i}=\De_{\ga_i,\ga_i},
\end{equation}
in particular the factors of $\psi_2$ commute. It follows from the definition
of $\De^{\ga_i}$ that
\begin{itemize}
\item[(1)] $\psi_2(1_U)=1$,
\item[(2)] $\psi_2\in A_q(\g)_{\ga_i,\ga_i}$,
\item[(3)] $e_j\cdot \psi_2=0=\psi_2\cdot f_j$ for all $j\in I$.
\end{itemize}
Here $1_U$ stands for the unit of $U_q(\g)$.
By Proposition~\ref{PW-thm} these properties characterize $\psi_2$ uniquely,
so it is sufficient to verify the properties (1) -- (3) for $\psi_1$.
\medskip
{\bf Property (1).\ } We use that for any $\psi,\phi\in A_q(\g)$ we have
$(\psi\,\phi)(1_U)=\psi(1_U)\phi(1_U)$ since $\De(1_U)=1_U\otimes 1_U$, and
note that $q^0=1_U$.
Now, $\De_{\vpi_i,s_i(\vpi_i)}=f_i\cdot\De^{\vpi_i}$, thus
$\De_{\vpi_i,s_i(\vpi_i)}(1_U)=\De^{\vpi_i}(f_i)=0$.
Similarly, $\De_{s_i(\vpi_i),\vpi_i}(1_U)=0$ and $\De_{\vpi_i,\vpi_i}(1_U)=1$.
Finally,
\[
\De_{s_i(\vpi_i),s_i(\vpi_i)}(1_U)=\De^{\vpi_i}(e_i\, f_i)=
{\De^{\vpi_i}(f_i\,1_U e_i)}+\De^{\vpi_i}\left(\frac{q^{h_i}-q^{-h_i}}{q-q^{-1}}\right)=0+1.
\]
Thus $\psi_1(1_U)=1$.
\medskip
{\bf Property (2).\ } Recall that $\De_{\ga,\de}\in A_q(\g)_{\ga,\de}$. Since
$A_q(\g)$ is $P\times P$-graded it follows from~\eqref{eq:gamma_i} that
both summands of $\psi_1$ belong to $A_q(\g)_{\ga_i,\ga_i}$.
\medskip
{\bf Property (3).\ } Since $\De(e_j)=e_j\otimes 1 +q^{h_j}\otimes e_j$ the
operator $e_j\cdot -$ acts on $A_q(\g)$ as a ``graded $q$-derivation'', \ie
for $\psi\in A_q(\g)_{\ga,\de}$ and $\psi'\in A_q(\g)_{\ga',\de'}$ we have
\begin{equation}
e_j\cdot (\psi\,\psi')= (e_j\cdot\psi)\,\psi'+
q^{\< h_j,\de\>}\psi\,(e_j\cdot\psi').
\end{equation}
Similarly,
\begin{equation}
(\psi\,\psi')\cdot f_j= \psi\ (\psi'\cdot f_j)+
q^{\< -h_j,\ga'\>}(\psi\cdot f_j)\,\psi'.
\end{equation}
Now, for $j\neq i$ we have $\< h_j,\vpi_i\>=0$ and
$\< h_j,s_i(\vpi_i)\>=\< h_j,\vpi_i -\a_i\>=-a_{ij}\geq 0$. Thus,
by~\eqref{eq:leftz},
the left multiplication by $e_j$ annihilates
all the minors appearing in
$\psi_1$, and as a consequence $e_j\cdot\psi_1=0$.
It remains to show that $e_i\cdot\psi_1=0$. To this end we observe first
that $e_i\cdot\De_{u(\vpi_i),s_i(\vpi_i)}=\De_{u(\vpi_i),\vpi_i}$ for any $u\in W$.
In fact,
\begin{multline*}
e_i\cdot\De_{u(\vpi_i),s_i(\vpi_i)}=(e_i\,f_i)\cdot\De_{u(\vpi_i),\vpi_i}=
{(f_i\, e_i)\cdot\De_{u(\vpi_i),\vpi_i}}\ +\
\frac{q^{h_i}-q^{-h_i}}{q-q^{-1}}\cdot\De_{u(\vpi_i),\vpi_i}\\
=0+\frac{q^{\< h_i,\vpi_i\>}-q^{\< -h_i,\vpi_i\>}}{q-q^{-1}}\De_{u(\vpi_i),\vpi_i}
=\De_{u(\vpi_i),\vpi_i}.
\end{multline*}
So, we can now calculate
\begin{multline} \label{eq:eikillpsi}
e_i\cdot\psi_1=
(e_i\cdot\De_{s_i(\vpi_i),s_i(\vpi_i)})\,\De_{\vpi_i,\vpi_i}-
q^{-1}(e_i\cdot\De_{s_i(\vpi_i),\vpi_i})\,\De_{\vpi_i,s_i(\vpi_i)}\\
+q^{\<h_i,s_i(\vpi)\>} \De_{s_i(\vpi_i),s_i(\vpi_i)})\,(e_i\cdot\De_{\vpi_i,\vpi_i})-
q^{\<h_i,\vpi_i\>-1}\De_{s_i(\vpi_i),\vpi_i}\,(e_i\cdot\De_{\vpi_i,s_i(\vpi_i)})\\
=\De_{s_i(\vpi_i),\vpi_i}\De_{\vpi_i,\vpi_i}-
q^0\De_{s_i(\vpi_i),\vpi_i}\De_{\vpi_i,\vpi_i}=0.
\end{multline}
Finally, we have to show that $\psi_i\cdot f_j=0$ for all $j\in I$.
Again, for $j\neq i$ we see by~\eqref{eq:rightz} that the map $x\mapsto x\cdot f_j$ annihilates
all the minors occuring in $\psi_1$. In order to see that
$\psi_1\cdot f_i=0$, we note that by Lemma~\ref{commute},
we have
$\De_{s_i(\vpi_i),\vpi_i}\,\De_{\vpi_i,s_i(\vpi_i)}=
\De_{\vpi_i,s_i(\vpi_i)}\,\De_{s_i(\vpi_i),\vpi_i}$.
Then we can proceed as in
\eqref{eq:eikillpsi}.
Proposition~\ref{qmin-id1} is proved.
\section{The quantum coordinate rings $A_q(\bb)$ and $A_q(\nn)$}
\subsection{The quantum coordinate ring $A_q(\bb)$}
Let $U_q(\bb)$ be the subalgebra of $U_q(\g)$ generated by $e_i\ (i\in I)$ and
$q^h\ (h\in P^*)$. We have
\begin{equation}
\De(U_q(\bb)) \subset U_q(\bb)\otimes U_q(\bb), \qquad
S(U_q(\bb)) \subseteq U_q(\bb),
\end{equation}
hence $U_q(\bb)$ is a Hopf subalgebra.
Therefore, as in (\ref{multA}), $U_q(\bb)^*$ has a multiplication
dual to the comultiplication of $U_q(\bb)$.
Clearly, the map $\rho\colon U_q(\g)^* \to U_q(\bb)^*$
given by restricting linear forms from $U_q(\g)$ to $U_q(\bb)$ is an
algebra homomorphism.
We define $A_q(\bb):= \rho(A_q(\g))$.
Let $Q_+ = \bigoplus_{i\in I} \N \a_i$.
For $\ga, \de \in P$, let $A_q(\bb)_{\ga,\de} = \rho(A_q(\g)_{\ga,\de})$.
Since $U_q(\bb)\subseteq \bigoplus_{\a \in Q_+} U_q(\g)_\a$, we have by Lemma~\ref{gradings},
\begin{equation}
A_q(\bb) = \bigoplus_{\ga-\de\in Q_+} A_q(\bb)_{\ga,\de}.
\end{equation}
\subsection{The quantum coordinate ring $A_q(\nn)$}\label{multAqn}
Recall that $U_q(\nn)$ is the subalgebra of $U_q(\g)$ generated by $e_i\ (i\in I)$.
Because of (\ref{comult}), this is \emph{not} a Hopf subalgebra. Nevertheless, we can endow $U_q(\nn)^*$
with a multiplication as follows. Recall that every $y\in U_q(\bb)$ can be written
as a $\Q(q)$-linear combination of elements of the form
$x\,q^h$ with $x\in U_q(\nn)$ and $h\in P^*$.
Given $\psi\in U_q(\nn)^*$, we define the linear form $\tpsi\in U_q(\bb)^*$ by
\begin{equation}\label{tilde}
\tpsi(x\,q^h) = \psi(x),\qquad (x\in U_q(\nn),\ h\in P^*).
\end{equation}
Clearly, $\iota\colon \psi \mapsto \tpsi$ is an injective linear map.
Moreover, since $\De(x\,q^h) = \sum x_{(1)}q^h\otimes x_{(2)}q^h$, it follows
immediately from (\ref{multA}) that
\begin{equation}
\left(\tpsi\cdot\tvphi\right)(x\,q^h) = \left(\tpsi\cdot\tvphi\right)(x),\qquad
(\psi, \vphi\in U_q(\nn)^*,\ x\in U_q(\nn),\ h\in P^*).
\end{equation}
Therefore $\iota(U_q(\nn)^*)$ is a subalgebra of $U_q(\bb)^*$, and we can define
\begin{equation}\label{defmult}
\psi\cdot\vphi = \iota^{-1}(\tpsi\cdot\tvphi).
\end{equation}
We have $U_q(\nn) = \bigoplus_{\a\in Q_+} U_q(\nn)_{\a}$, where
$U_q(\nn)_{\a} = U_q(\nn) \cap U_q(\g)_{\a}$ is finite-dimensional for every $\a\in Q_+$.
Let
\begin{equation}
A_q(\nn) = \bigoplus_{\a\in Q_+} \Hom_{\Q(q)}(U_q(\nn)_{\a},\Q(q))
= \bigoplus_{\a\in Q_+} A_q(\nn)_\a \subset U_q(\nn)^*
\end{equation}
denote the graded dual of $U_q(\nn)$.
It is easy to see that $A_q(\nn)$ is a subalgebra of $U_q(\nn)^*$ for the multiplication
defined in (\ref{defmult}). Moreover, $\iota(A_q(\nn)) \subset A_q(\bb)$, and more precisely
(\ref{tilde}) shows that
\begin{equation}
\iota(A_q(\nn)_\a) = A_q(\bb)_{\a,0},\quad (\a\in Q_+).
\end{equation}
To summarize, $A_q(\nn)$ can be identified with
the subalgebra $\iota(A_q(\nn))=\bigoplus_{\a\in Q_+} A_q(\bb)_{\a,0}$ of $A_q(\bb)$.
\subsection{The algebra isomorphism between $A_q(\nn)$ and $U_q(\nn)$}\label{isomAU}
For $i\in I$, let $\de_i \in \End_{\Q(q)}(U_q(\nn))$ be the $q$-derivation defined by
$\de_i(e_j)=\de_{ij}$ and
\begin{equation}\label{qderiv}
\de_i(xy)= \de_i(x)\,y + q^{\< h_i,\a\>}x\,\de_i(y),
\qquad
(x \in U_q(\nn)_\a,\ y\in U_q(\nn)).
\end{equation}
It is well known that there exists a unique nondegenerate symmetric
bilinear form on $U_q(\nn)$ such that $(1,1)=1$ and
\begin{equation}
(\de_i(x),\,y) = (x,\, e_iy),\qquad (x \in U_q(\nn),\ y\in U_q(\nn),\ i\in I).
\end{equation}
Denote by $\psi_x$ the linear form on $U_q(\nn)$ given by
$\psi_x(y) = (x,\,y)$.
Then, the map $\Psi\colon x\mapsto \psi_x$ is an isomorphism of
graded vector spaces from $U_q(\nn)$ to $A_q(\nn)$.
\begin{Prop}\label{isomPhi}
Let $A_q(\nn)$ be endowed with the multiplication (\ref{defmult}).
Then $\Psi$ is an isomorphism of algebras from $U_q(\nn)$ to $A_q(\nn)$.
\end{Prop}
\proof
We need to show that
\begin{equation}
\tpsi_{xy}(zq^h) = \left(\tpsi_x\cdot\tpsi_y\right)(zq^h),
\quad
(x,y,z\in U_q(\nn),\ h\in P^*).
\end{equation}
By linearity, we can assume that $z=e_{i_1}\cdots e_{i_m}$ for some $i_1,\ldots,i_m\in I$.
By definition,
\begin{equation}\label{calcul1}
\tpsi_{xy}(zq^h) = (xy,\,z) = (\de_{i_m}\cdots \de_{i_1}(xy),\,1).
\end{equation}
Let us assume, without loss of generality, that $x\in U_q(\nn)_\a$.
It follows from (\ref{qderiv}) that
\begin{equation}\label{prodder}
\de_{i_m}\cdots \de_{i_1}(xy) = \sum_{K} q^{\si(J,K)} \de_{i_{j_s}}\cdots \de_{i_{j_1}}(x)\,
\de_{i_{k_r}}\cdots \de_{i_{k_1}}(y)
\end{equation}
where the sum is over all subsets $K=\{k_1 < \cdots < k_r\}$ of $[1,m]$,
$J= \{j_1<\cdots < j_s\}$ is the complement of $K$ in $[1,m]$, and
\[
\si(J,K) = \left\<h_{k_1},\,\a - \sum_{j\in J,\, j<k_1}\a_j\right\>
+ \cdots +
\left\<h_{k_r},\,\a - \sum_{j\in J,\, j<k_r}\a_j\right\>.
\]
Moreover, a summand of the r.h.s. of (\ref{prodder}) can give a nonzero contribution
to (\ref{calcul1}) only if $\a = \sum_{j\in J} \a_j$. In this case we have
\begin{equation}\label{AJK}
\si(J,K) = \left\<h_{k_1},\,\sum_{j\in J,\, j>k_1}\a_j\right\>
+ \cdots +
\left\<h_{k_r},\,\sum_{j\in J,\, j>k_r}\a_j\right\>,
\end{equation}
and
\begin{equation}
(xy,\,z) = \sum_{K} q^{\si(J,K)} (e_{i_{j_1}}\cdots e_{i_{j_s}},\,x)\,
(e_{i_{k_1}}\cdots e_{i_{k_r}},\,y).
\end{equation}
On the other hand, it is easy to deduce from (\ref{comult}) that
\[
\De(z) = \sum_{K} q^{\si(J,K)} e_{i_{j_1}}\cdots e_{i_{j_s}}t_{i_{k_1}}\cdots t_{i_{k_r}}
\otimes
e_{i_{k_1}}\cdots e_{i_{k_r}}
\]
where $\si(J,K)$ is again given by (\ref{AJK}).
It then follows from the definition of $\tpsi_x$ and $\tpsi_y$ that
\[
\left(\tpsi_x\cdot\tpsi_y\right)(zq^h) = \left(\tpsi_x\cdot\tpsi_y\right)(z)
=
\sum_{K} q^{\si(J,K)} (e_{i_{j_1}}\cdots e_{i_{j_s}},\,x)\,
(e_{i_{k_1}}\cdots e_{i_{k_r}},\,y) = \tpsi_{xy}(zq^h).
\]
\cqfd
\section{Determinantal identities for unipotent quantum minors}
\subsection{Principal quantum minors}
Let $\la\in P_+$, and $u,v\in W$.
The quantum minor $\De_{u(\la),v(\la)}$ is called \emph{principal} when $u(\la)=v(\la)$.
In this case we have by Lemma~\ref{gradings}~(c) that $\De_{v(\la),v(\la)}(x) = 0$
if $x\not\in U_q(\g)_0$.
Therefore the restriction $\rho(\De_{v(\la),v(\la)}) \in A_q(\bb)$ is given by
\begin{equation}
\rho(\De_{v(\la),v(\la)})(xq^h) = \eps(x)q^{\<h,v(\la)\>},\qquad (x\in U_q(\n),\ h\in P^*).
\end{equation}
Define
\begin{equation}
\De_{v(\la),v(\la)}^* := \De_{v(\la),v(\la)}\circ S \in U_q(\g)^*,
\end{equation}
where $S$ is the antipode of $U_q(\g)$.
\begin{Lem}\label{qcentral}
\begin{itemize}
\item[(a)] The principal quantum minors $\rho(\De_{v(\la),\,v(\la)})\ (v\in W,\, \la\in P_+)$
are invertible in $A_q(\bb)$, with inverse equal to $\rho(\De_{v(\la),v(\la)}^*)$.
\item[(b)] The principal quantum minors $\rho(\De_{v(\la),\,v(\la)})$
are $q$-central in $A_q(\bb)$. More precisely, for $\psi\in A_q(\bb)_{\ga,\de}$ we have
\[
\rho(\De_{v(\la),\,v(\la)})\cdot \psi = q^{(v(\la),\ga-\de)} \psi \cdot \rho(\De_{v(\la),\,v(\la)}).
\]
\end{itemize}
\end{Lem}
\proof (a) First, we note that $\rho(\De_{v(\la),v(\la)}^*)$ belongs to $A_q(\bb)$.
Indeed, let $U_q(\bb_-)$ be the subalgebra of $U_q(\g)$ generated by $f_i\ (i\in I)$ and
$q^h\ (h\in P^*)$, and $U_q^0$ the subalgebra generated by
$q^h\ (h\in P^*)$.
Since $S$ is an anti-automorphism of $U_q(\g)$ which stabilizes
$U_q(\bb)$, $U_q(\bb_-)$, and $U_q^0$, it is clear that $\De_{v(\la),v(\la)}^*$
generates an integrable left (\resp right) submodule of $U_q(\g)^*$.
So $\De_{v(\la),v(\la)}^*\in A_q(\g)$, and $\rho(\De_{v(\la),v(\la)}^*)\in A_q(\bb)$.
Now, it follows easily from the definition that
\[
\De_{v(\la),v(\la)}^*(xq^h) = \eps(x) q^{-\<h,v(\la)\>}, \qquad (x\in U_q(\n),\ h\in P^*),
\]
so that
\[
\left(\De_{v(\la),\,v(\la)}\cdot \De_{v(\la),v(\la)}^*\right)(xq^h)
=
\sum \De_{v(\la),\,v(\la)}(x_{(1)}q^h) \De_{v(\la),v(\la)}^*(x_{(2)}q^h)
=
\eps(xq^h),
\]
which proves (a).
(b) As for (a), it is enough to evaluate each side of the equation at a typical element $xq^h$ of $U_q(\bb)$.
Since the equation relates two elements of $A_q(\bb)_{\ga+v(\la),\de+v(\la)}$, we may assume that
$x\in U_{\ga - \de}$. By linearity, we may further assume that
$x=e_{i_1}\cdots e_{i_k}$, where $\a_{i_1}+\cdots +\a_{i_k} = \ga-\de$, without loss of generality.
Using (\ref{comult}), we have
\[
\left(\De_{v(\la),\,v(\la)}\cdot \psi\right)(xq^h) = \sum \De_{v(\la),\,v(\la)}(x_{(1)}q^h) \psi(x_{(2)}q^h)
= \De_{v(\la),\,v(\la)}(t_{i_1}\cdots t_{i_k}q^h) \psi(xq^h),
\]
because $\De_{v(\la),\,v(\la)}(x_{(1)}q^h) \not = 0$ only if $x_{(1)} \in U_q(\g)_0$.
Hence
\[
\left(\De_{v(\la),\,v(\la)}\cdot \psi\right)(xq^h) = q^{(v(\la),\,\ga-\de)}\De_{v(\la),\,v(\la)}(q^h) \psi(xq^h).
\]
On the other hand it also follows from (\ref{comult}) that
\[
\left(\psi\cdot\De_{v(\la),\,v(\la)} \right)(xq^h) = \sum \psi(x_{(1)}q^h)\De_{v(\la),\,v(\la)}(x_{(2)}q^h)
= \psi(xq^h) \De_{v(\la),\,v(\la)}(q^h),
\]
hence the result.
\cqfd
\subsection{Unipotent quantum minors}
The quantum minor $\De_{u(\la),\,v(\la)}$
belongs to $A_q(\g)_{u(\la),v(\la)}$,
hence its restriction $\rho(\De_{u(\la),\,v(\la)})$ belongs to
$A_q(\bb)_{u(\la),\,v(\la)}$. Therefore, if $v(\la)\not = 0$, it does not belong to $\iota(A_q(\nn))$.
But a slight modification does, as we shall now see.
We define the \emph{unipotent quantum minor} $D_{u(\la),\,v(\la)}$ by
\begin{equation}
D_{u(\la),\,v(\la)} = \rho\left(\De_{u(\la),\,v(\la)}\cdot \De_{v(\la),\,v(\la)}^*\right).
\end{equation}
This is an element of $A_q(\bb)_{u(\la)-v(\la),\,0} \subset \iota(U_q(\nn)^*)$.
In fact, the same calculation as in the proof of Lemma~\ref{qcentral} shows that
for $x\in U_q(\nn)$ and $h\in P^*$, we have
\begin{equation}
D_{u(\la),\,v(\la)}(xq^h) = \De_{u(\la),\,v(\la)}(xq^h)\, \De_{v(\la),\,v(\la)}(q^{-h})
= \De_{u(\la),\,v(\la)}(x) = D_{u(\la),\,v(\la)}(x).
\end{equation}
Thus, we can regard $D_{u(\la),\,v(\la)}$ as the restriction to $U_q(\nn)$ of the quantum
minor $\De_{u(\la),\,v(\la)}$.
But we should be aware that this restriction is \emph{not} an algebra homomorphism.
For example, by Lemma~\ref{rem:MultQMin} we have
\[
\De_{u(\la),v(\la)}\cdot\De_{u(\mu),v(\mu)}= \De_{u(\mu),v(\mu)}\cdot\De_{u(\la),v(\la)} ,\quad
(u,v\in W,\ \la,\mu\in P_+).
\]
The corresponding commutation relation for unipotent quantum minors
is given by the following
\begin{Lem}\label{qcommute2}
For $u,v\in W$ and $\la,\mu\in P_+$ we have
\[
D_{u(\la),v(\la)}\cdot D_{u(\mu),v(\mu)}= q^{(v(\mu),\,u(\la))-(v(\la),\,u(\mu))}
D_{u(\mu),v(\mu)}\cdot D_{u(\la),v(\la)}.
\]
\end{Lem}
\proof Let us write for short $\rho(\De_{u(\la),\,v(\la)}) = \De_{u(\la),\,v(\la)}$
and $\rho(\De_{v(\la),\,v(\la)}^*) = \De^*_{v(\la),\,v(\la)}$.
We have, by Lemma~\ref{qcentral},
\begin{eqnarray*}
D_{u(\la),v(\la)} D_{u(\mu),v(\mu)}&=& \De_{u(\la),\,v(\la)}\De^*_{v(\la),\,v(\la)}
\De_{u(\mu),\,v(\mu)}\De^*_{v(\mu),\,v(\mu)}\\
&=&
q^{(v(\la),\,v(\mu)-u(\mu))}
\De_{u(\la),\,v(\la)}\De_{u(\mu),\,v(\mu)}
\De^*_{v(\la),\,v(\la)}\De^*_{v(\mu),\,v(\mu)}\\
&=&
q^{(v(\la),\,v(\mu)-u(\mu))}
\De_{u(\mu),\,v(\mu)}\De_{u(\la),\,v(\la)}
\De^*_{v(\mu),\,v(\mu)}\De^*_{v(\la),\,v(\la)}\\
&=&
q^{(v(\mu),\,u(\la))-(v(\la),\,u(\mu))}
\De_{u(\mu),\,v(\mu)}\De^*_{v(\mu),\,v(\mu)}
\De_{u(\la),\,v(\la)}\De^*_{v(\la),\,v(\la)}\\
&=& q^{(v(\mu),\,u(\la))-(v(\la),\,u(\mu))} D_{u(\mu),v(\mu)} D_{u(\la),v(\la)}
\end{eqnarray*}
\cqfd
Similarly, Lemma~\ref{commute} implies:
\begin{Lem}\label{qcommute}
Suppose that for $u,v\in W$ and $i\in I$ we have $l(us_i)=l(u)+1$ and
$l(vs_i)=l(v)+1$. Then
\[
D_{us_i(\vpi_i),\,v(\vpi_i)}\,D_{u(\vpi_i),\,vs_i(\vpi_i)}
=q^{(vs_i(\vpi_i),\,us_i(\vpi_i))-(v(\vpi_i),\,u(\vpi_i))}
D_{u(\vpi_i),\,vs_i(\vpi_i)}\,D_{us_i(\vpi_i),\,v(\vpi_i)}.
\]
\cqfd
\end{Lem}
We can also regard the quantum unipotent minors
as linear forms on $U_q(\nn)$ given by matrix coefficients of integrable
representations.
Indeed, using (\ref{matrixCoeff}), we have
\begin{equation}
D_{u(\la),v(\la)}(x) = (m_{u(\la)},\,xm_{v(\la)})_\la,\qquad (x\in U_q(\nn)).
\end{equation}
In particular, when $u=e$ we get the same quantum flag minors
\begin{equation}\label{matrix_coeff_flag}
D_{\la,v(\la)}(x) = (m_{\la},\,xm_{v(\la)})_\la,\qquad (x\in U_q(\nn)).
\end{equation}
as in \cite[\S6.1]{Ki} (up to a switch from $U_q(\n_-)$ to $U_q(\n)$).
\subsection{A family of identities for unipotent quantum minors}
We are now in a position to deduce from Proposition~\ref{qmin-id1} an algebraic
identity satisfied by unipotent quantum minors.
Later on, we will see that particular cases of this identity can be seen as
quantum exchange relations in certain quantum cluster algebras.
As in \S\ref{pf:qmin-id1}, let us write $\ga_i = \vpi_i+s_i(\vpi_i)$, so that
\begin{equation}\label{factorqminor}
\De_{u(\ga_i),v(\ga_i)} = \prod_{j\neq i}\De_{u(\vpi_j),v(\vpi_j)}^{-a_{ji}},
\end{equation}
where, by Lemma~\ref{rem:MultQMin}, the order of the factors in the
right-hand side is irrelevant.
\begin{Prop}\label{unip_minor_identity}
Suppose that for $u,v\in W$ and $i\in I$ we have $l(us_i)=l(u)+1$ and
$l(vs_i)=l(v)+1$. Then
\[
q^{A}\,
D_{us_i(\vpi_i),\,vs_i(\vpi_i)}\,D_{u(\vpi_i),\,v(\vpi_i)}=
q^{-1+B}\,
D_{us_i(\vpi_i),\,v(\vpi_i)}\,D_{u(\vpi_i),\,vs_i(\vpi_i)}
+
D_{u(\ga_i),v(\ga_i)}
\]
holds in $A_q(\nn)$, where
\[
A = (vs_i(\vpi_i),\,u(\vpi_i)-v(\vpi_i)),\qquad
B = (v(\vpi_i),\,u(\vpi_i)-vs_i(\vpi_i)).
\]
\end{Prop}
\proof
Again let us write for short $\rho(\De_{u(\la),\,v(\la)}) = \De_{u(\la),\,v(\la)}$
and $\rho(\De^*_{v(\la),\,v(\la)}) = \De^*_{v(\la),\,v(\la)}$.
We apply the restriction homomorphism $\rho$ to the equality of Proposition~\ref{qmin-id1},
and we multiply both sides from the right
by
\[
\De^*_{v(\vpi_i),\,v(\vpi_i)}\De^*_{v(s_i(\vpi_i)),\,v(s_i(\vpi_i))}
= \De^*_{v(\ga_i),v(\ga_i)}.
\]
Note that all these minors commute by Lemma~\ref{rem:MultQMin}.
The result then follows directly from the definition of unipotent quantum minors, and from
Lemma~\ref{qcentral}~(b).
\cqfd
It is sometimes useful to write the second summand of the right-hand side of Proposition~\ref{unip_minor_identity}
as a product. It is straightforward to deduce from (\ref{factorqminor}) and Lemma~\ref{qcentral}~(b)
that we have
\begin{equation}\label{prodD}
D_{u(\ga_i),v(\ga_i)}
=
q^C\,
\prod_{j\not = i}^{\longrightarrow} \left(D_{u(\vpi_j),\,v(\vpi_j)}
\right)^{-a_{ij}},
\end{equation}
where
\begin{equation}\label{eqC}
C =
\sum\limits_{\substack{j<k\\ j\not= i\not = k}}
a_{ij}a_{ik}(v(\vpi_j),\,u(\vpi_k)-v(\vpi_k))
+ \sum_{j\not = i} \left(\begin{matrix}-a_{ij} \cr 2\end{matrix}\right) (v(\vpi_j),\,u(\vpi_j)-v(\vpi_j)).
\end{equation}
\subsection{A quantum $T$-system}\label{ssectTsystem}
Let $\ii = (i_1,\ldots,i_r)\in I^r$ be such that $l(s_{i_1}\cdots s_{i_r})=r$.
We will now deduce from Proposition~\ref{unip_minor_identity} a system of identities
relating the unipotent quantum minors
\begin{equation}
D(k,l;j) := D_{s_{i_1}\cdots s_{i_k}(\vpi_j),\ s_{i_1}\cdots s_{i_l}(\vpi_j)},
\qquad
(0\le k < l \le r,\ j\in I).
\end{equation}
Here, we use the convention that $D(0,l;j)= D_{\vpi_j,\ s_{i_1}\cdots s_{i_l}(\vpi_j)}$,
a \emph{quantum flag minor}.
This system can be viewed as a $q$-analogue of a $T$-system, (see \cite{KNS}).
It will allow us to express every quantum minor $D(k,l;j)$ in terms of the
flag minors $D(0,m;i)$.
Note that, because of (\ref{sivpi}), every quantum minor $D(k,l;j)$ is equal to a minor of
the form $D(b,d;j)$ where $i_b=i_d=j$.
When this is the case, we can simply write $D(b,d;j) = D(b,d)$.
Note that in particular, $D(b,b)=1$ for every $b$.
By convention, we write $D(0,b) = D_{\vpi_{i_b},\ s_{i_1}\cdots s_{i_b}(\vpi_{i_b})}$.
We will also use the following shorthand notation:
\begin{eqnarray}
b^-(j) &:=& \max\left(\{s < b \mid i_s = j\}\cup\{0\}\right),\\
b^- &:=& \max\left(\{s < b \mid i_s = i_b\}\cup\{0\}\right),\\
\mu(b,j) &:=& s_{i_1}\cdots s_{i_b}(\vpi_j).\label{equamu}
\end{eqnarray}
In (\ref{equamu}) we understand that $\mu(0,j) = \vpi_j$.
Clearly, we have $D(b,d;j) = D(b^-(j),d^-(j))$.
\begin{Prop}\label{Tsystem}
Let $1\le b<d\le r$ be such that $i_b = i_d = i$.
There holds
\begin{equation}\label{eqTsystem}
q^A\, D(b,d) D(b^-,d^-) =
q^{-1+B}\, D(b,d^-)D(b^-,d)
\ +\ q^C
\prod_{j\not = i}^{\longrightarrow}D(b^-(j),d^-(j))^{-a_{ij}}
\end{equation}
where
\[
A = (\mu(d,i),\,\mu(b^-,i)-\mu(d^-,i)),\qquad
B = (\mu(d^-,i),\,\mu(b^-,i)-\mu(d,i)),
\]
and
\[
C
=
\sum\limits_{\substack{j<k\\ j\not= i\not = k}}
a_{ij}a_{ik}\left(\mu(d,j),\,\mu(b,k)-\mu(d,k)\right)
+\ \sum_{j\not = i} \left(\begin{matrix}-a_{ij} \cr 2\end{matrix}\right)
\left(\mu(d,j),\ \mu(b,j)-\mu(d,j)\right).
\]
\end{Prop}
\proof
This follows directly from Proposition~\ref{unip_minor_identity}, (\ref{prodD}), and (\ref{eqC}), by taking
\[
u = s_{i_1}\cdots s_{i_{b-1}},\qquad
v = s_{i_1}\cdots s_{i_{d-1}},\qquad
i=i_b=i_d.
\]
\cqfd
\section{Canonical bases}
\subsection{The canonical basis of $U_q(\nn)$}\label{def_can}
We briefly review Lusztig's definition of a canonical basis of $U_q(\nn)$.
Recall the scalar product $(\cdot,\cdot)$ on $U_q(\n)$ defined in \S\ref{isomAU}.
In \cite[Chapter 1]{Lu}, Lusztig defines a similar scalar product $(\cdot,\cdot)_L$,
using the same $q$-derivation $\de_i$ (denoted by ${}_ir$ in \cite[1.2.13]{Lu}) but with
a different normalization $(e_i,e_i)_L = (1-q^{-2})^{-1}$.
It it easy to see that $(x,y)=0$ if and only if $(x,y)_L=0$, and if $x,y\in U_q(\n)_\b$ then
\begin{equation}\label{scalarL}
(x,y)_L = (1-q^{-2})^{-\deg\b} (x,y),
\end{equation}
where, for $\b = \sum_i c_i\a_i$, we set $\deg\b = \sum_ic_i$. This slight difference
will not affect the definition of the canonical basis below,
and we will always use $(\cdot,\cdot)$ instead of $(\cdot,\cdot)_L$.
Let $\A = \Q[q,q^{-1}]$.
We introduce the $\A$-subalgebra $U_\A(\n)$ of $U_q(\n)$ generated by the divided powers
$e_i^{(k)}\ (i\in I,\ k\in \N)$. We define a ring automorphism $x \mapsto \overline{x}$ of
$U_q(\g)$ by
\begin{equation}
\overline{q} = q^{-1},\quad \overline{e_i} = e_i,\quad \overline{f_i}=f_i, \qquad (i\in I).
\end{equation}
This restricts to a ring automorphism of $U_q(\n)$.
The canonical basis $\B$ is an $\A$-basis of $U_\A(\n)$ such that
\begin{equation}\label{bar}
\overline{b}=b,\qquad (b\in \B).
\end{equation}
Moreover, for every $b, b'\in \B$ the scalar product $(b,b')\in \Q(q)$
has no pole at $q=\infty$, and
\begin{equation}\label{almost_ortho}
(b,\, b')|_{q=\infty} = \delta_{b,b'}.
\end{equation}
By this we mean that
\[
(b,b') = \frac{a_jq^j+\cdots a_1q+a_0}{a'_kq^k+\cdots+ a'_1q + a'_0},
\qquad (a_i,a'_i \in \Z,\ a_j\not = 0,\ a'_k\not = 0),
\]
with $j<k$ when $b\not = b'$, and $j=k, a_j=a'_k$ when $b=b'$.
It is easy to see that if an $\A$-basis of
$U_\A(\n)$ satisfies (\ref{bar}) and (\ref{almost_ortho}),
then it is unique up to sign (see \cite[14.2]{Lu}).
The existence of $\B$ is proved in \cite[Part 2]{Lu}, and a consistent choice of signs
is provided.
Of course, $\B$ is also a $\Q(q)$-basis of~$U_q(\n)$.
\subsection{The dual canonical basis of $U_q(\nn)$}\label{dualcan}
Let $\B^*$ be the basis of $U_q(\n)$ adjoint to $\B$ with respect to the
scalar product $(\cdot,\,\cdot)$. We call it the \emph{dual canonical basis}
of $U_q(\n)$, since
it can be identified via $\Psi$ with the dual basis of $\B$ in $A_q(\n)$.
Note that $\B^*$ is not invariant under the bar automorphism $x\mapsto \overline{x}$.
The property of $\B^*$ dual to (\ref{bar}) can be stated as follows.
Let $\si$ be the composition of the anti-automorphism $*$ and the
bar involution, that is, $\si$ is the ring \emph{anti}-automorphism
of $U_q(\nn)$ such that
\begin{equation}
\si(q) = q^{-1},\qquad \si(e_i) = e_i.
\end{equation}
For $\b \in Q_+$, define
\begin{equation}\label{defN(b)}
N(\b) := \frac{(\b,\,\b)}{2} - \deg\b.
\end{equation}
Then, if $b \in U_q(\nn)_\b$
belongs to $\B^*$, there holds
\begin{equation}\label{eq:sigma_b}
\si(b) = q^{N(\b)}\, b,
\end{equation}
(see \cite{Re,Ki}).
\subsection{Specialization at $q=1$ of $U_q(\n)$ and $A_q(\n)$}\label{specq1}
Recall that $U_\A(\n)$ is the $\A$-submodule of $U_q(\n)$ spanned by
the canonical basis $\B$.
If we regard $\C$ as an $\A$-module via the homomorphism $q \mapsto 1$,
we can define
\begin{equation}
U_1(\n) := \C \otimes_\A U_\A(\n).
\end{equation}
This is a $\C$-algebra isomorphic to the enveloping algebra $U(\n)$.
Similarly, let $A_\A(\n)$ be the $\A$-submodule of $A_q(\n)$ spanned by
the basis $\Psi(\B^*)$.
Define
\begin{equation}
A_1(\n) := \C \otimes_\A A_\A(\n).
\end{equation}
This is a $\C$-algebra isomorphic to the graded dual $U(\n)^*_{\rm gr}$.
This commutative ring can be identified with the coordinate ring $\C[N]$
of a pro-unipotent pro-group $N$ with Lie algebra the completion $\widehat{\n}$
of $\n$ (see \cite{GLS}).
\subsection{Global bases of $U_q(\n_-)$}
We shall also use Kashiwara's lower global basis $\B^{\rm low}$
of $U_q(\n_-)$, constructed in \cite{K1}. It was proved by Grojnowski and Lusztig that
$\vphi(\B) = \B^{\rm low}$, where $\vphi$ is the anti-automorphism
of (\ref{vphi}).
For $i\in I$, we introduce the $q$-derivations $e'_i$ and ${}_ie'$ of
$U_q(\n_-)$, defined by $e'_i(f_j)={}_ie'(f_j) = \de_{ij}$
and, for homogeneous elements $x, y \in U_q(\n_-)$,
\begin{eqnarray}\label{qderiv-}
e'_i(xy)&=& e'_i(x)\,y + q^{\< h_i,{\rm wt}(x)\>}x\,e'_i(y),\\[2mm]
{}_ie'(xy)&=& q^{\< h_i,{\rm wt}(y)\>}{}_ie'(x)\,y + x\,\,{}_ie'(y).
\end{eqnarray}
Note that ${}_ie' = *\circ e'_i\circ *$.
Let us denote by $(\cdot,\,\cdot)_K$ the Kashiwara scalar product
on $U_q(\n_-)$. It is the unique symmetric bilinear form such that
$(1,\, 1)_K = 1$, and
\begin{equation}
(f_i x,\,y)_K = (x,\,e'_i(y))_K,\qquad (x \in U_q(\nn_-),\ y\in U_q(\nn_-),\ i\in I).
\end{equation}
It also satisfies
\begin{equation}
(x f_i,\,y)_K = (x,\,{}_ie'(y))_K,\qquad (x \in U_q(\nn_-),\ y\in U_q(\nn_-),\ i\in I).
\end{equation}
Let $\bvphi$ be the composition of $\vphi$ and the bar involution, that is,
$\bvphi$ is the ring \emph{anti}-automorphism
of $U_q(\g)$ such that
\begin{equation}
\bvphi(q) = q^{-1},\quad \bvphi(e_i) = f_i,
\quad \bvphi(f_i) = e_i,
\quad \bvphi(q^h) = q^{-h}.
\end{equation}
The following lemma expresses the compatibility between the scalar products and
$q$-derivations on $U_q(\n)$ and $U_q(\n_-)$.
\begin{Lem}\label{lem+-}
\begin{itemize}
\item[(a)] For $i\in I$, we have ${}_ie'\circ \bvphi = \bvphi \circ \de_i$.
\item[(b)] For $x, y \in U_q(\n)$ we have $\overline{(x,\,y)} = (\bvphi(x),\,\bvphi(y))_K$.
\end{itemize}
\end{Lem}
\proof
As ${}_ie'\circ \bvphi$ and $\bvphi \circ \de_i$ are both linear, it is enough to prove that
${}_ie'\circ \bvphi(z) = \bvphi \circ \de_i(z)$ for
any homogeneous element $z$ of $U_q(\nn)$.
We prove this by induction on the degree of $z$. If the degree is
1, then this follows easily from the definition of all these maps.
Then assume that the degree
of $z$ is bigger than one. Then, without loss of generality, we can assume that
$z = xy$ with degrees of $x$ and $y$ smaller
than the degree of $z$. Now
\[
{}_ie'(\bvphi(xy))= {}_ie'(\bvphi(y)\bvphi(x))
= q^{-\< h_i,\b\>}{}_ie'(\bvphi(y))\,\bvphi(x) + \bvphi(y)\,\,{}_ie'(\bvphi(x)).
\]
By induction on the degrees of $x$ and $y$ we can assume that
${}_ie'(\vphi(x))=\bvphi(\de_i(x))$ and ${}_ie'(\bvphi(y))=\bvphi(\de_i(y))$,
so that
\[
{}_ie'(\bvphi(xy))=\bvphi\left(q^{\< h_i,\b\>}x\de_i(y) + \de_i(x)y)\right) = \bvphi(\de_i(xy)),
\]
which proves (a). Then
\[
(\bvphi(e_ix),\bvphi(y))_K = (\bvphi(x)f_i,\bvphi(y))_K=(\bvphi(x),{}_ie'(\bvphi(y)))_K
=(\bvphi(x),\bvphi(\de_i(y))_K.
\]
By induction on the degrees of $x$ and $y$ we can assume that
$(\bvphi(x),\bvphi(\de_i(y)))_K=\overline{(x,\de_i(y))}=\overline{(e_ix,y)}$, which proves (b).
\cqfd
Let $\B^{\rm up}$ denote the upper global basis of $U_q(\n_-)$.
This is the basis adjoint to $\B^{\rm low}$ with respect to $(\cdot,\,\cdot)_K$.
By Lemma~\ref{lem+-}(b), we also have $\B^{\rm up} = \bvphi(\B^*)$.
Let us denote by $\BB(\infty)$ the crystal of $U_q(\n_-)$, and by $b_\infty$ its
highest weight element.
The elements of $\B^{\rm low}$ (\resp $\B^{\rm up}$)
are denoted by $G^{\rm low}(b)\ (b\in \BB(\infty))$ (\resp $G^{\rm up}(b)$).
As usual, for every $i\in I$, one denotes by $\te_i$ and $\tf_i$ the Kashiwara
crystal operators of $\BB(\infty)$. We will need the following well known
property of the upper global basis, (see \cite[Lemma 5.1.1]{K}, \cite[Corollary 3.16]{Ki}):
\begin{Lem}\label{rmax}
Let $b\in \BB(\infty)$, and put $k=\max\{j\in\N\mid ({}_ie')^j(G^{\rm up}(b)) \not = 0\}$.
Then, denoting by $({}_ie')^{(k)}$ the $k$th $q$-divided power of ${}_ie'$, we have
$({}_ie')^{(k)}(G^{\rm up}(b))\in \B^{\rm up}$.
More precisely,
\[
({}_ie')^{(k)}(G^{\rm up}(b)) = G^{\rm up}((\te_i^*)^k b),
\]
where $\te_i^*$ is the crystal operator obtained from $\te_i$ by conjugating with the involution $*$
of (\ref{star}).
\end{Lem}
The integer $k=\max\{j\in\N\mid (e'_i)^j(G^{\rm up}(b)) \not = 0\} = \max\{j\in\N\mid (\te_i)^j(b) \not = 0\}$
is denoted by $\eps_i(b)$.
Similarly, the integer
$k=\max\{j\in\N\mid ({}_ie')^j(G^{\rm up}(b)) \not = 0\} = \max\{j\in\N\mid (\te_i^*)^j(b) \not = 0\}$
is denoted by $\eps_i^*(b)$.
\subsection{Unipotent quantum minors belong to $\B^*$}
Using the isomorphism $\Psi\colon U_q(\n) \to A_q(\n)$ of \ref{isomAU}, we can
regard the unipotent quantum minors $D_{u(\la),v(\la)}$ as elements of $U_q(n)$.
More precisely, let $d_{u(\la),v(\la)} = \Psi^{-1}(D_{u(\la),v(\la)})$ be the
element of $U_q(\n)$ such that
\begin{equation}
D_{u(\la),v(\la)}(x) = (d_{u(\la),v(\la)},\,x),\qquad (x\in U_q(\n)).
\end{equation}
By a slight abuse, we shall also call $d_{u(\la),v(\la)}$ a unipotent quantum minor.
In this section we show:
\begin{Prop}\label{qmincan}
For every $\la\in P_+$ and $u,v \in W$ such that $u(\la)-v(\la)\in Q_+$, the
unipotent quantum minor $d_{u(\la),v(\la)}$ belongs to $\B^*$.
More precisely, writing $u=s_{i_{l(u)}}\cdots s_{i_1}$, $v=s_{j_{l(v)}}\cdots s_{j_1}$
and defining $b_k$ and $c_l$ as in (\ref{eq-bc}), we have
\[
d_{u(\la),v(\la)} =
\bvphi\left(G^{\rm up}\left((\te_{i_{l(u)}}^*)^{b_{l(u)}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right)\right).
\]
\end{Prop}
\proof
We proceed in two steps, and first consider the case when $u=e$ is the unit of $W$,
that is, the case of unipotent quantum flag minors $d_{\la,v(\la)}$.
By (\ref{matrix_coeff_flag}), we have for $x\in U_q(\n)$,
\[
(d_{\la,v(\la)},\,x) = (m_\la,\,xm_{v(\la)})_\la = (\vphi(x)m_\la,\,m_{v(\la)})_\la.
\]
It is well known that the extremal weight vectors $m_{v(\la)}$ belong to Kashiwara's
lower global basis of $V(\la)$, and also to the upper global basis.
More precisely,
we have
\[
m_{v(\la)} = G^{\rm low}\left(\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\la\right)
= G^{\rm up}\left(\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\la\right),
\]
where $b_\la$ is the highest weight element of the crystal $\BB(\la)$ of $V(\la)$.
Hence we have
\begin{equation}\label{eq.5.14}
(d_{\la,v(\la)},\,x) =
\left(\vphi(x)m_\la,\, G^{\rm up}\left(\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\la\right)\right)_\la.
\end{equation}
It follows from (\ref{eq.5.14}) that $(d_{\la,v(\la)},\,x) = 1$ if
$\vphi(x) = G^{\rm low}\left(\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right)$,
and $(d_{\la,v(\la)},\,x) = 0$ if $\vphi(x)$ is any other element of $\B^{\rm low}$.
Therefore, $d_{\la,v(\la)}$ is the element of $\B^*$ given by
\[
d_{\la,v(\la)} = \bvphi\left(G^{\rm up}\left(\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right)\right).
\]
Now let us consider the general case when $u = s_{i_{l(u)}}\cdots s_{i_1}$ is non trivial.
For $k=1,\ldots, l(u)$, write $u_k = s_{i_{k}}\cdots s_{i_1}$.
Using Lemma~\ref{cons_def}, we have for $x\in U_q(\n)$,
\[
D_{u_k(\la),v(\la)}(x) = \De_{u_k(\la),v(\la)}(x) = \left(\De_{u_{k-1}(\la),v(\la)}\cdot e_{i_k}^{(b_k)}\right)(x)
\]
where $b_k=(\a_{i_k},u_{k-1}(\la)) = \max\{j \mid \De_{u_{k-1}(\la),v(\la)}\cdot e_{i_k}^{(j)} \not= 0\}$.
Now we can also write
\[
D_{u_k(\la),v(\la)}(x) = \left( d_{u_{k-1}(\la),v(\la)},\, e_{i_k}^{(b_k)}x\right) = \left( (\de_{i_k})^{(b_k)}d_{u_{k-1}(\la),v(\la)},\, x\right),
\]
hence
\[
d_{u_k(\la),v(\la)} = (\de_{i_k})^{(b_k)}d_{u_{k-1}(\la),v(\la)},
\]
where $(\de_i)^{(b)}$ means the $b$th $q$-divided power of the $q$-derivation $\de_i$.
Since $u_k(\lambda)-v(\la)\in Q_+$, the restriction $\rho(\De_{u_k(\la),v(\la)})$ is nonzero,
hence $b_k= \max\{j \mid (\de_{i_k})^{(b)}d_{u_{k-1}(\la),v(\la)}\not= 0\}$.
Applying $\bvphi$ and assuming by induction on $k$ that
\[
d_{u_{k-1}(\la),v(\la)} =
\bvphi\left(G^{\rm up}\left((\te_{i_{k-1}}^*)^{b_{k-1}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right)\right)
\]
we get by Lemma~\ref{lem+-}(a) that
\[
\bvphi(d_{u_k(\la),v(\la)}) =
({}_{i_{k}}e')^{(b_k)}\left(G^{\rm up}\left((\te_{i_{k-1}}^*)^{b_{k-1}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right)\right)
\]
and
\[
b_k= \eps_{i_k}^*\left((\te_{i_{k-1}}^*)^{b_{k-1}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty)\right).
\]
Thus, applying Lemma~\ref{rmax}, we get that
\[
\bvphi(d_{u_k(\la),v(\la)}) =
G^{\rm up}\left((\te_{i_{k}}^*)^{b_{k}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{c_{l(v)}}\cdots \tf_{j_1}^{c_1}b_\infty\right),
\]
and the statement follows by induction on $k$.
\cqfd
\section{Quantum unipotent subgroups}
In this section we provide a quantum version of the coordinate
ring $\C[N(w)]$ studied in \cite{GLS}, following \cite{Lu,S,Ki}.
\subsection{The quantum enveloping algebra $U_q(\n(w))$}
We fix $w\in W$, and we denote by $\De_w^+$ the subset of positive roots $\a$
of $\g$ such that $w(\a)$ is a negative root. This gives rise to a finite-dimensional
Lie subalgebra
\[
\n(w) := \bigoplus_{\a\in \De_w^+} \n_\a
\]
of $\n$, of dimension $l(w)$. The graded dual $U(\n(w))^*_{\rm gr}$ can be identified
with the coordinate ring $\C[N(w)]$ of a unipotent subgroup $N(w)$ of the Kac-Moody group
$G$ with ${\rm Lie}(N(w)) = \n(w)$. (For more details, see \cite{GLS}.)
In order to define a $q$-analogue of $U(\n(w))$, one introduces Lusztig's braid group
operation on $U_q(\g)$ \cite{Lu}. For $i\in I$, Lusztig has proved the existence
of a $\Q(q)$-algebra automorphism $T_i$ of $U_q(\g)$ satisfying
\begin{eqnarray}
T_i(q^h) &=& q^{s_i(h)},\\
T_i(e_i) &=& -t_i^{-1}f_i,\\
T_i(f_i) &=& -e_it_i,\\
T_i(e_j) &=& \sum_{r+s=-\<h_i,\a_j\>}(-1)^rq^{-r}e_i^{(r)}e_je_i^{(s)}\qquad (j\not=i),\\
T_i(f_j) &=& \sum_{r+s=-\<h_i,\a_j\>}(-1)^rq^{r}f_i^{(s)}f_jf_i^{(r)}\qquad (j\not=i).
\end{eqnarray}
(This automorphism is denoted by $T'_{i,-1}$ in \cite{Lu}.)
For a fixed reduced decomposition $w=s_{i_r}\cdots s_{i_1}$, let us set, as in
(\ref{defbeta}),
\begin{equation}\label{def_beta}
\b_k = s_{i_1}\cdots s_{i_{k-1}}(\a_{i_k}),\qquad (1\le k \le r).
\end{equation}
Then $\De_w^+ = \{\b_1,\ldots,\b_r\}$.
We define following Lusztig, the corresponding quantum root vectors:
\begin{equation}
E(\b_k) := T_{i_1}\cdots T_{i_{k-1}}(e_{i_k}),\qquad (1\le k \le r).
\end{equation}
It is known that $E(\b_k) \in U_q(\n)_{\b_k}$. For $\aa = (a_1,\ldots,a_r)\in \N^r$,
set
\begin{equation}
E(\aa) := E(\b_1)^{(a_1)}\cdots E(\b_r)^{(a_r)},
\end{equation}
where $E(\b_k)^{(a_k)}$ denotes the $a_k$th $q$-divided power of $E(\b_k)$.
Lusztig has shown that the subspace of $U_q(\n)$ spanned by
$\{E(\aa)\mid \aa\in\N^r\}$ is independent of the choice of the reduced word
$\ii=(i_r,\ldots,i_1)$ for~$w$. We denote it by $U_q(\n(w))$.
Moreover,
\begin{equation}
\P_\ii:=\{E(\aa)\mid \aa\in\N^r\}
\end{equation}
is a basis of $U_q(\n(w))$, which we call the \emph{PBW-basis} attached to $\ii$.
In fact, $U_q(\n(w))$ is even a subalgebra of $U_q(\n)$. This follows from a formula
due to Leven\-dor\-skii-Soibelman (see \cite[4.3.3]{Ki}).
\subsection{The quantum coordinate ring $A_q(\n(w))$}
Using the algebra isomorphism $\Psi\colon U_q(\n) \to A_q(\n)$ of \ref{isomAU},
we can define $A_q(\n(w)) := \Psi(U_q(\n(w))$. This is a subalgebra of $A_q(\n)$.
Lusztig \cite[38.2.3]{Lu} has shown that $\P_\ii$ is orthogonal, that is $(E(\aa),E(\bbf))=0$ if
$\aa \not = \bbf$. Moreover
\begin{equation}
(E(\b_k),E(\b_k))= (1-q^{-2})^{\deg\b_k-1},\qquad (1\le k\le r),
\end{equation}
and
\begin{equation}
(E(\aa),E(\aa)) = \prod_{k=1}^r \frac{(E(\b_k),E(\b_k))^{a_k}}{\{a_k\}!},
\end{equation}
where by definition
\begin{equation}
\{a\}! = \prod_{j=1}^a \frac{1-q^{-2j}}{1-q^{-2}}.
\end{equation}
Denote by $\P^*_\ii$ the basis
\begin{equation}
E^*(\aa) = \frac{1}{(E(\aa),E(\aa))}\ E(\aa),\qquad (\aa \in \N^r)
\end{equation}
of $U_q(\n(w))$ adjoint to $\P_\ii$.
We call $\P^*_\ii$ the \emph{dual PBW-basis} of $U_q(\n(w))$ since
it can be identified via $\Psi$ with the basis of $A_q(\n(w))$ dual to $\P_\ii$.
In particular we have the dual PBW generators:
\begin{equation}
E^*(\b_k) = (1-q^{-2})^{-\deg\b_k+1}E(\b_k),\qquad (1\le k\le r).
\end{equation}
\subsection{Action of $T_i$ on unipotent quantum minors}
\begin{Prop}\label{Tminor}
Let $\la \in P^+$, and $u, v \in W$ be such that $u(\la) - v(\la) \in Q_+$,
and consider the unipotent quantum minor $d_{u(\la),v(\la)}$.
Suppose that $l(s_iu) = l(u)+1$, and $l(s_iv)= l(v)+1$. Then
\[
T_i\left( d_{u(\la),v(\la)}\right)
= (1-q^{-2})^{(\a_i,v(\la)-u(\la))}\,d_{s_iu(\la),\,s_iv(\la)}.
\]
\end{Prop}
The proof will use Proposition~\ref{qmincan} and the following lemmas.
\begin{Lem}\label{Tphi}
We have $T_i \circ \bvphi = \bvphi\circ T_i$.
\end{Lem}
\proof This follows immediately from the definitions of $\bvphi$ and of $T_i$.
\cqfd
The next lemma is a restatement of a result of Kimura \cite[Theorem 4.20]{Ki},
based on previous results of Saito \cite{S} and Lusztig \cite{Lu2}.
Note that our $T_i$ is denoted by $T_i^{-1}$ in \cite{Ki}.
\begin{Lem}\label{lemKimura}
Let $b\in\BB(\infty)$ be such that $\eps_i(b) = 0$. Then
\[
T_i(G^{\rm up}(b)) =
(1-q^2)^{(\a_i,{\rm wt}(b))}\,
G^{\rm up}\left(\tf_i^{\ \vphi_i^*(b)}(\te^*_i)^{\eps_i^*(b)} b\right),
\]
where $\vphi_i^*(b) := \eps_i^*(b) + (\a_i, {\rm wt}(b))$.
\end{Lem}
\medskip\noindent {\it Proof of Proposition~\ref{Tminor} --- \ }
By Lemma~\ref{Tphi} and Proposition~\ref{qmincan}, we have
\[
\bvphi\left(T_i(d_{u(\la),v(\la)})\right) =
T_i\left(G^{\rm up}\left((\te_{i_{l(u)}}^*)^{b_{l(u)}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{\ c_{l(v)}}\cdots \tf_{j_1}^{\ c_1}b_\infty\right)\right).
\]
Let us write for short $b := (\te_{i_{l(u)}}^*)^{b_{l(u)}}\cdots (\te_{i_1}^*)^{b_1}\tf_{j_{l(v)}}^{\ c_{l(v)}}\cdots \tf_{j_1}^{\ c_1}b_\infty \in \BB(\infty)$.
Then
\[
\eps_i(b) = \max\{s \mid e_i^s\cdot \De_{u(\la),v(\la)} \not = 0 \}.
\]
The assumption $l(s_iv)=l(v)+1$ implies that $(\a_i, v(\la))\ge 0$, so
by (\ref{eq:leftz}), we have $\eps_i(b) = 0$, and we
are in a position to apply Lemma~\ref{lemKimura}.
Because of the assumption $l(s_iu)=l(u)+1$
we have
\[
\eps^*_i(b) = \max\{s \mid \De_{u(\la),v(\la)}\cdot e_i^s \not = 0 \} = (\a_i,u(\la)),
\]
and
\[
\vphi_i^*(b) = \eps^*_i(b) + (\a_i,{\rm wt}(b)) = (\a_i,u(\la)) + (\a_i,v(\la)-u(\la)) = (\a_i,v(\la)).
\]
Thus, again by Proposition~\ref{qmincan}, we have
\begin{align*}
T_i(d_{u(\la),v(\la)})&= \bvphi\left(T_i(G^{\rm up}b)\right)\\
&=\bvphi\left((1-q^2)^{(\a_i,v(\la)-u(\la))}
G^{\rm up}\left({\tf_i}^{\ \vphi_i^*(b)}(\te^*_i)^{\eps_i^*(b)} b\right)\right)\\
&= (1-q^{-2})^{(\a_i,v(\la)-u(\la))}
\bvphi\left(
G^{\rm up}\left((\te^*_i)^{\eps_i^*(b)}(\te_{i_{l(u)}}^*)^{b_{l(u)}}\cdots (\te_{i_1}^*)^{b_1}{\tf_i}^{\ \vphi_i^*(b)}\tf_{j_{l(v)}}^{\ c_{l(v)}}\cdots \tf_{j_1}^{\ c_1}b_\infty\right)\right)\\
&= (1-q^{-2})^{(\a_i,v(\la)-u(\la))}d_{s_iu(\la),\,s_iv(\la)}.
\end{align*}
\cqfd
\subsection{Dual PBW generators are unipotent quantum minors}
Recall from (\ref{def_beta}) the definition of the roots $\b_k$.
\begin{Prop}\label{PBWminors}
For $k = 1,\ldots,r$, we have
\[
E^*(\b_k) = d_{s_{i_1}\cdots s_{i_{k-1}}(\vpi_{i_k}),\ s_{i_1}\cdots s_{i_{k}}(\vpi_{i_k})}.
\]
\end{Prop}
\proof
We have $e_{i_k} = d_{\vpi_{i_k},\ s_{i_k}(\vpi_{i_k})}$,
hence
\[
E(\b_k) = T_{i_1}\cdots T_{i_{k-1}}\left(d_{\vpi_{i_k},\ s_{i_k}(\vpi_{i_k})}\right).
\]
Applying $k-1$ times Proposition~\ref{Tminor}, we get
\[
E(\b_k) = (1-q^{-2})^N d_{s_{i_1}\cdots s_{i_{k-1}}(\vpi_{i_k}),\ s_{i_1}\cdots s_{i_{k}}(\vpi_{i_k})},
\]
where
\[
N = -(\a_{i_{k-1}},\a_{i_k}) - (\a_{i_{k-2}},\, s_{i_{k-1}}(\a_{i_k})) - \cdots
-(\a_{i_1},\, s_{i_2}\cdots s_{i_{k-1}}(\a_{i_k})).
\]
Now,
\begin{align*}
\b_k &= s_{i_1}\cdots s_{i_{k-1}}(\a_{i_k})\\
& = \a_{i_k} + \sum_{n=1}^{k-1} (s_{i_n}\cdots s_{i_{k-1}}(\a_{i_k})-s_{i_{n+1}}\cdots s_{i_{k-1}}(\a_{i_k}))\\
& = \a_{i_k} - \sum_{n=1}^{k-1} (\a_{i_n},\, s_{i_{n+1}}\cdots s_{i_{k-1}}(\a_{i_k}))\a_{i_n},
\end{align*}
so that $\deg\b_k = 1 + N$. Hence,
\[
d_{s_{i_1}\cdots s_{i_{k-1}}(\vpi_{i_k}),\ s_{i_1}\cdots s_{i_{k}}(\vpi_{i_k})}
= (1-q^{-2})^{1-\deg\b_k}\, E(\b_k) = E^*(\b_k).
\]
\cqfd
\subsection{The skew field $F_q(\nn(w))$}\label{skewfield}
It is well known that $A_q(\nn(w))$ is an Ore domain.
This follows for example from the fact that it has polynomial growth
(see \cite[Appendix]{BZ}).
Hence $A_q(\nn(w))$ embeds in its skew field of fractions, which
we shall denote by $F_q(\nn(w))$.
Recall the shorthand notation of \S\ref{ssectTsystem} for unipotent quantum minors.
\begin{Prop}\label{flagFq}
Let $0\le b < d \le r$ be such that $i_b = i_d$. Then the unipotent quantum
minor $D(b,d)$
belongs to $F_q(\nn(w))$. In particular, the quantum flag minors
$D(0,d)= D_{\vpi_{i_d},\ s_{i_1}\cdots s_{i_{d}}(\vpi_{i_d})}$ belong to $F_q(\nn(w))$.
\end{Prop}
\proof
If $b=d^-$, by Proposition~\ref{PBWminors}, we have $D(d^-,d)= \Psi(E^*(\b_{d}))$,
so
\[
D(d^-,d)\in A_q(\n(w)) \subset F_q(\n(w)).
\]
Recall the determinantal identity (\ref{eqTsystem}).
Arguing as in \cite[Corollary 13.3]{GLS}, we can order the set of minors
$D(b,d)\ (0\le b < d \le r)$ so that (i) the minors $D(d^-,d)$
are the smallest elements, and (ii) the minor $D(b^-,d)$
is strictly bigger than all the other minors occuring in
(\ref{eqTsystem}).
This allows to express, recursively $D(b^-,d)$ as a rational
expression in the minors $D(c^-,c)\ (1\le c \le r)$,
and shows that $D(b^-,d)\in F_q(\n(w))$.
\cqfd
We will show later (see Corollary~\ref{minorpol}) that, in fact, all quantum minors $D(b,d)$
are \emph{polynomials} in the dual PBW-generators $D(c^-,c)$.
Hence, they belong to $A_q(\n(w))$.
\subsection{Specialization at $q=1$ of $A_q(\n(w))$}\label{specialq1}
Let $A_\A(\n(w))$ denote the free $\A$-submodule of $A_q(\n(w))$
with basis $\Psi(\P^*_\ii)$. This integral form of $A_q(\n(w))$ is
an $\A$-algebra, independent of the choice of the reduced word $\ii$.
Moreover, if we regard $\C$ as an $\A$-module via the homomorphism $q \mapsto 1$,
we can define
\begin{equation}
A_1(\n(w)) := \C \otimes_\A A_\A(\n(w)).
\end{equation}
This is a $\C$-algebra isomorphic to the coordinate ring $\C[N(w)]$, (see \cite[Theorem 4.39]{Ki}).
In particular, if $D_{u(\la),v(\la)}$ is a unipotent quantum minor
in $A_q(\n(w))$, the element $1\otimes D_{u(\la),v(\la)}$ can be identified
with the corresponding classical minor in $\C[N(w)]$.
\section{Quantum cluster algebras}
In this section we recall, following Berenstein and Zelevinsky \cite{BZ}, the definition
of a quantum cluster algebra.
\subsection{Based quantum tori}
Let $L = (\la_{ij})$ be a skew-symmetric $r\times r$-matrix with integer entries.
The {\em based quantum torus} $\T(L)$
is the $\Z[q^{\pm 1/2}]$-algebra generated by
symbols $X_1,\ldots,X_r,X_1^{-1},\ldots,X_r^{-1}$
submitted to the relations
\begin{equation}\label{qT}
X_iX_i^{-1}=X_i^{-1}X_i=1,\qquad
X_i X_j = q^{\la_{ij}} X_j X_i, \qquad (1\le i,j\le r).
\end{equation}
For $\aa=(a_1,\ldots,a_r)\in \Z^r$, set
\begin{equation}\label{eqX^a}
X^\aa:=q^{\frac{1}{2}\sum_{i>j}a_ia_j\la_{ij}}X_1^{a_1}\cdots X_r^{a_r}.
\end{equation}
Then $\{X^\aa\mid \aa\in \Z^r\}$ is a $\Z[q^{\pm 1/2}]$-basis
of $\T(L)$, and we have for $\aa, \bbf\in\Z^r$,
\begin{equation}\label{multBZ}
X^\aa X^\bbf=q^{\frac{1}{2}\sum_{i>j}(a_ib_j-b_ia_j)\la_{ij}}X^{\aa+\bbf}
= q^{\sum_{i>j}(a_ib_j-b_ia_j)\la_{ij}}X^\bbf X^\aa.
\end{equation}
Since $\T(L)$ is an Ore domain, it can be embedded in its skew field of fractions $\F$.
\subsection{Quantum seeds}
Fix a positive integer $n< r$.
Let $\tB = (b_{ij})$ be an $r\times (r-n)$-matrix with integer coefficients.
The submatrix $B$ consisting of the first $r-n$ rows of $\tB$ is called the
\emph{principal part} of $\tB$. We will require $B$ to be skew-symmetric.
We call $\tB$ an \emph{exchange matrix}.
We say that the pair $(L,\tB)$ is \emph{compatible} if we have
\begin{equation}\label{def_compatible}
\sum_{k=1}^r b_{kj}\la_{ki} = \de_{ij}d,\qquad (1\le j\le r-n,\ 1\le i\le r)
\end{equation}
for some positive integer $d$.
If $(L,\tB)$ is compatible, the datum $\SC=((X_1,\ldots,X_r),L,\tB)$ is called a
\emph{quantum seed} in $\F$. The set $\{X_1,\ldots,X_r\}$ is called the \emph{cluster}
of $\SC$, and its elements the \emph{cluster variables}. The cluster variables
$X_{r-n+1},\ldots, X_r$ are called \emph{frozen variables}, since they will not be
affected by the operation of mutation to be defined now.
The elements $X^\aa$ with $\aa\in\N^r$ are called \emph{quantum cluster monomials}.
\subsection{Mutations}
For $k=1,\ldots,r-n$, we define the \emph{mutation} $\mu_k(L,\tB)$ of
a compatible pair $(L,\tB)$.
Let $E$ be the $r\times r$-matrix with entries
\begin{equation}
e_{ij} =
\begin{cases}
\de_{ij} &\text{ if } j\not = k,\\
-1 &\text{ if } i=j=k,\\
\max(0,-b_{ik}) &\text{ if } i\not = j = k.
\end{cases}
\end{equation}
Let $F$ be the $(r-n)\times (r-n)$-matrix with entries
\begin{equation}
f_{ij} =
\begin{cases}
\de_{ij} &\text{ if } i\not = k,\\
-1 &\text{ if } i=j=k,\\
\max(0,b_{kj}) &\text{ if } i = k \not = j.
\end{cases}
\end{equation}
Then $\mu_k(L,\tB) = (\mu_k(L),\mu_k(\tB))$ where
\begin{equation}
\mu_k(L) := E^{T} L E, \qquad \mu_k(\tB) := E\tB F.
\end{equation}
Note that the mutation $\mu_k(\tB)$ of the exchange
matrix is a reformulation of the classical one defined in \cite{FZ2}.
It is easy to check that $\mu_k(L,\tB)$ is again a compatible pair,
with the same integer $d$ as in~(\ref{def_compatible}).
Define $\aa' = (a'_1,\ldots,a'_r)$ and $\aa'' = (a''_1,\ldots,a''_r)$ by
\begin{equation}\label{qmut1}
a'_i =
\begin{cases}
-1 &\text{ if } i = k,\\
\max(0,b_{ik}) &\text{ if } i\not = k,
\end{cases}
\qquad
a''_i =
\begin{cases}
-1 &\text{ if } i = k,\\
\max(0,-b_{ik}) &\text{ if } i\not = k.
\end{cases}
\end{equation}
One then defines
\begin{equation}\label{qmut2}
\mu_k(X_i) =
\begin{cases}
X^{\aa'} + X^{\aa''},&\text{ if } i = k,\\
X_i&\text{ if } i\not = k.
\end{cases}
\end{equation}
Berenstein and Zelevinsky show that the elements $X'_i := \mu_k(X_i)$
satisfy
\begin{equation}
X'_i X'_j = q^{\la'_{ij}} X'_j X'_i, \qquad (1\le i,j\le r),
\end{equation}
where $\mu_k(L) = (\la'_{ij})$. Moreover they form a \emph{free generating set}
of $\F$, that is, one can write $X'_i = \theta(X^{\cc^i})$ where $\theta$ is a
$\Q(q^{1/2})$-linear automorphism of $\F$,
and $(\cc^1,\ldots,\cc^r)$ is a $\Z$-basis of $\Z^r$.
Therefore
\begin{equation}
\mu_k(\SC):=((\mu_k(X_1),\ldots,\mu_k(X_r)),\ \mu_k(L),\ \mu_k(\tB))
\end{equation}
is a new quantum seed in $\F$,
called the \emph{mutation of $\SC$ in direction $k$}.
Moreover,
the mutation operation is involutive, that is, $\mu_k(\mu_k(\SC))=\SC$.
\begin{Def}
The quantum cluster algebra $\AA_{q^{1/2}}(\SC)$ is the $\Z[q^{\pm 1/2}]$-subalgebra of
the skew field~$\F$ generated by the union of clusters of all quantum seeds
obtained from $\SC$ by any sequence of mutations.
\end{Def}
The following basic result is called the \emph{quantum Laurent phenomenon}.
\begin{Thm}[\cite{BZ}]
The quantum cluster algebra $\AA_{q^{1/2}}(\SC)$ is contained in the based quantum torus
generated by the quantum cluster variables of any given quantum seed $\SC'$ mutation equivalent to $\SC$.
\end{Thm}
In the next sections we are going to construct a class of quantum cluster
algebras attached to some categories of representations of preprojective algebras.
\section{The category $\CC_w$}
We recall the main facts about the category $\CC_w$ and its maximal rigid objects,
following \cite{BIRS, GLSUni1, GLS}.
\subsection{The preprojective algebra}
Let $Q$ be a finite connected quiver without
oriented cycles, with vertex set $I$, and
arrow set $Q_1$.
We can associate with $Q$ a symmetric generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$,
where $a_{ij}=2$ if $i=j$, and otherwise $a_{ij}$ is minus the number of edges between
$i$ and $j$ in the underlying unoriented graph.
We will assume that $A$ is the Cartan matrix attached to the Kac-Moody algebra $\g$, that is,
\begin{equation}
a_{ij} = \<h_i,\a_j\> = (\a_i,\a_j),\qquad (i,j\in I).
\end{equation}
Let $\C\overline{Q}$ be the path algebra of the
{\it double quiver } $\overline{Q}$
of $Q$, which is obtained
from $Q$ by adding to each arrow $a\colon i \to j$ in $Q_1$ an arrow
$a^*\colon j \to i$ pointing in the opposite direction.
Let $(c)$ be
the two-sided ideal of $\C\overline{Q}$ generated by the element
\[
c = \sum_{a \in Q_1} (a^*a - aa^*).
\]
The algebra
$$
\L := \C\overline{Q}/(c)
$$
is called the {\it preprojective algebra} of $Q$.
Recall that for all $X,Y \in \md(\L)$ we have
\begin{equation}
\dm \Ext_\L^1(X,Y) = \dm \Ext_\L^1(Y,X).
\end{equation}
This follows for example from the following important formula
\begin{equation}\label{CBform}
\dim\Ext^1_\L(X,Y)=\dim\Hom_\L(X,Y) + \dim\Hom_\L(Y,X) - (\dimv\, X,\ \dimv\, Y),
\end{equation}
where we identify the dimension vector $\dimv\, X$ with an element of $Q_+$
in the standard way.
We denote by $\hI_i\ (i\in I)$ the indecomposable injective $\L$-module
with socle $S_i$.
Here, $S_i$ is the one-dimensional simple $\L$-module
supported on the vertex $i$ of $Q$.
Note that the modules $\hI_i$ are infinite-dimensional if $Q$ is not
a Dynkin quiver.
For a $\L$-module $M$, let
$\soc_{(j)}(M)$ be the sum of all submodules
$U$ of $M$ with $U \cong S_j$.
For $(j_1,\ldots,j_s)\in I^s$, there is a unique sequence
$$
0 = M_0 \subseteq M_1 \subseteq \cdots \subseteq M_s \subseteq M
$$
of submodules of $M$ such that $M_p/M_{p-1} = \soc_{(j_p)}(M/M_{p-1})$.
Define $\soc_{(j_1,\ldots,j_s)}(M) := M_s$.
(In this definition, we do not assume that $M$ is finite-dimensional.)
\subsection{The subcategory $\CC_w$}
Let $\ii = (i_r,\ldots,i_1)$ be a reduced expression of $w\in W$.
For $1 \le k \le r$, let
\begin{equation}
V_k := V_{\ii,k} := \soc_{(i_k,\ldots,i_1)}\left(\hI_{i_k}\right),
\end{equation}
and set
$V_\ii := V_1 \oplus \cdots \oplus V_r$.
The module $V_\ii$ is dual to the cluster-tilting object
constructed in \cite[Section II.2]{BIRS}.
Define
$$
\CC_\ii := \Fac(V_\ii) \subseteq \md(\L).
$$
This is the full subcategory of $\md(\L)$ whose objects are
quotient modules of a direct sum of a finite number of copies of $V_\ii$.
For $j\in I$, let $k_j := \max\{ 1 \le k \le r \mid i_k = j \}$.
Define
$I_{\ii,j} := V_{\ii,k_j}$
and set
$$
I_\ii := I_{\ii,1} \oplus \cdots \oplus I_{\ii,n}.
$$
The category $\CC_\ii$ and the module $I_\ii$ depend only on $w$, and not
on the chosen reduced expression $\ii$ of $w$.
Therefore, we define
$$
\CC_w := \CC_\ii, \qquad I_w := I_\ii.
$$
\begin{Thm}[{\cite[Theorem II.2.8]{BIRS}}]
\label{main1}
For any $w\in W$,
the following hold:
\begin{itemize}
\item[(a)]
$\CC_w$ is a Frobenius category. Its
stable category $\underline{\CC}_w$ is a 2-Calabi-Yau category.
\item[(b)]
The indecomposable $\CC_w$-projective-injective modules are
the indecomposable direct summands of $I_w$.
\item[(c)]
$\CC_w = \Fac(I_w)$.
\end{itemize}
\end{Thm}
Note that in the case when $Q$ is a Dynkin quiver, that is, $\g$ is a simple Lie algebra of type $A, D, E$,
and $w=w_0$ is the longest element in $W$, then $\CC_{w_0} = \md(\L)$.
\subsection{Maximal rigid objects}\label{maximal_rigid}
For a $\L$-module $T$, we denote by $\add(T)$ the additive closure
of $T$, that is, the full subcategory of $\md(\L)$ whose objects
are isomorphic to direct sums of direct summands of $T$.
A $\L$-module $T$ is called {\it rigid} if $\Ext_\L^1(T,T) = 0$.
Let $T\in \CC_w$ be rigid. We say that
\begin{itemize}
\item
$T$ is $\CC_w$-{\it maximal rigid} if
$
\Ext_\L^1(T \oplus X,X) = 0
$
with $X \in \CC_w$ implies $X \in \add(T)$;
\item
$T$ is a $\CC_w$-{\it cluster-tilting module} if
$
\Ext_\L^1(T,X) = 0
$
with $X \in \CC_w$
implies $X \in \add(T)$.
\end{itemize}
\begin{Thm}[{\cite[Theorem I.1.8]{BIRS}}]
\label{main9}
For a rigid $\L$-module $T$ in $\CC_w$ the following are equivalent:
\begin{itemize}
\item[(a)]
$T$ has $r$ pairwise non-isomorphic indecomposable direct summands;
\item[(b)]
$T$ is $\CC_w$-maximal rigid;
\item[(c)]
$T$ is a $\CC_w$-cluster-tilting module.
\end{itemize}
\end{Thm}
Note that, given Theorem~\ref{main1}, the proof of \cite[Theorem 2.2]{GLSRigid}
carries over to this more general situation (see \cite[Theorem 2.2]{GLSUni1} in the case when $w$ is adaptable).
\subsection{The quiver $\G_T$ and the matrix $\tB_T$}\label{subsecGammaT}
Let $T = T_1\oplus \cdots \oplus T_r$ be a $\CC_w$-maximal rigid module, with each
summand $T_i$ indecomposable. Clearly each indecomposable $\CC_w$-injective module
$I_{\ii,j}$ is isomorphic to one of the $T_k$'s, so, up to relabelling, we can assume that
$T_{r-n+j}\cong I_{\ii,j}$ for $j=1,\ldots, n$.
Consider the endomorphism algebra $A_T:=\End_\L(T)^{\rm op}$. This is a basic algebra,
with indecomposable projective modules
\begin{equation}
P_{T_i} := \Hom_\L(T,T_i)\ (1\le i\le r).
\end{equation}
The simple $A_T$-modules are the heads $S_{T_i}$ of the projectives $P_{T_i}$.
One defines a quiver $\G_T$ with vertex set $\{1,\ldots,r\}$, and
$d_{ij}$ arrows from $i$ to $j$, where $d_{ij} = \dim \Ext^1_{A_T}(S_{T_i},S_{T_j})$.
(This is known as the Gabriel quiver of $A_T$.)
Most of the information contained
in $\G_T$ can be encoded in an $r\times (r-n)$-matrix $\tB_T = (b_{ij})_{1\le i\le r,\ 1\le j\le r-n}$,
given by
\begin{equation}
b_{ij} = (\text{number of arrows } j\to i \text{ in } \G_T)
- (\text{number of arrows } i\to j \text{ in } \G_T).
\end{equation}
Note that $\tB_T$ can be regarded as an exchange matrix, with skew-symmetric principal part.
The next theorem gives an explicit description of the quiver $\G_{T}$
(hence also of the matrix $\tB_{T}$) for certain $\CC_w$-maximal rigid modules.
Following \cite{BFZ}, we define a quiver
$\G_\ii$ as follows.
The vertex set of $\G_\ii$ is equal to $\{1,\ldots,r\}$.
For $1 \le k \le r$, let
\begin{align}\label{kminus}
k^- &:= \max\left(\{0\}\cup \{1 \le s \le k-1 \mid i_s = i_k \}\right),\\
k^+ &:= \min\left(\{ k+1 \le s \le r \mid i_s = i_k \} \cup \{r+1\}\right).
\end{align}
For $1 \le s,\,t \le r$ such that $i_s \not = i_t$,
there are $|a_{i_s,i_t}|$ arrows from $s$ to $t$ provided
$t^+ \ge s^+ > t > s$.
These are called the {\it ordinary arrows} of $\G_\ii$.
Furthermore, for each $1 \le s \le r$ there is an arrow $s \to s^-$
provided $s^- > 0$.
These are the {\it horizontal arrows} of $\G_\ii$.
The following result generalizes \cite[Theorem 1]{GLSAus} (see \cite[Theorem 2.3]{GLSUni1} in the case when $w$ is adaptable).
\begin{Thm}[{\cite[Theorem II.4.1]{BIRS}}]
\label{main6}
The $\L$-module $V_\ii$ is a $\CC_w$-maximal rigid module,
and we have $\G_{V_\ii} = \G_\ii$.
\end{Thm}
\subsection{Mutations of maximal rigid objects}
We consider again an arbitrary $\CC_w$-maximal rigid module $T$, and
we use the notation of \ref{subsecGammaT}.
\begin{Thm}[{\cite[Theorem I.1.10]{BIRS}}]\label{th_mutation}
Let $T_k$ be a non-projective indecomposable direct summand of $T$.
\begin{itemize}
\item[(a)] There exists a unique indecomposable module $T_k^*\not\cong T_k$
such that $(T/T_k)\oplus T_k^*$ is $\CC_w$-maximal rigid.
We call $(T/T_k)\oplus T_k^*$ the \emph{mutation of $T$ in direction $k$},
and denote it by $\mu_k(T)$.
\item[(b)]
We have $\tB_{\mu_k(T)} = \mu_k(\tB_T)$.
\item[(c)] We have $\dim\Ext^1_\L(T_k,T_k^*) = 1$.
Let $0\to T_k \to T'_k \to T_k^* \to 0$ and
$0\to T_k^* \to T''_k \to T_k \to 0$ be non-split short exact sequences.
Then
\[
T'_k \cong \bigoplus_{b_{jk}<0} T_j^{-b_{jk}},\qquad
T''_k \cong \bigoplus_{b_{jk}>0} T_j^{b_{jk}}.
\]
\end{itemize}
\end{Thm}
Note that again, given Theorem~\ref{main1}, the proof of \cite[\S2.6]{GLSRigid}
carries over to this more general situation.
\subsection{The modules $M[l,k]$}
In this section we assume that the reduced word $\ii$ is fixed.
So for simplicity we often omit it from the notation.
Thus, for $k = 1,\ldots,r$, we may write $V_k$ instead of $V_{\ii,k}$.
Moreover, we use the convention that $V_0 = 0$.
For $1\le k\le l\le r$ such that $i_k=i_l=i$, we have a natural embedding
of $\L$-modules $V_{k^-} \to V_{l}$.
Following \cite[\S9.8]{GLS}, we define $M[l,k]$ as the cokernel
of this embedding, that is,
\begin{equation}
M[l,k] := V_{l} / V_{k^-}.
\end{equation}
In particular, we set $M_k:= M[k,k]$, and
\begin{equation}
M=M_\ii := M_1\oplus \cdots \oplus M_r.
\end{equation}
We will use the convention that $M[l,k]=0$ if $k>l$.
Every module $M[l,k]$ is indecomposable and rigid.
But note that $M$ is \emph{not} rigid.
Define
\begin{align}\label{kmin}
k_{\min} &:= \min\{1 \le s \le r \mid i_s = i_k \},\\
k_{\max} &:= \max\{ 1 \le s \le r \mid i_s = i_k \}.
\end{align}
Then $V_k = M[k,k_{\min}]$ corresponds to an \emph{initial interval}.
The direct sum of all modules $M[k_{\max},k]$ corresponding to
\emph{final intervals} is also a $\CC_w$-maximal rigid module, denoted
by $T_\ii$.
By \cite[\S10]{GLS}, for every module $X\in\CC_w$, there exists
a chain
\begin{equation}
0 = X_0 \subseteq X_1 \subseteq \cdots \subseteq X_r = X
\end{equation}
of submodules of $X$ with $X_k/X_{k-1} \cong M_k^{m_k}$, for some
uniquely determined non-negative integers $m_k$.
The $r$-tuple $\mm(X) := (m_1, \ldots, m_r)$ will be called the
\emph{$M$-dimension vector} of $X$.
\subsection{Dimensions of Hom-spaces}
Let $R=R_\ii=(r_{kl})$ the $r\times r$-matrix with entries
\begin{equation}
r_{kl} =
\begin{cases}
0& \text{ if } k<l,\\
1& \text{ if } k=l,\\
(\b_k,\b_l)& \text{ if } k>l,
\end{cases}
\end{equation}
\begin{Prop} \label{prp:HomDim}
Suppose that $X,Y\in \CC_w$ satisfy $\Ext^1_\L(X,Y)=0$. Then
\[
\dim\Hom_\L(X,Y)= \mm(X)\, R\, \mm(Y)^T.
\]
\end{Prop}
\proof
By \cite[Proposition~10.5]{GLS},
for all $Y\in\CC_w$ with $M$-dimension vector $\mm(Y)$,
we have
\begin{equation}
\dim\Hom_\L(V_k,Y)=\dim\Hom_\L\left(V_k,\ \bigoplus_k M_k^{(\mm(Y))_k}\right).
\end{equation}
Moreover, the $M$-dimension vector of $V_k$ is given by
\begin{equation}\label{aaV}
(\mm(V_k))_j=\begin{cases} 1 &\text{ if } i_j=i_k\text{ and } j\leq k,\\
0 &\text{ else.}
\end{cases}
\end{equation}
Therefore, we can restate~\cite[Lemma~9.8]{GLS} as
\begin{equation} \label{eq:Lem}
\dim\Hom_\Lam(V_k,Y)= \mm(V_k)\, R\,\mm(Y)^T.
\end{equation}
Now, for $X\in\CC_w$ we can find a short exact sequence
$0\ra V''\ra V'\ra X\ra 0$ with $V',V''\in\add(V_\ii)$.
Since $\Hom_\Lam(V_\ii,-)$ is exact on this sequence we conclude that
\begin{equation}\label{eq:xvv}
\mm(X)=\mm(V')-\mm(V'').
\end{equation}
Finally, since $\Ext^1_\L(X,Y)=0$, the sequence
\[
0\ra\Hom_\Lam(X,Y)\ra\Hom_\Lam(V',Y)\ra\Hom_\Lam(V'',Y)\ra 0
\]
is exact. Thus we can calculate
\begin{align*}
\dim\Hom_\Lam(X,Y) &=\dim\Hom_\Lam(V',Y)-\dim\Hom_\Lam(V'',Y)\\
&
=(\mm(V')-\mm(V''))\, R\, \mm(Y)^T\\
&
=\mm(X)\, R\, \mm(Y)^T.
\end{align*}
\cqfd
\begin{Lem} \label{lem8.6}
Let $1\le b < d \le r$ be such that $i_b = i_d=i$.
There holds
\[
\dim\Hom_\Lam(M[d,b^+],\,M[d^-,b])=
\mm(M[d,b])\, R\, \mm(M[d^-,b])^T.
\]
\end{Lem}
\proof
We have a short exact sequence
\[
0\ra M[d^-,b]\ra M[d,b]\oplus M[d^-,b^+]\ra M[d,b^+]\ra 0
\]
with
\begin{equation} \label{eq:MutAdd}
\mm(M[d^-,b])-(\mm(M[d,b])+\mm(M[d^-,b^+]))+
\mm(M[d,b^+])= 0.
\end{equation}
Since
\[
\Ext^1_\L(M[d,b^+],M[d^-,b]) = 1, \quad
\Ext^1_\L(M[d,b^+],M[d,b]) = \Ext^1_\L(M[d,b^+],M[d^-,b^+]) = 0,
\]
this yields an exact sequence
\begin{multline*}
0\ra\Hom_\Lam(M[d,b^+],M[d^-,b])\ra\Hom_\Lam(M[d,b^+],M[d,b]\oplus M[d^-,b^+])
\ra \\
\ra\Hom_\Lam(M[d,b^+],M[d,b^+])\ra \C\ra 0.
\end{multline*}
Applying Proposition~\ref{prp:HomDim}
to the second and third non-trivial terms
yields the required equality,
if we take into account \eqref{eq:MutAdd} and the fact that $r_{bb}=1$.
\cqfd
\begin{Prop}\label{propcalculhom}
Let $1\le b < d \le r$ be such that $i_b = i_d=i$.
Let $j,k\in I$ with $j\not = i$ and $k \not = i$.
There holds
\begin{align*}
&\dim\Hom_\Lam(M[d,b^+],M[d^-,b]) = (\dimv\, V_{d},\, \dimv\, M[d^-,b]) - (\dimv\, M[d^-,b])_i,\\[2mm]
&\dim\Hom_\Lam(M[d,b],M[d^-,b^+])= (\dimv\, V_{d},\, \dimv\, M[d^-,b^+]) -(\dimv\, M[d^-,b^+])_i,\\[2mm]
&\dim\Hom_\Lam(M[d^-(j),(b^-(j))^+],\,M[d^-(k),(b^-(k))^+])= (\dimv\, V_{d^-(j)},\, \dimv\, M[d^-(k),(b^-(k))^+])\\[2mm]
&\hskip 7.8cm -\ (\dimv\, M[d^-(k),(b^-(k))^+])_j.
\end{align*}
\end{Prop}
\proof
By Lemma~\ref{lem8.6},
\[
\dim\Hom_\Lam(M[d,b^+],M[d^-,b]) =
\dim\Hom_\Lam(M[d,b],M[d^-,b]).
\]
Moreover, Proposition~\ref{prp:HomDim} shows that
\[
\dim\Hom_\Lam(M[d,b],M[d^-,b])
=
\dim\Hom_\Lam(V_{d},M[d^-,b]).
\]
Now, it follows from (\ref{CBform}) that
\[
\dim\Hom(V_{d},M[d^-,b])= (\dimv\,V_{d},\,\dimv\, M[d^-,b]) - (\dimv\, M[d^-,b])_i.
\]
Indeed,
$V_{d}$ and $M[d^-,b]$ belong to the subcategory $\CC_{s_{i_{d}}\cdots s_{i_1}}$,
and $V_{d}$ is projective-injective in this category, with socle $S_i$. Therefore
\[
\dim\Hom(M[d^-,b],V_{d}) =(\dimv\, M[d^-,b])_i \quad \text{ and } \quad
\Ext^1_\Lam(M[d^-,b],V_{d})=0.
\]
This proves the first equation.
For the remaining two equations we note that again, by Proposition~\ref{prp:HomDim},
we can replace in the left hand side
$M[d,b]$ by $V_{d}$, and $M[d^-(j),(b^-(j))^+]$ by $V_{d^-(j)}$.
The claim follows.
\cqfd
\section{The quantum cluster algebra associated with $\CC_w$}
\subsection{The cluster algebra $\AA(\CC_w)$}
Following \cite{GLSUni1}, \cite{BIRS}, \cite{GLS}, we can associate with $\CC_w$
a (classical \ie not quantum) cluster algebra. This is given by the initial seed
\begin{equation}
\Sigma_{V_\ii} := ((x_1,\ldots,x_r), \tB_{V_\ii}).
\end{equation}
Although this seed depends on the choice of a reduced expression
$\ii$ for $w$, one can show that any two matrices $\tB_{V_\ii}$ and $\tB_{V_\jj}$
are connected by a sequence of mutations. Therefore this cluster algebra
is independent of this choice, and we denote it by $\AA(\CC_w)$.
Moreover, every seed of $\AA(\CC_w)$ is of the form
\[
\Sigma_T = ((x_{T_1},\ldots,x_{T_r}),\tB_T),
\]
for a unique $\CC_w$-maximal rigid module $T=T_1\oplus\cdots \oplus T_r$, and some Laurent polynomials
$x_{T_1},\ldots,x_{T_r}$ in the variables $x_1 = x_{V_{\ii,1}},\ldots,x_r = x_{V_{\ii,r}}$.
These modules $T$ are those which
can be reached from $V_\ii$ using a sequence of mutations, and we called them \emph{reachable}.
(It is still an open problem whether every $\CC_w$-maximal rigid module is reachable or not.)
If $\jj$ is another reduced expression for $w$, it is known that $V_\jj$ is reachable
from $V_\ii$ \cite{BIRS}. Therefore, the collection of reachable $\CC_w$-maximal rigid modules does
not depend on the choice of $\ii$.
It was shown in \cite{GLSUni1,GLS} that there is a natural isomorphism from
$\AA(\CC_w)$ to the coordinate ring $\C[N(w)]$, mapping the cluster monomials
to a subset of Lusztig's dual semicanonical basis of $\C[N(w)]$.
\subsection{The matrix $L_T$}
Let $T=T_1\oplus\cdots\oplus T_r$ be a $\CC_w$-maximal rigid module
as in \S\ref{subsecGammaT}.
Let $L_T=(\la_{ij})$ be the $r\times r$-matrix with entries
\begin{equation}\label{def_lambda}
\la_{ij} := \dim\Hom_\L(T_i,T_j) - \dim\Hom_\L(T_j,T_i),\qquad (1\le i, j\le r).
\end{equation}
Note that $L_T$ is skew-symmetric.
From now on we will use the following convenient shorthand notation.
Given two $\L$-modules $X$ and $Y$, we will write
\begin{equation}
[X,Y] := \dim\Hom_\L(X,Y), \qquad [X,Y]^1 := \dim\Ext^1_\L(X,Y).
\end{equation}
Thus, we shall write $\la_{ij} = [T_i,T_j]-[T_j,T_i]$.
\begin{Prop}\label{Tcomp}
The pair $(L_T,\tB_T)$ is compatible.
\end{Prop}
\proof
For $1\le j\le r-n$, and $1\le i,\ k\le r$, by Theorem~\ref{th_mutation} (c) we have:
\[
\sum_{k=1}^r b_{kj}\la_{ki} = [T_i,T'_j]+[T''_j,T_i]-[T'_j,T_i]-[T_i,T''_j].
\]
Let us first assume that $i\not = j$.
Applying the functor $\Hom_\L(T_i,-)$ to the short exact sequence
\[
0 \to T_j \to T'_j \to T_j^* \to 0,
\]
and taking into account that $[T_i,T_j]^1=0$, we get a short exact sequence
\[
0 \to \Hom_\L(T_i,T_j) \to \Hom_\L(T_i,T'_j) \to \Hom_\L(T_i,T_j^*) \to 0,
\]
therefore
\begin{equation}
[T_i,T'_j] = [T_i,T_j^*] + [T_i,T_j].
\end{equation}
Similarly, applying the functor $\Hom_\L(-,T_i)$ to the same short exact sequence
and taking into account that $[T_j^*,T_i]^1=0$, we get a short exact sequence
\[
0 \to \Hom_\L(T^*_j,T_i) \to \Hom_\L(T'_j,T_i) \to \Hom_\L(T_j,T_i) \to 0,
\]
therefore
\begin{equation}
[T'_j,T_i] = [T^*_j,T_i] + [T_j,T_i].
\end{equation}
It follows that
\begin{equation}\label{1steq}
[T_i,T'_j] - [T'_j,T_i] = [T_i,T_j^*] - [T^*_j,T_i] + \la_{ij}.
\end{equation}
Arguing similarly with the short exact sequence
\[
0 \to T^*_j \to T''_j \to T_j \to 0,
\]
we obtain
\begin{equation}\label{2ndeq}
[T_j'',T_i] - [T_i,T_j''] = [T_j^*,T_i] - [T_i,T^*_j] + \la_{ji}.
\end{equation}
Hence
$\sum_{k=1}^r b_{kj}\la_{ki} = \la_{ij} + \la_{ji} = 0$.
Assume now that $i=j$. Using that $[T_j,T_j^*]^1=1$ and $[T_j,T_j]^1=[T_j,T_j']^1=[T_j,T_j'']^1=0$,
and arguing as above, we easily obtain the relations
\begin{align}
[T_j,T'_j] &= [T_j,T_j^*] + [T_j,T_j], \label{eq9.7}\\
[T'_j,T_j] &= [T^*_j,T_j] + [T_j,T_j]-1,\label{eq9.8}\\
[T_j'',T_j] &= [T_j^*, T_j] + [T_j,T_j],\label{eq9.9}\\
[T_j,T_j''] &= [T_j,T_j^*] + [T_j,T_j] -1\label{eq9.10}.
\end{align}
It follows that
$\sum_{k=1}^r b_{kj}\la_{kj} = 2.$
Thus, in general,
\[
\sum_{k=1}^r b_{kj}\la_{ki} = 2\de_{ij},
\]
which proves the proposition.
\cqfd
Let $k\le r-n$, so that the mutation $\mu_k(T)$ is well-defined.
\begin{Prop}\label{LTmut}
We have $\mu_k(L_T) = L_{\mu_k(T)}$.
\end{Prop}
\proof
Put $\mu_k(L_T)=(\la'_{ij})$.
By definition, $\la'_{ij}=\la_{ij}$ if $i$ or $j$ is different from $k$.
Similarly, since $T$ and $\mu_k(T)$ differ only by the replacement of $T_k$
by $T_k^*$, all entries in $L_{\mu_k(T)}$ not situated on row $k$ or column $k$
are equal to the corresponding entries of $L_T$.
If $i=k$, we have by definition of $\mu_k(L_T)$ and by Theorem~\ref{th_mutation}~(c)
\[
\la'_{kj} = \sum_{s=1}^r e_{sk}\la_{sj} = [T'_k,T_j] - [T_j,T'_k] -[T_k,T_j]+[T_j,T_k] - \la_{kj}.
\]
Thus, by (\ref{1steq}), we get
\[
\la'_{kj} = [T_k^*,T_j] - [T_j, T_k^*],
\]
as claimed. The case $j=k$ follows by skew-symmetry. \cqfd
Let $H_\ii=(h_{kl})$ be the $r\times r$-matrix with entries
\begin{equation}
h_{kl} =
\begin{cases}
1& \text{ if } l = k^{(-m)} \text{ for some } m\ge 0,\\
0& \text{ otherwise.}
\end{cases}
\end{equation}
\begin{Prop}\label{propLV}
The matrix $L_{V_\ii}$ has the following explicit expression
\[
L_{V_\ii} = H_\ii (R_\ii - R_\ii^T) H_\ii^T.
\]
\end{Prop}
\proof This follows immediately from Proposition~\ref{prp:HomDim}, if we
note that the $k$th row of $H_\ii$ is equal to the $M$-dimension vector $\mm(V_{\ii,k})$. \cqfd
\subsection{The quantum cluster algebra $\AA_{q^{1/2}}(\CC_w)$}
Proposition~\ref{Tcomp} and Proposition~\ref{LTmut} show that the family
$(L_T,\tB_T)$, where $T$ ranges over all $\CC_w$-maximal rigid modules reachable
from $V_\ii$, gives rise to a quantum analogue of the cluster algebra $\AA(\CC_w)$.
We shall denote it by $\AA_{q^{1/2}}(\CC_w)$.
For every reduced expression $\ii$ of $w$, an explicit initial quantum seed is given by
\[
\SC_{V_\ii}:=((X_{V_{\ii,1}},\ldots,X_{V_{\ii,r}}),\ L_{V_\ii},\ \tB_{V_\ii}),
\]
where the matrices $L_{V_\ii}$ and $\tB_{V_\ii}$ have been computed in
Proposition~\ref{propLV}, and in Section~\ref{subsecGammaT}.
The quantum seed corresponding to a reachable $\CC_w$-maximal rigid module
$T=T_1\oplus\cdots\oplus T_r$
will be denoted by
\begin{equation}
\SC_T = ((X_{T_1},\ldots,X_{T_r}),\ L_T,\ \tB_T).
\end{equation}
For every $\aa = (a_1,\ldots, a_r)\in \N^r$, we have a rigid module
$T^\aa := T_1^{a_1}\oplus \cdots \oplus T_r^{a_r}$ in the additive closure
$\add(T)$ of $T$.
Following (\ref{eqX^a}) and writing $L_T = (\la_{ij})$, we put
\begin{equation}
X_{T^\aa} = X^\aa:=q^{\frac{1}{2}\sum_{i>j}a_ia_j\la_{ij}}X_{T_1}^{a_1}\cdots X_{T_r}^{a_r}.
\end{equation}
Thus, denoting by $\R_w$ the set of rigid modules
$R$ in the additive closure of some reachable $\CC_w$-maximal rigid module,
we obtain a canonical labelling
\begin{equation}
X_R, \qquad (R\in \R_w).
\end{equation}
of the quantum cluster monomials of $\AA_{q^{1/2}}(\CC_w)$.
\subsection{The elements $Y_R$}\label{eltsY}
It will be convenient in our setting to proceed to a slight rescaling
of the elements $X_R$. For $R\in\R_w$, we define
\begin{equation}\label{XM}
Y_R := q^{-[R,R]/2}\, X_R.
\end{equation}
In particular, writing $R = T^\aa$ as above, an easy calculation gives
\begin{equation}
Y_{R} = q^{-\a(R)}
Y^{a_1}_{T_1}\cdots Y^{a_r}_{T_r},
\end{equation}
where
\begin{equation}
\a(R) := \sum_{i<j}a_ia_j[T_i,T_j] + \sum_i{a_i\choose2}[T_i,T_i].
\end{equation}
Note that $\a(R)$ is an integer (not a half-integer), so that $ Y_R$
belongs to $\Z[q^{\pm1}][Y_{T_1},\cdots,Y_{T_r}]$.
Moreover, we have the following easy lemma.
\begin{Lem}\label{YMN}
Let $T$ be a reachable $\CC_w$-maximal rigid module.
For any $R, S$ in $\add(T)$ we have
\[
Y_R Y_S = q^{[R,S]} Y_{R\oplus S}.
\]
\end{Lem}
\proof
Write $R = T_1^{a_1} \oplus \cdots \oplus T_r^{a_r}$
and $S = T_1^{b_1} \oplus \cdots \oplus T_r^{b_r}$.
We have
\[
Y_R Y_S = q^{-\a(R)-\a(S)} Y^{a_1}_{T_1}\cdots
Y^{a_r}_{T_r} Y^{b_1}_{T_1}\cdots Y^{b_r}_{T_r}
= q^{-\a(R)-\a(S)+\sum_{i>j}a_ib_j\la_{ij}}
Y^{a_1+b_1}_{T_1}\cdots Y^{a_r+b_r}_{T_r}.
\]
On the other hand, using the obvious identity
\[
{a+b\choose 2} = {a\choose 2} + {b\choose 2} + ab,
\]
we see that
\[
\a(R\oplus S) = \a(R) + \a(S) +
\sum_{i<j} (a_ib_j + a_jb_i) [T_i,T_j] +
\sum_i a_ib_i[T_i,T_i].
\]
Hence, taking into account (\ref{def_lambda}),
\begin{eqnarray*}
-\a(R)-\a(S)+\sum_{i>j}a_ib_j\la_{ij} &=&
-\a(R\oplus S) + \sum_i a_ib_i[T_i,T_i]
+\sum_{i>j} a_ib_j[T_i,T_j]
+\sum_{i<j} a_ib_j[T_i,T_j]\\
&=&-\a(R\oplus S) + [R,S],
\end{eqnarray*}
and the result follows.
\cqfd
Note that because of (\ref{CBform}) and the fact that every
$R\in\add(T)$ is rigid, we have
\begin{equation}
[R,R]=(\dimv\, R,\ \dimv\, R)/2,
\end{equation}
hence the exponent of $q$ in (\ref{XM}) depends only on
the dimension vector $\dimv\,R$ of $R$.
\subsection{Quantum mutations}\label{mutY}
We now rewrite formulas (\ref{qmut1}) (\ref{qmut2}) for quantum mutations in $\AA_{q^{1/2}}(\CC_w)$,
using the rescaled quantum cluster monomials $Y_R$.
Let $T$ be a $\CC_w$-maximal rigid module, and let $T_k$ be a non-projective
indecomposable direct summand of $T$. Let $\mu_k(T) = (T/T_k)\oplus T_k^*$ be
the mutation of $T$ in direction $k$.
By Theorem~\ref{th_mutation}, we have two short exact sequences
\begin{equation}
0\to T_k\to T'_k \to T_k^* \to 0, \qquad 0\to T_k^*\to T''_k \to T_k \to 0,
\end{equation}
with $T'_k, T''_k \in \add(T/T_k)$.
The quantum exchange relation between the quantum cluster variables $Y_{T_k}$
and $Y_{T_k^*}$ can be written as follows:
\begin{Prop}\label{Propqmut}
With the above notation, we have
\[
Y_{T_k^*}Y_{T_k} = q^{[T_k^*,T_k]}(q^{-1}Y_{T'_k} + Y_{T''_k}).
\]
\end{Prop}
\proof
We have
\begin{eqnarray*}
Y_{T_k^*}Y_{T_k}&=& q^{-([T_k^*,T_k^*]+[T_k,T_k])/2}X_{T_k^*}X_{T_k}\\
&=&q^{-([T_k^*,T_k^*]+[T_k,T_k])/2}
\left(X^{\aa'}+ X^{\aa''}\right)X_{T_k},
\end{eqnarray*}
where, if we write
\[
T'_k = \bigoplus_{i\not = k} T_i^{a'_i},\qquad
T''_k = \bigoplus_{i\not = k} T_i^{a''_i},
\]
the multi-exponents $\aa'$ and $\aa''$ are given by
\[
\aa' = (a'_1\ldots,a'_{k-1},-1,a'_{k+1},\ldots,a'_r),\quad
\aa'' = (a''_1\ldots,a''_{k-1},-1,a''_{k+1},\ldots,a''_r).
\]
Using (\ref{qT}), (\ref{eqX^a}), and (\ref{def_lambda}), one obtains
easily that
\[
X^{\aa'}X_{T_k} = q^{([T'_k,T_k]-[T_k,T'_k])/2} X_{T'_k},\qquad
X^{\aa''}X_{T_k} = q^{([T''_k,T_k]-[T_k,T''_k])/2} X_{T''_k}.
\]
Now, using twice (\ref{CBform}), we have
\begin{eqnarray}
[T'_k,T'_k]&=&(\dimv\, T'_k\,,\ \dimv\, T'_k)/2\\
&=&(\dimv\, T_k + \dimv\, T_k^*\,,\ \dimv\, T_k + \dimv\, T_k^*)/2\\
&=&[T_k,T_k]+[T_k^*,T_k^*]+(\dimv\, T_k\,,\ \dimv\, T_k^*)\\
&=&[T_k,T_k]+[T_k^*,T_k^*]+[T_k,T_k^*]+[T_k^*,T_k]-1,\label{relationhom}
\end{eqnarray}
and this is also equal to $[T''_k,T''_k]$.
Therefore, the exponent of $q^{1/2}$ in front of $Y_{T'_k}$ in the product
$Y_{T_k^*}Y_{T_k}$ is equal to
\begin{eqnarray*}
[T'_k,T_k]-[T_k,T'_k]+[T'_k,T'_k]-[T_k^*,T_k^*]-[T_k,T_k]
&=&
[T_k,T_k^*]+[T_k^*,T_k]-1-[T_k,T'_k]+[T'_k,T_k]\\
&=& 2([T_k^*,T_k]-1),
\end{eqnarray*}
where the second equality follows from (\ref{eq9.7}) and (\ref{eq9.8}).
Similarly, using (\ref{eq9.9}) and (\ref{eq9.10}), we see that
the exponent of $q^{1/2}$ in front of $Y_{T''_k}$ in the product
$Y_{T_k^*}Y_{T_k}$ is equal to $2[T^*_k,T_k]$, as claimed.
\cqfd
\subsection{The rescaled quantum cluster algebras $\AA_q(\CC_w)$ and $\AA_\A(\CC_w)$}
By definition, $\AA_{q^{1/2}}(\CC_w)$ is a $\Z[q^{\pm1/2}]$-algebra.
It follows from Lemma~\ref{YMN} and Proposition~\ref{Propqmut} that if we replace the
quantum cluster variables $X_{T_i}$ by their rescaled versions $Y_{T_i}$
we no longer need half-integral powers of $q$.
So we are led to introduce the rescaled quantum cluster algebra
$\AA_q(\CC_w)$.
This is defined as the $\Z[q^{\pm1}]$-subalgebra of $\AA_{q^{1/2}}(\CC_w)$
generated by the elements $Y_{T_i}$, where $T_i$ ranges over the
indecomposable direct summands of all reachable $\CC_w$-maximal rigid modules.
It turns out that in the sequel, in order to have a coefficient ring which
is a principal ideal domain, it will be convenient to slightly extend coefficients
from $\Z[q^{\pm1}]$ to $\A = \Q[q^{\pm1}]$. We will denote by
\[
\AA_\A(\CC_w) := \A \otimes_{\Z[q^{\pm1}]} \AA_q(\CC_w)
\]
the corresponding quantum cluster algebra. Of course, the based quantum
tori of $\AA_q(\CC_w)$ (\resp $\AA_\A(\CC_w)$) will be defined over
$\Z[q^{\pm1}]$ (\resp over $\A$).
\subsection{The involution $\si$} \label{sectinvolsigma}
We now define an involution of $\AA_\A(\CC_w)$, which will turn out to be related
to the involution $\si$ of $U_q(\n)$ defined in \S\ref{dualcan}. For this reason
we shall also denote it by $\si$. This involution is a twisted-bar involution,
as defined by Berenstein and Zelevinsky \cite[\S6]{BZ}.
We first define an additive group automorphism $\si$
of the $\A$-module $\A[Y_R\mid R\in \add(V_\ii)]$ by setting
\begin{eqnarray}
\si(f(q)Y_R) &=& f(q^{-1})\, q^{[R,R] - \dim R} Y_R, \qquad (f \in \A,\ R \in \add(V_\ii))\label{barYM}.
\end{eqnarray}
Clearly, $\si$ is an involution.
\begin{Lem}\label{baraa}
$\si$ is a ring anti-automorphism.
\end{Lem}
\proof
We have
\begin{eqnarray*}
\si{(Y_R Y_S)}&=&q^{-[R,S]}\si(Y_{R\oplus S})\\
&=&q^{-[R,S]-\dim(R\oplus S)+[R\oplus S,\ R\oplus S]}Y_{R\oplus S}\\
&=&q^{-\dim R - \dim S + [R,R] +[S,S] + [S,R]}Y_{R\oplus S}\\
&=&q^{-\dim R - \dim S + [R,R] + [S,S]} Y_S Y_R\\
&=&\si(Y_S)\si(Y_R).
\end{eqnarray*}
\cqfd
The involution $\si$ extends to an anti-automorphism of the based quantum torus
\begin{equation}
\V_\ii := \A[Y_{V_i}^{\pm 1}\mid 1\le i\le r]
\end{equation}
which we still denote by $\si$.
Moreover, we have
\begin{Lem}\label{lemsigma}
For every indecomposable reachable rigid module $U$ we have
\[
\si(Y_U)= q^{[U,U] - \dim U} Y_U.
\]
\end{Lem}
\proof
By induction on the length of the mutation sequence, we may assume
that the result holds for all indecomposable direct summands $T_j$
of a $\CC_w$-maximal rigid module $T$. We then have to check that
it also holds for $U = T_k^* = \mu_k(T_k)$.
By Proposition~\ref{Propqmut}, we have
\[
q^{[T_k,T_k] - \dim T_k} Y_{T_k} \si(Y_{T_k^*}) =
q^{-[T_k^*,T_k]}\left(q^{[T'_k,T'_k]-\dim T'_k +1} Y_{T'_k} +
q^{[T''_k,T''_k]-\dim T''_k} Y_{T''_k}\right).
\]
Using that $\dim T'_k=\dim T''_k = \dim T_k + \dim T^*_k$, and
(\ref{relationhom}), this becomes
\[
Y_{T_k} \si(Y_{T_k^*}) = q^{[T^*_k,T_k^*] - \dim T^*_k + [T_k, T_k^*]}(Y_{T'_k} + q^{-1} Y_{T''_k})
= q^{[T^*_k,T_k^*] - \dim T^*_k} Y_{T_k} Y_{T_k^*},
\]
where the last equality comes again from Proposition~\ref{Propqmut}, since quantum mutations
are involutive. It follows that $\si(Y_{T_k^*}) = q^{[T^*_k,T_k^*] - \dim T^*_k} Y_{T_k^*}$,
as claimed, and this proves the lemma.
\cqfd
By Lemma~\ref{lemsigma}, $\si$ restricts to an automorphism of $\AA_\A(\CC_w)$, that we again
denote by $\si$. Moreover, for any quantum cluster monomial, that is for every element
$Y_U$ where $U$ is a reachable, non necessarily indecomposable, rigid module in $\CC_w$,
there holds
\begin{equation}\label{sigmaY}
\si(Y_U)= q^{[U,U] - \dim U} \, Y_U = q^{(\dimv\, U,\ \dimv\, U)/2 - \dim U} \, Y_U
= q^{N(\dimv\, U)} Y_U,
\end{equation}
where, for $\b\in Q_+$, $N(\b)$ is defined in (\ref{defN(b)}).
\section{Based quantum tori}
In this section, we fix again a reduced word $\ii:=(i_r,\ldots,i_1)$ for $w$,
and we set
\begin{equation}
\la_k = s_{i_1}\cdots s_{i_k}(\vpi_{i_k}),\qquad (k\le r).
\end{equation}
\subsection{Commutation relations}
The next lemma is a particular case of \cite[Theorem~10.1]{BZ}. We include
a brief proof for the convenience of the reader.
\begin{Lem}\label{LemDe}
For $1\le k<l \le r$, we have in $A_q(\g)$:
\[
\De_{\vpi_{i_k},\,\la_k} \De_{\vpi_{i_l},\,\la_l} =
q^{(\vpi_{i_k},\vpi_{i_l})-(\la_k,\la_l)}
\De_{\vpi_{i_l},\,\la_l} \De_{\vpi_{i_k},\,\la_k}.
\]
\end{Lem}
\proof
Since $k<l$, we have $\la_l = s_{i_1}\cdots s_{i_k}(\nu)$, where $\nu = s_{i_{k+1}}\cdots s_{i_l}(\vpi_{i_l})$.
For $x\in U_q(\n)$ and $y\in U_q(\n_-)$, we have
\[
x\cdot \De_{\vpi_{i_k},\vpi_{i_k}} = \eps(x) \De_{\vpi_{i_k},\vpi_{i_k}},
\qquad
\De_{\vpi_{i_l},\nu}\cdot y = \eps(y)\De_{\vpi_{i_l},\nu}.
\]
Therefore, using again \cite[Lemma 10.2]{BZ} as in the proof of Lemma~\ref{commute},
we obtain that
\[
\De_{\vpi_{i_k},\vpi_{i_k}} \De_{\vpi_{i_l},\nu} =
q^{(\vpi_{i_k},\vpi_{i_l}) -(\vpi_{i_k},\nu)}
\De_{\vpi_{i_l},\nu} \De_{\vpi_{i_k},\vpi_{i_k}}.
\]
Now, using $l-k$ times Lemma~\ref{lem:QMinIdRed}, we deduce from this equality that
\[
\De_{\vpi_{i_k},\, s_{i_1}\cdots s_{i_k}(\vpi_{i_k})}\, \De_{\vpi_{i_l},\, s_{i_1}\cdots s_{i_k}(\nu)} =
q^{(\vpi_{i_k},\vpi_{i_l}) -(\vpi_{i_k},\nu)}
\De_{\vpi_{i_l},\, s_{i_1}\cdots s_{i_k}(\nu)}\, \De_{\vpi_{i_k},\, s_{i_1}\cdots s_{i_k}(\vpi_{i_k})},
\]
and taking into account that
$(\vpi_{i_k},\,\nu) = (s_{i_1}\cdots s_{i_k}(\vpi_{i_k}),\,s_{i_1}\cdots s_{i_k}(\nu))= (\la_k,\,\la_l)$,
we get the claimed equality.
\cqfd
\begin{Lem}\label{qtor1}
For $1\le k<l \le r$, we have in $A_q(\n)$:
\[
D_{\vpi_{i_k},\,\la_k} D_{\vpi_{i_l},\,\la_l} =
q^{(\vpi_{i_k}-\la_k,\ \vpi_{i_l}+\la_l)}
D_{\vpi_{i_l},\,\la_l} D_{\vpi_{i_k},\,\la_k}.
\]
\end{Lem}
\proof This follows from Lemma~\ref{LemDe} using the same type of calculations as in the proof of Lemma~\ref{qcommute2}.
We leave the easy verification to the reader.
\cqfd
\begin{Lem}\label{qtor3}
For $1\le k<l \le r$, we have:
\[
(\vpi_{i_k}-\la_k,\ \vpi_{i_l}+\la_l) = [V_{\ii,k}, V_{\ii,l}] - [V_{\ii,l}, V_{\ii,k}].
\]
\end{Lem}
\proof
We have
\[
(\vpi_{i_k}-\la_k,\ \vpi_{i_l}+\la_l) = (\dimv\,V_{\ii,k},\,2\vpi_{i_l} -\dimv\,V_{\ii,l}).
\]
Since $[V_{\ii,k},\,V_{\ii,l}]^1 = 0$, using (\ref{CBform}) we have
\[
(\dimv\,V_{\ii,k},\,\dimv\,V_{\ii,l}) = [V_{\ii,k}, V_{\ii,l}] + [V_{\ii,l}, V_{\ii,k}].
\]
Hence,
\[
(\vpi_{i_k}-\la_k,\ \vpi_{i_l}+\la_l)
= 2(\dimv\,V_{\ii,k},\,\vpi_{i_l}) - [V_{\ii,k}, V_{\ii,l}] - [V_{\ii,l}, V_{\ii,k}],
\]
and we are reduced to show that
\begin{equation}\label{eqinterest}
(\dimv\,V_{\ii,k},\, \vpi_{i_l}) = [V_{\ii,k}, V_{\ii,l}],\qquad (k<l).
\end{equation}
By Proposition~\ref{propLV}, the right-hand side is equal to
\[
\sum_{m\le 0}\left( \sum\limits_{\substack{s<k^{(m)}\\ i_s = i_l}} (\b_{k^{(m)}},\,\b_s) + \de_{i_k i_l}\right).
\]
Since the left-hand side is equal to
$\sum_{m\le 0} (\b_{k^{(m)}}, \vpi_{i_l})$, it is enough to show that for every $m\ge 0$,
\begin{equation}\label{eqabove}
(\b_{k^{(m)}}, \vpi_{i_l}) = \sum\limits_{\substack{s<k^{(m)}\\ i_s = i_l}} (\b_{k^{(m)}},\,\b_s) + \de_{i_k i_l}.
\end{equation}
Let $t:=\max\{s < k^{(m)}\mid i_s = i_l\}$.
Eq. (\ref{eqabove}) can be rewritten
\begin{equation}\label{eqjustabove}
(\b_{k^{(m)}},\,\la_t) = \de_{i_k i_l}.
\end{equation}
But the left-hand side of (\ref{eqjustabove}) is equal to
\[
(s_{i_1}\cdots s_{i_{k^{(m)}-1}}(\a_{i_k}),\ s_{i_1}\cdots s_{i_t}(\vpi_{i_l}))
=
(s_{i_1}\cdots s_{i_{k^{(m)}-1}}(\a_{i_k}),\ s_{i_1}\cdots s_{i_{k^{(m)}-1}}(\vpi_{i_l}))
=
(\a_{i_k},\,\vpi_{i_l})
= \de_{i_k i_l},
\]
so (\ref{eqjustabove}) holds, and this proves (\ref{eqinterest}).
\cqfd
\subsection{An isomorphism of based quantum tori}
By Proposition~\ref{flagFq}, the quantum flag minors $D_{\vpi_{i_k},\la_k}=D(0,k)$
belong to $F_q(\n(w))$.
Let $\T_\ii$ be the $\A$-subalgebra of $F_q(\n(w))$ generated by the
$D_{\vpi_{i_k},\la_k}^{\pm 1}\ (1\le k\le r)$.
\begin{Lem}\label{qtor2}
The algebra $\T_\ii$ is a based quantum torus over $\A$.
\end{Lem}
\proof
By Lemma~\ref{qtor1}, the generators $D_{\vpi_{i_k},\la_k}$ pairwise $q$-commute.
So we only have to show that the monomials
\[
D^\aa := \prod_k^{\longrightarrow} D_{\vpi_{i_k},\la_k}^{a_k},\qquad (\aa=(a_1,\ldots,a_r)\in \N^r),
\]
are linearly independent over $\A$.
Suppose that we have a non-trivial relation
\[
\sum_{\aa} \ga_\aa(q) D^\aa = 0,
\]
for some $\ga_\aa(q)\in\A$. Dividing this equation (if necessary) by the largest power of $q-1$ which
divides all the coefficients $\ga_\aa(q)$, we may assume that at least one of these
coefficients is not divisible by $q-1$. Using \ref{specq1}, we see that by specializing
this identity at $q=1$ we get a non-trivial $\C$-linear relation between monomials in the
corresponding classical flag minors of $\C[N]$. But all these monomials belong to the dual semicanonical
basis of $\C[N]$ (see \cite[Corollary 13.3]{GLS}), hence they are linearly independent, a contradiction.
\cqfd
We note that Lemma~\ref{qtor2} also follows from \cite[Theorem 6.20]{Ki}.
\begin{Prop}\label{isomqtori}
The assignment $Y_{V_{\ii,k}}\mapsto D_{\vpi_{i_k},\,\la_k}\ (1\le k\le r)$ extends to an
algebra isomorphism from $\V_\ii$ to the based quantum torus $\T_\ii$.
\end{Prop}
\proof
By definition of the cluster algebra $\AA_\A(\CC_w)$, the elements
$Y_{V_{\ii,k}}^{\pm 1}$ generate a based quantum torus over $\A$,
with $q$-commutation relations
\[
Y_{V_{\ii,k}}Y_{V_{\ii,l}} = q^{[V_{\ii,k},V_{\ii,l}] - [V_{\ii,l},V_{\ii,k}]}
Y_{V_{\ii,l}}Y_{V_{\ii,k}}, \qquad (1\le k<l \le r).
\]
The proposition then follows immediately from Lemma~\ref{qtor1} and Lemma~\ref{qtor3}.
\cqfd
\section{Cluster structures on quantum coordinate rings}
\subsection{Cluster structures on quantum coordinate rings of unipotent subgroups}
Consider the following diagram of homomorphisms of $\A$-algebras:
\begin{equation}
\AA_\A(\CC_w) \longrightarrow \V_\ii \stackrel{\sim}{\longrightarrow}
\T_\ii \longrightarrow F_q(\n(w)).
\end{equation}
Here, the first arrow denotes the natural embedding given by
the (quantum) Laurent phenomenon \cite{BZ}, the second arrow is
the isomorphism of Proposition~\ref{isomqtori}, and the third
arrow is the natural embedding.
The composition $\k \colon \AA_\A(\CC_w) \longrightarrow F_q(\n(w))$ of these
maps is therefore injective.
Recall the notation $A_\A(\n(w))$ of \S\ref{specialq1}.
\begin{Prop}\label{surj}
The image $\k\left(\AA_\A(\CC_w)\right)$ contains $A_\A(\n(w))$.
\end{Prop}
\proof
Since $\k$ is an algebra map, it is enough to show that its image
contains the dual PBW-generators $E^*(\b_k)\ (1\le k \le r)$.
It was shown in \cite[\S13.1]{GLS} that there is an explicit sequence
of mutations of $\CC_w$-maximal rigid modules starting from $V_\ii$ and
ending in $T_\ii$. Each step of this sequence consists of the mutation
of a module $M[d^-,b]$ into a module $M[d,b^+]$, for some
$1\le b < d \le r$ with $i_b = i_d = i$.
The corresponding pair of short exact sequences is
\[
0 \to M[d^-,b] \to M[d^-,b^+] \oplus M[d,b] \to M[d,b^+] \to 0,
\]
\[
0 \to M[d,b^+] \to \bigoplus_{j\not = i} M[d^-(j),(b^-(j))^+]^{-a_{ij}} \to M[d^-,b] \to 0.
\]
Write
\[
T'_{bd} := M[d^-,b^+] \oplus M[d,b]
\]
and
\[
T''_{bd} := \bigoplus_{j\not = i} M[d^-(j),(b^-(j))^+]^{-a_{ij}}.
\]
Then, by Proposition~\ref{Propqmut}, in $\AA_\A(\CC_w)$ we have the
corresponding mutation relation:
\begin{equation}\label{elementary_mut}
Y_{M[d,b^+]} Y_{M[d^-,b]} = q^{[M[d,b^+],\,M[d^-,b]]}\left(q^{-1} Y_{T'_{bd}} + Y_{T''_{bd}}\right).
\end{equation}
Moreover,
\[
Y_{T'_{bd}} = q^{-\a(T'_{bd})} Y_{M[d^-,b^+]} Y_{M[d,b]},
\qquad
Y_{T''_{bd}} = q^{-\a(T''_{bd})} \prod_{j\not = i}^{\longrightarrow} Y_{M[d^-(j),(b^-(j))^+]}^{-a_{ij}}
\]
where
$\a(T'_{bd}) = [M[d^-,b^+],\, M[d,b]]$, and
\begin{align*}
\a(T''_{bd}) = & \sum_{j<k}a_{ij}a_{ik} [M[d^-(j),(b^-(j))^+],M[d^-(k),(b^-(k))^+]] \\
&+\ \sum_{j\not = i}{-a_{ij}\choose2} [M[d^-(j),(b^-(j))^+],M[d^-(j),(b^-(j))^+]].
\end{align*}
Note that for $1\le k \le l \le r$, and $i_k = i_l =j$, one has
\begin{equation}
\dimv\,M[l,k] = \mu(k^-,j) - \mu(l,j).
\end{equation}
Therefore, using Proposition~\ref{propcalculhom} and
the notation of Proposition~\ref{Tsystem}, we see that
\begin{align*}
[M[d,b^+],\,M[d^-,b]] &= (\dimv\, V_{d},\, \dimv\, M[d^-,b]) - (\dimv\, M[d^-,b])_i\\[2mm]
&= (\vpi_i- \mu(d,i),\, \mu(b^-,i)-\mu(d^-,i)) - (\vpi_i,\, \mu(b^-,i)-\mu(d^-,i))\\[2mm]
&= - (\mu(d,i),\, \mu(b^-,i)-\mu(d^-,i))\\[2mm]
&= - A.
\end{align*}
Similarly we obtain that
\[
\a(T'_{bd}) = -B,\qquad \a(T''_{bd}) = -C.
\]
Therefore (\ref{elementary_mut}) can be rewritten as
\begin{equation}\label{eqYsystem}
q^A Y_{M[d,b^+]} Y_{M[d^-,b]} = q^{-1+B} Y_{M[d^-,b^+]}Y_{M[d,b]} + q^{C}\prod_{j\not = i}^{\longrightarrow} Y_{M[d^-(j),(b^-(j))^+]}^{-a_{ij}}.
\end{equation}
We observe that this has exactly the same form as (\ref{eqTsystem}).
By definition of $\k$ one has
\[
\k(Y_{V_{\ii,k}}) = \k(Y_{M[k,k_{\min}]}) = D(0,k).
\]
Hence all the initial variables of the systems of equations (\ref{eqYsystem})
and (\ref{eqTsystem}) are matched under the algebra homomorphism $\k$.
Therefore, by induction, it follows that $\k(Y_{M[d,b^+]}) = D(b,d)$ for
every $b<d$. In particular, $\k(Y_{M[k,k]}) = D(k^-,k)=E^*(\b_k)$,
by Proposition~\ref{PBWminors}.
\cqfd
\begin{Prop}\label{rank}
The algebra $\AA_\A(\CC_w)$ is a $Q_+$-graded free $\A$-module,
with homogeneous components of finite rank.
Moreover,
\[
\rk \left(\AA_\A(\CC_w)_\a\right) = \rk \left(A_\A(\n(w))_\a\right), \qquad (\a\in Q_+).
\]
\end{Prop}
\proof
The $\A$-module $\V_\ii$ is free, hence
its submodule $\AA_\A(\CC_w)$ is projective, and therefore free
since $\A$ is a principal ideal domain.
It has a natural $Q_+$-grading given by
\[
\deg Y_R := \dimv\, R
\]
for every indecomposable reachable rigid module $R$ in $\CC_w$.
The rank of $\AA_\A(\CC_w)_\a$ is equal to the dimension of the
$\C$-vector space $\C\otimes_\A \AA_\A(\CC_w)_\a$, which is
equal to the dimension of the corresponding
homogeneous component of the (classical) cluster algebra
$\C\otimes_\Z \AA(\CC_w)$.
Now, by \cite{GLS}, this is equal to the dimension of
$\C[N(w)]_\a$, that is, by \S\ref{specialq1}, to the rank of
$A_\A(\n(w))_\a$.
\cqfd
Let $\AA_{\Q(q)}(\CC_w) := \Q(q)\otimes_\A \AA_{\A}(\CC_w)$.
The $\A$-algebra homomorphism $\k$ naturally extends to a $\Q(q)$-algebra
homomorphism from $\AA_{\Q(q)}(\CC_w)$ to $F_q(\n(w))$,
which we continue to denote by~$\k$.
We can now prove our main result:
\begin{Thm}\label{main}
$\k$ is an isomorphism from the quantum cluster algebra $\AA_{\Q(q)}(\CC_w)$ to
the quantum coordinate ring $A_q(\n(w))$.
\end{Thm}
\proof
By construction, $\k$ is injective. By Proposition~\ref{surj},
the image of $\k$ contains $A_q(\n(w))$.
Finally, since $\k$ preserves the $Q_+$-gradings, and since the
homogeneous components of $A_{\Q(q)}(\CC_w)$ and
$A_q(\n(w))$ have the same dimensions, by Proposition~\ref{rank},
we see that
\[
\k\left(\AA_{\Q(q)}(\CC_w)\right) = A_q(\n(w)).
\]
\cqfd
The following Corollary proves the claim made at the end of \S\ref{skewfield}.
\begin{Cor}\label{minorpol}
All quantum minors $D(b,d)\ (1\le b<d\le r)$ belong to $A_q(\n(w))$,
and therefore are polynomials in the dual PBW generators $E^*(\b_k)=D(k^-,k)=Y_{M_k}$.
\end{Cor}
\proof As shown in the proof of Proposition~\ref{surj}, $D(b,d) = \k(Y_{M[d,b^+]})$,
and so belongs to $\k\left(\AA_{\Q(q)}(\CC_w)\right) = A_q(\n(w))$, by
Theorem~\ref{main}.
\cqfd
The following Corollary is a $q$-analogue of \cite[Theorem 3.2 (i) (ii)]{GLS}.
\begin{Cor}
\begin{itemize}
\item[(a)] $\AA_{\Q(q)}(\CC_w)$ is an iterated skew polynomial ring.
\item[(b)] The set $\{Y_M(\aa):= Y_{M_1}^{a_1}\cdots Y_{M_r}^{a_r} \mid \aa =(a_1,\ldots, a_r)\in \N^r\}$
is a $\Q(q)$-linear basis of $\AA_{\Q(q)}(\CC_w)$.
\end{itemize}
\end{Cor}
\proof
This follows via the isomorphism $\k$ from known properties of $A_q(\n(w))$.
\cqfd
\begin{Cor}
Let $\g$ be a simple Lie algebra of type $A_n, D_n, E_n$\,, and let $\n$ be
a maximal nil\-po\-tent subalgebra of $\g$. Let $\L$ be the corresponding
preprojective algebra of type $A_n, D_n, E_n$. Then
\begin{itemize}
\item[(a)] The quantum cluster algebra $\AA_{\Q(q)}(\md\L)$
is isomorphic to the quantum enveloping algebra $U_q(\n)$.
\item[(b)] In this isomorphism, the Chevalley generators $e_i\ (1\le i\le n)$
are identified with the quantum cluster variables $Y_{S_i}$ attached to
the simple $\L$-modules $S_i$.
\end{itemize}
\end{Cor}
\proof
(a) Take $w$ to be the longest element $w_0$ of the Weyl group. Hence
$\n(w_0) = \n$, so
\[
A_q(\n(w_0)) = A_q(\n) \cong U_q(\n),
\]
by Proposition~\ref{isomPhi}. On the other hand $\CC_{w_0} = \md(\L)$,
and this proves (a). Property (b) is then obvious.
\cqfd
We believe that Theorem~\ref{main} can be strengthened as follows.
\begin{Conj}\label{integral}
The map $\k$ restricts to an isomorphism from the integral form $\AA_{\A}(\CC_w)$ of the
quantum cluster algebra to the integral
form $A_\A(\n(w))$ of the quantum coordinate ring.
\end{Conj}
\subsection{Example}\label{exampleA3}
We illustrate our arguments on a simple example. We take $\g$ of type $A_3$
and $w=w_0$. We choose the reduced word $\ii = (i_6,i_5,i_4,i_3,i_2,i_1) = (1,2,3,1,2,1)$.
Therefore
\[
\b_1=\a_1,\ \
\b_2 = \a_1+\a_2,\ \
\b_3 = \a_2,\ \
\b_4 = \a_1+\a_2+\a_3,\ \
\b_5 =\a_2+\a_3,\ \
\b_6 = \a_3.
\]
Using the convention of \cite[\S2.4]{GLS} for visualizing $\L$-modules,
we can represent the summands of $V_\ii$ as follows:
\begin{align*}
V_1 &= {\bsm1\esm} &
V_2 &= {\bsm1\\&2\esm} &
V_3 &= {\bsm&2\\1\esm} \\[2mm]
V_4 &= {\bsm1\\&2\\&&3\esm} &
V_5 &= {\bsm&2\\1&&3\\&2\esm} &
V_6 &= {\bsm&&3\\&2\\1\esm}
\end{align*}
and the summands of $M_\ii$ as follows:
\begin{align*}
M_1 &= {\bsm1\esm} &
M_2 &= {\bsm1\\&2\esm} &
M_3 &= {\bsm2\esm} \\[2mm]
M_4 &= {\bsm1\\&2\\&&3\esm} &
M_5 &= {\bsm2\\&3\esm} &
M_6 &= {\bsm3\esm}
\end{align*}
The sequence of mutations of \cite[\S13]{GLS} consists here in 4 mutations.
\medskip\noindent
\emph{Mutation 1:} One mutates at $V_1$ in the maximal rigid module $V_\ii$.
One has
\[
\mu_1(V_1) = \mu_1(M[3^-,1]) = M[3,1^+] = M[3,3]=M_3.
\]
This gives rise to the two short exact sequences (which can be read from the
graph $\G_\ii$):
\[
0\to V_1 \to V_3 \to M_3 \to 0,\qquad 0\to M_3 \to V_2 \to V_1\to 0.
\]
By Proposition~\ref{mutY}, we thus have
\[
Y_{M_3}Y_{V_1} = q^{[M_3,V_1]}\left(q^{-1} Y_{V_3} + Y_{V_2}\right) = q^{-1} Y_{V_3} + Y_{V_2},
\]
since $[M_3,V_1] = 0$. Using the notation $M[l,k]$, this can be rewritten
\begin{equation}\label{eqY1}
Y_{M[3,3]}Y_{M[1,1]} = q^{-1}Y_{M[3,1]} + Y_{M[2,2]}.
\end{equation}
\medskip\noindent
\emph{Mutation 2:} One mutates at $V_3$ in the maximal rigid module $\mu_1(V_\ii)$.
One has
\[
\mu_3(V_3) = \mu_3(M[6^-,1]) = M[6,1^+] = M[6,3] = {\bsm&3\\2\esm} .
\]
This gives rise to the two short exact sequences (which can be read from the
graph $\mu_1(\G_\ii)$):
\[
0\to V_3 \to M_3\oplus V_6 \to M[6,3] \to 0,\qquad 0\to M[6,3] \to V_5 \to V_3\to 0.
\]
By Proposition~\ref{mutY}, we thus have
\[
Y_{M[6,3]}Y_{V_3} = q^{[M[6,3],V_3]}\left(q^{-1} Y_{M_3\oplus V_6} + Y_{V_5}\right)
= q^{-1} Y_{M_3}Y_{V_6} + Y_{V_5},
\]
since $[M[6,3],V_3] = 0$, and $[M_3,V_6] = 0$.
Using the notation $M[l,k]$, this can be rewritten
\begin{equation}\label{eqY2}
Y_{M[6,3]}Y_{M[3,1]} = q^{-1}Y_{M[3,3]}Y_{M[6,1]} + Y_{M[5,2]}.
\end{equation}
\medskip\noindent
\emph{Mutation 3:} One mutates at $V_2$ in the maximal rigid module $\mu_3\mu_1(V_\ii)$.
One has
\[
\mu_2(V_2) = \mu_2(M[5^-,2]) = M[5,2^+] = M[5,5] = M_5.
\]
This gives rise to the two short exact sequences (which can be read from the
graph $\mu_3\mu_1(\G_\ii)$):
\[
0\to V_2 \to V_5 \to M_5 \to 0,\qquad 0\to M_5 \to M_3\oplus V_4 \to V_2\to 0.
\]
By Proposition~\ref{mutY}, we thus have
\[
Y_{M_5}Y_{V_2} = q^{[M_5,V_2]}\left(q^{-1} Y_{V_5} + Y_{M_3\oplus V_4}\right)
= Y_{V_5} + q Y_{M_3}Y_{V_4},
\]
since $[M_5,V_2] = 1$, and $[M_3,V_4] = 0$.
Using the notation $M[l,k]$, this can be rewritten
\begin{equation}\label{eqY3}
Y_{M[5,5]}Y_{M[2,2]} = Y_{M[5,2]} + qY_{M[3,3]}Y_{[4,4]}.
\end{equation}
\medskip\noindent
\emph{Mutation 4:} One mutates at $M_3$ in the maximal rigid module $\mu_2\mu_3\mu_1(V_\ii)$.
One has
\[
\mu_1(M_3) = \mu_1(M[6^-,3]) = M[6,3^+] = M[6,6] = M_6.
\]
This gives rise to the two short exact sequences (which can be read from the
graph $\mu_2\mu_3\mu_1(\G_\ii)$):
\[
0\to M_3 \to M[6,3] \to M_6 \to 0,\qquad 0\to M_6 \to M_5 \to M_3\to 0.
\]
By Proposition~\ref{mutY}, we thus have
\[
Y_{M_6}Y_{M_3} = q^{[M_6,M_3]}\left(q^{-1} Y_{M[6,3]} + Y_{M_5}\right)
= q^{-1} Y_{M[6,3]} + Y_{M_5},
\]
since $[M_6,M_3] = 0$.
Using the notation $M[l,k]$, this can be rewritten
\begin{equation}\label{eqY4}
Y_{M[6,6]}Y_{M[3,3]} = q^{-1}Y_{M[6,3]} + Y_{M[5,5]}.
\end{equation}
Now, in $A_q(\n)$ we have the following quantum $T$-system given by Proposition~\ref{Tsystem}:
\begin{align}
D(1,3)D(0,1) &= q^{-1}D(0,3) + D(0,2),\label{eqT1}\\[2mm]
D(1,6)D(0,3) &= q^{-1}D(1,3)D(0,6) + D(0,5),\label{eqT2}\\[2mm]
D(2,5)D(0,2) &= D(0,5) + qD(1,3)D(0,4),\label{eqT3}\\[2mm]
D(3,6)D(1,3) &= q^{-1}D(1,6) + D(2,5).\label{eqT4}
\end{align}
By definition, the homomorphism $\k \colon \AA_{\Q(q)}(\md\L) \to F_q(\n)$ satisfies
\begin{align*}
&\k(Y_{M[1,1]}) = D(0,1),\quad
\k(Y_{M[3,1]}) = D(0,3),\quad
\k(Y_{M[6,1]}) = D(0,6),
\\[2mm]
&\k(Y_{M[2,2]}) = D(0,2),\quad
\k(Y_{M[5,2]}) = D(0,5),\quad
\k(Y_{M[4,4]}) = D(0,4).
\end{align*}
Thus, comparing (\ref{eqY1}) and (\ref{eqT1}) we see that $\k(Y_{M[3,3]})=D(1,3)$.
Next, comparing (\ref{eqY2}) and (\ref{eqT2}) we see that $\k(Y_{M[6,3]})=D(1,6)$.
Next, comparing (\ref{eqY3}) and (\ref{eqT3}) we see that $\k(Y_{M[5,5]})=D(2,5)$.
Finally, comparing (\ref{eqY4}) and (\ref{eqT4}) we see that $\k(Y_{M[6,6]})=D(3,6)$.
So in particular, we have
\[
\k(Y_{M_k}) = D(k^-,k) = E^*(\b_k),\qquad (1\le k \le 6).
\]
This shows that $\k\left(\AA_{\Q(q)}(\md\L)\right)$ contains $A_q(\n)$.
Comparing dimensions of $Q_+$-homogeneous components, as in Proposition~\ref{rank},
shows that $\k\left(\AA_{\Q(q)}(\md\L)\right) = A_q(\n) \cong U_q(\n)$.
In this isomorphism, the Chevalley generators $e_1, e_2, e_3$ of $U_q(\n)$
correspond to the quantum cluster variables $Y_{M_1}, Y_{M_3}, Y_{M_6}$,
respectively.
The quantum Serre relation $e_1e_3 = e_3e_1$ corresponds to the fact that
\[
[M_1, M_6] = [M_6,M_1] = 0,\qquad [M_1, M_6]^1 = 0.
\]
To recover, for instance, the relation $e_1^2e_2 -(q+q^{-1})e_1e_2e_1 + e_2e_1^2 = 0$,
we start from the mutation relations
\[
Y_{M_3}Y_{M_1} = q^{-1} Y_{V_3} + Y_{V_2},
\qquad
Y_{M_1}Y_{M_3} = q^{-1} Y_{V_2} + Y_{V_3}.
\]
The first one is our first mutation above, and the second one is the mutation back,
in the opposite direction (mutations are involutive).
Eliminating $Y_{V_2}$ between these two equations, we get
\[
Y_{M_1}Y_{M_3} - q^{-1}Y_{M_3}Y_{M_1} = (1-q^{-2})Y_{V_3}.
\]
Since $[M_1,V_3]^1 = 0$, $[M_1,V_3] = 1$, and $[V_3,M_1]= 0$, we have
\[
Y_{M_1}Y_{V_3} = qY_{V_3}Y_{M_1},
\]
which yields
\[
Y_{M_1}^2 Y_{M_3} - q^{-1}Y_{M_1}Y_{M_3}Y_{M_1}
= qY_{M_1}Y_{M_3}Y_{M_1} - Y_{M_3}Y_{M_1}^2,
\]
as expected.
\subsection{Canonical basis of $\AA_{\Q(q)}(\CC_w)$}\label{canonbasACw}
By \cite[\S4.7]{Ki}, the subalgebra $U_q(\n(w))$ is spanned
by a subset $\B^*(w)$ of the dual canonical basis $\B^*$ of $U_q(\n)$.
Using the isomorphism $\k\colon \AA_{\Q(q)}(\CC_w) \stackrel{\sim}{\to} A_q(\n(w)) \cong U_q(\n(w))$,
we can pull back $\B^*(w)$ and obtain a $\Q(q)$-basis of $\AA_{\Q(q)}(\CC_w)$,
which we shall call the \emph{canonical basis of $\AA_{\Q(q)}(\CC_w)$}, and denote
by $\BB(w) = \{b(\aa) \mid \aa\in\N^r\}$.
It may be characterized as follows.
Recall the involution $\si$ of $\AA_\A(\CC_w)$ defined in \S\ref{sectinvolsigma}.
We will also denote by $\si$ its extension to $\AA_{\Q(q)}(\CC_w)$.
\begin{Prop}\label{caracter}
For $\aa=(a_1,\ldots,a_r)\in\N^r$, the vector $b(\aa)$ is uniquely determined by the following
conditions:
\begin{itemize}
\item[(a)] the expansion of $b(\aa)$ on the basis $\{Y_M(\cc)\mid \cc\in \N^r\}$ is of the
form
\[
b(\aa) = Y_M(\aa) + \sum_{\cc\not = \aa} \ga_{\aa,\cc}(q)\, Y_M(\cc)
\]
where $\ga_{\aa,\cc}(q) \in q^{-1}\Q[q^{-1}]$ for every $\cc\not = \aa$;
\item[(b)] Let $\b(\aa) := \sum_{1\le k\le r}a_k \b_k$. Then $\si(b(\aa)) = q^{N(\b(\aa))}b(\aa)$.
\end{itemize}
\end{Prop}
\proof
This is a restatement in our setup of \cite[Theorem 4.26]{Ki}.
The same result was previously obtained in \cite{La} in a particular case.
\cqfd
It follows from (\ref{sigmaY}) that all quantum cluster monomials satisfy
condition (b) of Proposition~\ref{caracter}.
This is similar to \cite[Proposition 10.9 (2)]{BZ}.
Unfortunately, it is not easy
to prove that quantum cluster monomials satisfy (a), so one can only conjecture
\begin{Conj}\label{conj}
All quantum cluster monomials $Y_R$, where $R$ runs over the set
of reachable rigid modules in $\CC_w$, belong to $\BB(w)$.
\end{Conj}
It follows from the original work of Berenstein and Zelevinsky \cite{BZ0}
that the conjecture holds in the prototypical Example~\ref{exampleA3},
namely, for $\g= \Sl_4$ the dual canonical basis of $U_q(\n)$ is equal to the set of
quantum cluster monomials. In this case, there are 14 clusters and
12 cluster variables (including the frozen ones), which all are
unipotent quantum minors.
We note that the conjecture is satisfied when $R=M[b,a]$ is one
of the modules occuring in the quantum determinantal identities.
Indeed, $Y_{M[b,a]}$ is then a quantum minor, and belongs to $\BB(w)$
by Proposition~\ref{qmincan}.
It is proved in \cite{La,La2} that the conjecture holds for all quantum
cluster variables when $\AA_{\Q(q)}(\CC_w)$ is associated with the
Kronecker quiver or a Dynkin quiver of type $A$, and $w$ is the square of a Coxeter element.
Conjecture~\ref{conj} would imply the open orbit conjecture
of \cite[\S18.3]{GLS} for reachable rigid modules, by specializing~$q$ to $1$.
It also appears as Conjecture~1.1 (2) in \cite{Ki}.
\subsection{Quantum coordinate rings of matrices}
Let $\g$ be of type $A_{n}$.
Let $j$ be a fixed integer between $1$ and $n$, and set $k=n+1-j$.
Let $w$ be the Weyl group element with
reduced decomposition
\[
w= (s_j s_{j-1} \cdots s_1)(s_{j+1} s_j \cdots s_2)\cdots (s_n s_{n-1} \cdots s_{k}).
\]
We denote by $\ii=(i_{jk},i_{jk-1},\ldots, i_1)$ the corresponding word.
It is well known (see \cite{MC}) that for this particular choice
of $\g$ and $w$, $U_q(\n(w))$ is isomorphic to
the quantum coordinate ring $A_q(\Mat(k,j))$ of the space of $k\times j$-matrices.
In this case, $\CC_w$ is the subcategory of $\mod(\L)$ generated by the indecomposable
projective $P_k$ with simple top $S_k$.
For $1\le a \le k$ and $1\le b \le j$,
the module $P_k$ has a unique quotient $X_{ab}$
with dimension vector
$\dimv\, X_{ab} = \a_a+\a_{a+1}+\cdots + \a_{n+1-b}$.
It is not difficult to check that
\[
M_\ii = \bigoplus\limits_{\substack{1\le a\le k\\[1mm] 1\le b\le j}} X_{ab}.
\]
Then, setting $x_{ab} := Y_{X_{ab}} \in \AA_q(\CC_w)$, we have that the
elements $x_{ab}$ satisfy the defining relations of $A_q(\Mat(k,j))$,
namely
\begin{align}
x_{ab}x_{ac} & = qx_{ac}x_{ab},&(b <c),\\
x_{ac}x_{bc} & = qx_{bc}x_{ac},&(a <b),\\
x_{ab}x_{cd} & = x_{cd}x_{ab}, &(a<c,\ b>d),\\
x_{ab}x_{cd} & = x_{cd}x_{ab} + (q-q^{-1})x_{ad}x_{cb}, &(a<c,\ b<d).
\end{align}
Thus, Theorem~\ref{main} gives immediately
\begin{Cor}
The quantum coordinate ring $A_q(\Mat({k,j}))$ is isomorphic to the
quantum cluster algebra $\AA_{\Q(q)}(\CC_w)$.
\end{Cor}
\begin{example}\label{ex11_8}
{\rm
The quantum coordinate ring $A_q(\Mat({3,3}))$ is isomorphic to
$U_q(\n(w))$ for $\g$ of type $A_5$ and $w= s_3s_2s_1s_4s_3s_2s_5s_4s_3$.
The category $\CC_w$ has finite representation type $D_4$.
Taking $\ii = (3,2,1,4,3,2,5,4,3)$, the direct indecomposable summands
of $V_\ii$ are
\begin{align*}
V_1 &= {\bsm3\esm} &
V_2 &= {\bsm3\\&4\esm} &
V_3 &= {\bsm3\\&4\\&&5\esm} \\[2mm]
V_4 &= {\bsm&3\\2\esm} &
V_5 &= {\bsm&3\\2&&4\\&3\esm} &
V_6 &= {\bsm&3\\2&&4\\&3&&5\\&&4\esm} \\[2mm]
V_7 &= {\bsm&&3\\&2\\1\esm} &
V_8 &= {\bsm&&3\\&2&&4\\1&&3\\&2\esm} &
V_9 &= {\bsm&&3\\&2&&4\\1&&3&&5\\&2&&4\\&&3\esm}
\end{align*}
The direct indecomposable summands
of $M_\ii$ are
\begin{align*}
M_1 &= {\bsm3\esm} &
M_2 &= {\bsm3\\&4\esm} &
M_3 &= {\bsm3\\&4\\&&5\esm} \\[2mm]
M_4 &= {\bsm&3\\2\esm} &
M_5 &= {\bsm&3\\2&&4\esm} &
M_6 &= {\bsm&3\\2&&4\\&&&5\esm} \\[2mm]
M_7 &= {\bsm&&3\\&2\\1\esm} &
M_8 &= {\bsm&&3\\&2&&4\\1&&\esm} &
M_9 &= {\bsm&&3\\&2&&4\\1&&&&5\esm}
\end{align*}
In the identification of $A_q(\Mat({3,3}))$ with $\AA_{\Q(q)}(\CC_w)$, we have
\begin{align*}
x_{11}& = Y_{M_9},& x_{12} & = Y_{M_8}, &x_{13} &= Y_{M_7}, \\[2mm]
x_{21}& = Y_{M_6},& x_{22} & = Y_{M_5}, &x_{23} &= Y_{M_4}, \\[2mm]
x_{31}& = Y_{M_3},& x_{32} & = Y_{M_2}, &x_{33} &= Y_{M_1}.
\end{align*}
The quantum cluster algebra $\AA_{\Q(q)}(\CC_w)$ has finite cluster type $D_4$.
Its ring of coefficients is the skew polynomial ring in the variables
$Y_{V_3}, Y_{V_6}, Y_{V_7}, Y_{V_8}, Y_{V_9}$.
It has 16 non frozen cluster variables, namely the 9 canonical generators $Y_{M_i}$,
the 4 unipotent quantum minors $Y_{T_i}$ attached to the modules:
\begin{align*}
T_1 &= {\bsm&&3\\&2&&4\\1&&3&&5\esm}&
T_2 &= {\bsm&&3\\&2&&4\\1&&3&&5\\&2\esm} &
T_3 &= {\bsm&&3\\&2&&4\\1&&3&&5\\&&&4\esm}&
T_4 &= {\bsm&&3\\&2&&4\\1&&3&&5\\&2&&4\esm}
\end{align*}
and the elements $Y_{U_i}$ attached to the 2 modules
\begin{align*}
U_1 &= {\bsm&&33\\&22&&44\\1&&3&&5\esm}&
U_2 &= {\bsm&&3\\&2&&4\\1&&33&&5\\&2&&4\esm}
\end{align*}
with dimension vector $\a_1+2\a_2+3\a_3+2\a_4+\a_5$.
These cluster variables form 50 clusters.
}
\end{example}
\subsection{Quantum coordinate rings of open cells in partial
flag varieties}
In this section, we assume that $\g$ is a simple Lie algebra of simply-laced
type.
We briefly review some classical material, using the notation of \cite{GLSflag,GLS}.
Let $G$ be a simple simply connected complex algebraic group
with Lie algebra $\g$.
Let $H$ be a maximal torus of $G$, and $B, B^-$ a pair of opposite
Borel subgroups containing $H$ with unipotent radicals $N, N^-$.
We denote by $x_i(t)\ (i\in I,\, t\in\C)$ the simple root subgroups
of $N$, and by $y_i(t)$ the corresponding simple root subgroups of
$N^-$.
We fix a non-empty subset $J$ of $I$ and
we denote its complement by $K=I\setminus J$.
Let $B_K$ be the standard parabolic
subgroup of $G$ generated by $B$
and the one-parameter subgroups
\[
y_k(t),\qquad (k\in K,\,t\in\C).
\]
We denote by $N_K$ the unipotent radical of $B_K$.
Similarly, let $B_K^-$ be the parabolic subgroup of $G$ generated by $B^-$
and the one-parameter subgroups
\[
x_k(t),\qquad (k\in K,\,t\in\C).
\]
The projective variety $B_K^-\backslash G$ is called
a \emph{partial flag variety}.
The natural projection map
$$
\pi\colon G \ra B_K^-\backslash G
$$
restricts to an embedding of $N_K$ into
$B_K^-\backslash G$ as a dense open subset.
Let $W_K$ be the subgroup of the Weyl group $W$ generated by the
reflections $s_k\ (k\in K)$. This is a finite Coxeter group and
we denote by $w_0^K$ its longest element. The longest element
of $W$ is denoted by $w_0$.
It is easy to check that
\begin{equation}
N_K = N\left(w_0w_0^K\right),
\end{equation}
see \cite[Lemma 17.1]{GLS}.
It follows that $\C[N\left(w_0w_0^K\right)]$ can be identified with the
coordinate ring of the affine open chart $\O_K:=\pi(N_K)$ of $B_K^-\backslash G$.
Therefore, $A_q(\n(w_0w_0^K))$ can be regarded as the quantum coordinate
ring $A_q(\O_K)$ of $\O_K$, and Theorem~\ref{main} implies:
\begin{Cor}
The quantum coordinate ring $A_q(\O_K)$ is isomorphic to the
quantum cluster algebra $\AA_{\Q(q)}(\CC_{w_0w_0^K})$.
\end{Cor}
\begin{example}
{\rm Take $G=SL(6)$, and $J=\{3\}$.
Then $B_K^-\backslash G$ is the Grassmannian $\Gr(3,6)$ of $3$-dimensional subspaces
of $\C^6$, and $N_K = N(w_0w_0^K)$, where $w=w_0w_0^K$ is as in Example~\ref{ex11_8}.
Here, $N(w)$ can be identified with the open cell $\O$ of $\Gr(3,6)$ given by
the non-vanishing of the first Pl\"ucker coordinate.
The quantum cluster algebra $\AA_{\Q(q)}(\CC_w)$ of
Example~\ref{ex11_8} can be regarded as the quantum coordinate ring of $\O$.
}
\end{example}
\begin{example}
{\rm Take $G=SO(8)$, of type $D_4$. We label the vertices of the Dynkin diagram
from $1$ to $4$ so that the central node is $3$. Let $J=\{4\}$.
Then $B_K^-\backslash G$ is a smooth projective quadric ${\mathcal Q}$ of dimension 6,
and $N_K$ can be regarded as an open cell $\O$ in ${\mathcal Q}$.
Here $N_K=N(w)$ where $w = w_0w_0^K = s_4s_3s_1s_2s_3s_4$.
It is easy to check that the elements $Y_{M_k}\ (1\le k \le 6)$
satisfy
\begin{align}
Y_{M_j}Y_{M_i} & = q Y_{M_i}Y_{M_j}, \qquad (i<j,\ i+j\not = 7),\\
Y_{M_4}Y_{M_3} & = Y_{M_3}Y_{M_4},\\
Y_{M_5}Y_{M_2} & = Y_{M_2}Y_{M_5} + (q-q^{-1})Y_{M_3}Y_{M_4},\\
Y_{M_6}Y_{M_1} & = Y_{M_1}Y_{M_6} + (q-q^{-1})(Y_{M_2}Y_{M_5} - q^{-1} Y_{M_3}Y_{M_4}),
\end{align}
and that this is a presentation of the quantum coordinate ring $A_q(\O)$.
This shows that $A_q(\O)$ is isomorphic to the quantum coordinate ring of the
space of $4\times 4$-skew-symmetric matrices, introduced by Strickland \cite{St}.
The category $\CC_w$ has finite representation type $A_1\times A_1$.
Hence, there are 4 cluster variables $Y_{M_1}$, $Y_{M_2}$, $Y_{M_5}$, $Y_{M_6}$, together
with 4 frozen ones, namely, $Y_{V_3}=Y_{M_3}$, $Y_{V_4}=Y_{M_4}$, and
\[
Y_{V_5} = Y_{M_2}Y_{M_5} - q^{-1}Y_{M_3}Y_{M_4},\qquad
Y_{V_6} = Y_{M_1}Y_{M_6} -q^{-1}Y_{M_2}Y_{M_5} + q^{-2} Y_{M_3}Y_{M_4}.
\]
Observe that $Y_{V_6}$ coincides with the quantum Pfaffian of \cite{St}.
There are 4 clusters
\[
\{Y_{M_1},\ Y_{M_2}\},\quad
\{Y_{M_1},\ Y_{M_5}\},\quad
\{Y_{M_6},\ Y_{M_2}\},\quad
\{Y_{M_6},\ Y_{M_5}\}.
\]
Note that since all quantum cluster variables belong to the basis
$\{Y_M(\cc)\mid \cc\in \N^r\}$, Conjecture~\ref{conj} is easily verified
in this case.
}
\end{example}
\subsection*{Acknowledgments}
We thank the Hausdorff Center for Mathematics in Bonn for organizing
a special trimester on representation theory in the spring of 2011,
during which this paper was finalized.
The first author acknowledges support from
the grants PAPIIT IN117010-2 and CONACYT 81948,
and thanks the Max Planck Institute in Bonn for hospitality.
The second author is grateful to the University of Bonn for
several invitations.
The third author also thanks the
Transregio SFB/TR 45 for support.
| 212,826
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Newswise — After a national search, Children’s Hospital Los Angeles has named Alexandra Carter, MBA, CFRE, Senior Vice President and Chief Development Officer effective September 25.
Carter will direct the hospital’s fundraising and development programs, providing strategic leadership to support CHLA’s mission and to leverage philanthropy to enhance the hospital’s capacity to innovate and transform pediatric medicine. Carter will also support the Advancement Committee of the Board of Trustees, which determines fundraising priorities for the hospital.
“Alexandra will be an incredible asset to CHLA,” says Children’s Hospital Los Angeles President and CEO Paul S. Viviano. “Her experience as a leader with nonprofit organizations, combined with her impressive accomplishments as a fundraiser, will allow her to provide strategic guidance to our Foundation department to support CHLA’s stature as a top-ranked pediatric academic medical center.”
Carter most recently served as a fundraising advisor to Verity Health System, Albertina Kerr Centers and the Boys and Girls Clubs of Metro Los Angeles. Previous to her consulting for these nonprofit organizations, she served as Chief Development Officer, Academic Institutes, for Cedars-Sinai Medical Center during a $600 million campaign. She oversaw frontline fundraising teams including major gifts, planned giving and foundation and corporate relations, and served on the institution’s leadership team for its partnership with the Women’s Heart Alliance, a national advocacy organization.
Prior to joining Cedars-Sinai, Carter served as Vice President of Public Affairs and Development at John Wayne Cancer Institute at Saint John’s Health Center in Santa Monica, California. She has also held leadership roles at Pepperdine University and at the American Red Cross in Santa Barbara, California.
Carter earned a bachelor’s degree in Communication and Linguistics from the University of California, Santa Barbara, and a master’s degree in Business Administration from Pepperdine University. She currently serves on the Executive Council for the Women’s Leadership Network of Jewish Vocational Services in Los Angeles and recently completed her term as a member of the Board of Directors at the National Association of Cancer Center Development Officers. Carter also served on the Board of Directors for the literacy and service charity Milk+Bookies, and has volunteered in support of dog rescue initiatives. () and our research blog ().
| 383,984
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Snail Bob is a series of games that tells stories about an interesting snail named Bob. Here’s some information about the Snail Bob games.
Snail Bob Overview
This snail game gets good reviews online at Snail Bob. The site has 8 games that have different themes to it. For example, the first game is about Bob trying to get through construction on his way home. There are many different obstacles that you have to navigate through, and you also have to try out a lot of different tools. It’s definitely a puzzle type game where you have lots of levers and counter levers, and a lot of puzzle navigation to get past. It makes you think carefully in order to make your way through everything. It’s an intellectually challenging game, and it emphasizes patience. By the end of it, you might just have the patience of a snail.
Making Your Way Through the Game
The second game is about bob trying to get to his grandfather’s party for his birthday. The perfect gift for Bob’s grandpa is a flower. The problem is that Bob has to move through a tricky forest to get there. The trickiest part of all is the evil animals of the forest that are trying to trip you up. Again, the key here is careful strategy and observation, You don’t want to rush through anything.
As you move through the third game, you navigate with bob’s grandpa through to Egypt by way of a magical gateway. There are a lot of difficulties to get through here and it’s a visually engaging situation with many different magical dangers at play. Every move you make will be of great consequence, and the hasty are the ones who are going to fail. But, there is a way through.
The midpoint of the series is the fourth game, which is about space. Bob wants to be an astronaut, and he gets his chance in this game. The game is about making your way through space tests in order to achieve bob’s goal, There are a lot of brain teasers in this game, and there’s no doubt that you will need to think carefully to make it through them all.
The rest of the games in the series involve Bob falling in love and making it through more forests, celebrating a Christmas event, and fighting a villain named Mr. Grinch who kidnaps one of Bob’s friends. There’s also number 7 where Bob’s interest in ghost stories leads to a world that’s just like a nightmare. Eventually, Bob has to make his way through puzzles to fight and defeat a dragon.
The last game is about bob and his grandfather as they go ice fishing. Bob gets stuck on a bit of ice in the middle of the ocean. There’s more to it than this, however, and you have to make your way through the game to uncover the mystery and emerge victorious in the end.
| 353,189
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TITLE: How to prove that there is no 5-ary operation in a clone satisfying certain conditions
QUESTION [3 upvotes]: Question:
How to prove that there is no 5-ary operation $f \in \mathcal{C}$ satisfying
$f(2, 1, 3, 4, 3) = 1$ and
$f(2, 1, 1, 4, 3) = 2$?
The $\mathcal{C} = Clo(\textbf{A})$ is a clone of $\textbf{A}$, where $\textbf{A} = ({1, 2, 3, 4}, \ast)$ with
\begin{array}{ |c|c|c|c|c| }
\hline
∗& 1& 2& 3& 4 \\
\hline
1 &2 &3& 2& 1\\
\hline
2 &1& 4& 3& 4\\
\hline
3& 2& 1& 2& 1\\
\hline
4& 3 &4& 3& 2\\
\hline
\end{array}
My thoughts:
I want to use invariant relations somehow, but not sure, how to proceed with that.
According to the Theorem (Geiger; Bodnarcuk, Kaluznin, Kotov, Romov), $Clo(\textbf{A})=Pol(Inv(\textbf{A}))$ for finite set $\textbf{A}$. (Where $Pol({A})$ = the clone of polynomial operations on $\textbf{A}$, $Inv({A})$ = all relations invariant under every $f$ in $\textbf{A}$).
So maybe I should prove that if $f$ is in $Pol(Inv(\textbf{A}))$ then it cannot be 5-ary?
Any advice is appreciated.
REPLY [4 votes]: The algebra $A$ in question has universe $\{1,2,3,4\}$ and a single binary operation given by the table
$$
\begin{array}{|c||c|c|c|c|}
\hline
∗& 1& 2& 3& 4 \\
\hline
\hline
1 &2 &3& 2& 1\\
\hline
2 &1& 4& 3& 4\\
\hline
3& 2& 1& 2& 1\\
\hline
4& 3 &4& 3& 2\\
\hline
\end{array}
$$
The question is whether the clone of $A$ contains an operation $f$ satisfying both $f(2, 1, 3, 4, 3) = 1$ and
$f(2, 1, 1, 4, 3) = 2$. If the clone of $A$ contained such an operation,
then we could apply it to the pairs
$(2,2), (1,1), (3,1), (4,4), (3,3)\in A\times A$ to produce $(1,2)$.
That is, the two equations involving $f$ acting on individual elements of $A$ can be combined into a statement about $f$ acting on pairs:
$$
f((2,2), (1,1), (3,1), (4,4), (3,3))=(1,2).
$$
If there were such an $f$, then it could be used to prove that the pair $(1,2)$ belongs to the subalgebra of $A^2$ (or `binary invariant relation' of $A$) that is generated by $(2,2), (1,1), (3,1), (4,4), (3,3)$. The subalgebra of $A^2$ that is generated by $(2,2), (1,1), (3,1), (4,4), (3,3)$
is exactly the congruence $\theta$ of $A$ that is generated by $(1,3)$. So, if $f$ existed, then $(1,2)$ would have to be a member of $\theta$.
But you can compute $\theta$ easily and see that it is the equivalence relation corresponding to the partition $13/2/4$, so $(1,2)\notin \theta$, so $f$ is not in the clone of $A$.
| 146,614
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Адрес:
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The hotel gets it’s name because of the roses in the garden. There is a wide range of roses that have been collected from all around of Turkey, The Garden and common open air areas are covered with Basalt Stone ( Lava Stone ) This kind of stone material has the special features which other stone material doesn’t have for its individual holes, such as sound absorption and insulation, heat insulation and reserve, frozen proof etc. It has good sightseeing for the continuous big and small holes, with primitive simplicity and elegant style.
Rose Garden Suites is a unique place with it’s two garden in the same location one of the Gardens is covered by roses and other Garden is for lunch and dinner
| 215,570
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As you’ve no doubt heard, former Carolina Panthers quarterback Cam Newton has signed a one year deal with the New England Patriots. Fans of fantasy football have been waiting for Newton to sign somewhere ever since Panthers head coach Matt Rhule released him in April. Pro Football Network’s own Crissy Froyd speculated a few weeks ago that the Patriots could sign him even though their cap space could present an issue, and they appear to have found a way around that issue. Now that he’s got a team, let’s examine what that means for Cam Newton’s dynasty value heading into the 2020!
Cam Newton’s Dynasty Value After Signing with Patriots
The contract itself and the Patriots lack of cap space
The Patriots have less than $1 million in cap space but Newton’s contract is reportedly worth up to $7.5 million for a one-year deal. How is that possible? Simply put, this contract has a very low base salary and is heavily tilted toward incentives. This means that the Patriots aren’t risking much with the deal and if Newton doesn’t play much the Patriots won’t have to pay him much.
To me, this seems like a low risk, high reward type of move for the Patriots. They already have Jarrett Stidham and Brian Hoyer in their QB room, but adding a player like Newton definitely doesn’t hurt. Newton is a far cry from the MVP version of himself after suffering multiple injuries in the last few seasons but he’s still a valuable asset in the locker room and can definitely help a team as a bench player as the season wears along.
Cam’s various injuries and playing style
Newton isn’t the speedster he used to be. His injuries have definitely slowed him down as he’s gotten older. Newton suffered a foot injury in the offseason that eventually sabotaged his 2019 season where he only played in two games before being benched due to the injury. Newton also has a history of knee and shoulder injuries sustained while playing football and a back injury from a car accident in 2014. He’s not exactly a spring chicken either, having turned 31 in May of 2020.
Related | Chargers and Bears, Oh My! Who has the best odds to sign Cam Newton?
While it’s true that quarterbacks can play well into their 30s and beyond, such as former Patriots QB Tom Brady, father time is always chasing us. Add in all of his injury risk and it wasn’t surprising that it took so long for a team to sign him. He might be healthy now, sure, but how long can that really last? His past makes me nervous.
In his free-agent QB piece, Oliver Hodgkinson noticed some stats about Newton that also make me worried about his potential this season, stating “Newton, in 2017 and 2018, was making plays behind the sticks, relying on the players around him to keep the offense moving.” The Patriots don’t have the same weapons that the Panthers did. Sure, running back James White can catch the ball out of the backfield and wide receiver Julien Edelman is a great short pass-catching option as well, but can Cam adjust to his new teammates quickly enough to be a viable option anyway?
Buy/Hold/Sell and Stidham could still start
From a fantasy standpoint, it’s difficult to know what Cam Newton’s dynasty value is now that he’s finally on a team. If you’ve held him this long, then odds are that you’re not looking to trade him away just yet. He could be a league winner, especially in SuperFlex leagues, if he’s even 80% of what he used to be. However, he also has a chance of never really catching on and retiring in September due to yet another injury. His low floor and high ceiling make for a risky addition to any team right now.
On the flip side, this news makes me think that now is a great time to buy Jarrett Stidham. He still has a solid shot at being the starter, and it’s possible that his current dynasty owner is frustrated at the thought of the team signing Newton. He’s a perfect buy-low target depending on what you have to give up to get him, and a great hold option if you’ve already rostered him. Might as well take a shot, right?
As for the other Patriots weapons, it’s difficult to predict what the right move is just yet. For the RBs like White and Sony Michel, Newton could provide a much-needed boost to their value if he is indeed the starter. For the WRs like Edelman and second-year player N’Keal Harry, they too could see a boost if that offense is as powerful as the Panthers used to be. If you’re looking to add risk to your dynasty team you might be able to get some of them on the cheap as dynasty owners digest the news. That being said, it’s pretty much buyer beware for anyone outside of the QBs in my opinion.
Where do you stand on this debate? Who will be the starter for the Patriots, and a better question, which QB will help your fantasy team more this year? Feel free to hit us up on Twitter to discuss!.
| 342,248
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The amount of credit relies on the month to month pay of borrower. As a quick fix for a transient money issue, payday credits may support a magnificent arrangement. Regardless of whether it is an individual credit, pay day loan, or house advance, each increase will accompany two intrigue costs.
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On the off chance that you may pay your advances on the following compensation day, at that point, you’re in karma. It’s so near being advance sharking I can’t trust it’s permitted. In case you’re keen on ICS payday credits, you need to look at your money related condition and see whether you can pay the advance off with insignificant bother to you.
| 135,966
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Exeter Hospital Emergency Room ReopensThe emergency room at Exeter Hospital reopened Saturday after a thorough scrubbing by a cleaning crew.
19 Exeter Hospital Staff Get Sick In Operating RoomNineteen staff members were evacuated from the Exeter Hospital operating room in New Hampshire after they began to feel dizzy.
3 NH High School Students Hurt In Head-On Crash With Drunk Driver, Police SayThree high school students were hurt in a head-on collision with a drunk driver in Exeter, New Hampshire on Friday night, police say.
NH Hospital Tech Sentenced To 39 Years For Infecting Patients With Hepatitis CA traveling medical technician was sentenced Monday to 39 years in prison for stealing painkillers and infecting dozens of patients in four states with hepatitis C through tainted syringes.
.
| 30,069
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Choosing a Web Designer
A hip or girly web design is so much more than just the look and style of your site.
To stay competitive in today’s market, you must consider whether your business website meets current W3C web standards, which are simply standardized best practices for building web sites. These standards not only ensure compatibility with both the latest browsers and conventional browsers, but are crucial to your web site’s success as they directly affect the accessibility and usability of your website.
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You may not even realize it, but your customers are already expecting these standards each time they access the internet through their mobile devices. An even more compelling reason to consider web standards is the role they play in assisting people with disabilities.
By integrating web standards into your website design, you will gain an advantage over your competition. Your customers will also appreciate the care you have put into creating a web site with them in mind.
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| 196,920
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TITLE: I am searching for the name of a partition (if it already exists)
QUESTION [11 upvotes]: I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a family of sets. However, I did not find anything suitable on google or on wikipedia.
Let a family of sets, say $A_1, \ldots, A_n$, be given. To avoid misunderstanding I will call them modules. This family induces a unique partition on the union set $A = \bigcup_{i=1}^{n} A_i$ in the following way: I call building block a maximal subset $B$ of $A$ such that do not exist 2 different modules $A_i$ and $A_j$ with:
$B \cap A_i \not= \emptyset$,
$B \cap A_j \not= \emptyset$ and
$B\not\subseteq A_i \cap A_j$.
For example, if my family consists of 2 different overlapping modules $A_1$ and $A_2$, I can partition the set $A = A_1 \cup A_2$ as:
(elements in $A_1 \cap A_2$);
(elements in $A_1$ but not in $A_2$);
(elements in $A_2$ but not in $A_1$).
I know that in logic there is something similar, but I am searching for something in set theory. Moreover, I want to underline the dependency of this uniquely derived partition from the family of sets I am given.
See also http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics#Given_a_family_of_sets.2C_I_can_generate_a_partition_of_their_union_set.2C_by_looking_at_their_overlapping...
Thanks to all!
A newcomer
REPLY [19 votes]: Your building blocks are known as the atoms in the Boolean algebra or field of sets generated by the $A_i$. Each building block will consist of points that have the same pattern of answers for membership in the various $A_i$. To see this, observe first that by maximality any building block will respect this equivalence and therefore be a union of such atoms. Conversely, if a set has points with two different patterns of answers, then it will contain points from two $A_i$ without being contained in their intersection. (Specifically, if $x,y\in B$ and $x\in A_i$ but $y\notin A_i$, then pick $j$ such that $y\in A_j$, and observe that $B$ meets both $A_i$ and $A_j$, buit is not contained in $A_i\cap A_j$.) So the building blocks are the atoms.
If your family is finite as you indicated, then the Boolean algebra generated by the $A_i$ consists precisely of the unions of blocks. This is the representation theorem showing that every finite Boolean algebra is isomorphic to a finite power set---the power set of the atoms.
In the infinite case, however, the Boolean algebra generated by the $A_i$ may be atomless---it may have no atoms at all, and this is a fascinating case. Nevertheless, your blocks still form a partition, and are precisely the atoms in the infinitary-generated field of sets, still determined by my argument above by the patttern-of-answers to membership in the $A_i$.
| 95,137
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W for the attack on Monday.
Watch WBAL's video for more details.
| 194,057
|
TITLE: Calculating the number of variables and constants in a term
QUESTION [2 upvotes]: I'm reading Kees Doets's Basic Model Theory and couldn't get around the first exercise, which is rather simple (no doubt my lack of arithmetical skills played a part). Let $t$ be a term and let $n_i$ ($i = 1, 2, 3, \dots$) be the number of $i$-ary function symbols that occur in $t$. The exercise is to provem by term induction, that the number of variables and constants in $t$ is $1+\sum_i(i-1) n_i$, that is, $1 + n_2 + 2n_3 + \dots$.
The base case is obvious. As for the induction step, suppose $t$ is $f(t_1, \dots, t_k)$. The induction hypothesis hold for each $t_l (1 \leq l \leq k)$, so we know that, for each such $t_l$, the above formula holds. Further, we know that the number of variables and constants in $t$ is the sum of the number of constants and variables in each $t_l$ (right?). Here comes my problem: doesn't that result in the number $k + \sum_i(i-1)n_i$? Where did I go wrong? It's probably something very simple, but I couldn't figure out what, exactly, is the problem.
REPLY [1 votes]: Ihe number of occurrences of $k$-ary function symbols in $f(t_1,\dots,t_k)$ is $1$ more than the sum of the numbers of occurrences of $k$-ary function symbols in the $t_i$.
More formally, if $n_k$ is the combined number of occurrences of $k$-ary function symbols in the $t_i$, then the number $n'_k$ of occurrences of $k$-ary function symbols in $f(t_1,\dots,t_k)$ is given by $n'_k=1+n_k$.
In the formula, the summand $(k-1)n_k$ becomes $(k-1)n'_k$, that is, $(k-1)+(k-1)n_k$.
That yields an additional $k-1$ in the sum, and your $k$ in front is $1+(k-1)$.
| 192,862
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\begin{document}
\title[On the ascending and descending chain conditions]{On the ascending and descending chain conditions in the lattice of monoid varieties}
\thanks{Supported by the Ministry of Education and Science of the Russian Federation (project 1.6018.2017/8.9) and by Russian Foundation for Basic Research (grant 17-01-00551).}
\author{S.\,V.\,Gusev}
\address{Ural Federal University, Institute of Natural Sciences and Mathematics, Lenina 51, 620000 Ekaterinburg, Russia}
\email{sergey.gusb@gmail.com}
\date{}
\begin{abstract}
In this work we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element. We found an example of two varieties of monoids with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition.
\end{abstract}
\keywords{monoid, variety, lattice of varieties, ascending chain condition, descending chain condition}
\subjclass{Primary 20M07, secondary 08B15}
\maketitle
\section{Introduction and summary}
\label{introduction}
This paper is devoted to the examination of the lattice $\mathbb{MON}$ of all monoid varieties (referring to monoid varieties, we consider monoids as algebras with an associative binary operation and the nullary operation that fixes the identity element). There are a lot of articles about the monoid varieties. However, these articles are devoted mainly to the examination of identities of monoids. At the same time, although the first results about the lattice $\mathbb{MON}$ was found a long time ago (see~\cite{Head-68,Pollak-81,Wismath-86}), only a little information was recently known about the lattice $\mathbb{MON}$.{\sloppy
}
The situation has recently changed. There are papers, devoted to examination of identities of monoids, that contain also some non-trivial results about the lattice $\mathbb{MON}$ (see~\cite{Jackson-05,Lee-08,Lee-12,Lee-14}, for instance). Several works devoted to the examination of the lattice $\mathbb{MON}$ was published in 2018~\cite{Gusev-18-IzVUZ,Gusev-18-AU,Gusev-Vernikov-18}. In these articles, several restrictions on the lattices of monoid varieties formulated in terms that are somehow connected with lattice identities were studied. At the same time, when studying lattices of varieties of algebras of various types, much attention has been also paid to the \emph{finiteness conditions}, i.e., conditions that hold in every finite lattice (see~\cite[Section~10]{Shevrin-Vernikov-Volkov-09}, for instance).
The subvariety lattice of a variety $\mathbf V$ is denoted by $L(\mathbf V)$. A variety is called \emph{finitely generated} if it is generated by a finite algebra. In~\cite[Theorems~2.3,~2.4 and~2.5]{Lvov-73} I.V.L'vov proved that for an associative ring variety $\mathbf V$, the following are equivalent:
\begin{itemize}
\item[a)] the lattice $L(\mathbf V)$ is finite;
\item[b)] the lattice $L(\mathbf V)$ satisfies the ascending chain condition;
\item[c)] $\mathbf V$ is finitely generated.
\end{itemize}
A similar result does not hold for group varieties. Indeed, there are only countably many finitely generated group varieties. At the same time, there are uncountably many periodic non-locally finite varieties of groups with subvariety lattice isomorphic to the 3-element chain~\cite{Kozhevnikov-12}. As far as we know, in the group case the question about the equivalence of the claims a) and b) still remains open. In the semigroup case the claim c) is not equivalent to the claims a) and b). This follows from the folklore fact that the 2-element semigroup with zero multiplication with a new identity element adjoined generates a semigroup variety with countably infinitely many subvarieties~\cite[Fig.~5(b)]{Evans-71}). In the semigroup case the claims a) and b) are not equivalent too. This fact follows from the results of the work~\cite{Sapir-91} where an example of semigroup varieties whose subvariety lattice satisfies the descending chain condition but violates the ascending chain condition is given. Also, this example shows that the classes of semigroup varieties whose subvariety lattices are finite or satisfy the descending chain condition are not closed with respect to the join of the varieties and to coverings. The similar questions about the class of semigroup varieties whose subvariety lattices satisfy the ascending chain condition remain open.
In~\cite[Subsection~3.2]{Jackson-Lee-18} two monoid varieties $\mathbf U$ and $\mathbf W$ are exhibited such that the subvariety lattices of both varieties are finite, while the lattice $L(\mathbf U\vee \mathbf W)$ is uncountably infinite and does not satisfy the ascending chain condition. Moreover, it follows from the proof of Theorem~3.4 in~\cite{Jackson-Lee-18} that $L(\mathbf U\vee \mathbf W)$ violates the descending chain condition. So, the classes of monoid varieties whose subvariety lattices are finite, satisfy the descending chain condition or satisfy the ascending chain condition are not closed with respect to the join of the varieties. At the same time, the results of~\cite{Jackson-Lee-18} leave open the question about stability of the these classes of varieties with respect to coverings.
In this work we find two monoid varieties with finite subvariety lattices such that their join covers one of them and has a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition. Thus, we give a negative answer to the question noted in the previous paragraph.
In order to formulate the main result of the article, we need some definitions and notation. We denote by $F$ the free semigroup over a countably infinite alphabet. As usual, elements of $F$ and the alphabet are called \emph{words} and \emph{letters} respectively. The words and the letters are denoted by small Latin letters. However, the words unlike the letters are written in bold. The symbol $F^1$ stands for the semigroup $F$ with a new identity element adjoined. We treat this identity element as the empty word and denote it by $\lambda$. Expressions like $\mathbf u\approx\mathbf v$ are used for identities, whereas $\mathbf{u=v}$ means that the words $\mathbf u$ and $\mathbf v$ coincide. One can introduce notation for the following three identities:
$$
\sigma_1:\ xyzxty\approx yxzxty,\quad\sigma_2:\ xtyzxy\approx xtyzyx,\quad\sigma_3:\ xzxyty \approx xzyxty.
$$
Note that the identities $\sigma_1$ and $\sigma_2$ are dual to each other. A letter is called \emph{simple} [\emph{multiple}] \emph{in a word} $\mathbf w$ if it occurs in $\mathbf w$ once [at least twice]. Note also that the identity $\sigma_1$ [respectively, $\sigma_2$] allows us to swap the adjacent non-latest [respectively, non-first] occurrences of two multiple letters, while the identity $\sigma_3$ allows us to swap a non-latest occurrence and a non-first occurrence of two multiple letters whenever these occurrences are adjacent to each other. Put
$$
\Phi=\{x^2y\approx yx^2,\,x^2yz\approx xyxzx,\,\sigma_3\}.
$$
The trivial variety of monoids is denoted by $\mathbf T$, while $\mathbf{SL}$ denotes the variety of all semilattice monoids. For an identity system $\Sigma$, we denote by $\var\,\Sigma$ the variety of monoids given by $\Sigma$. Let us fix notation for the following varieties:
\iffalse
\begin{align*}
&\mathbf C=\var\{x^2\approx x^3,\,xy\approx yx\},\\
&\mathbf D_1=\var\{x^2\approx x^3,\,x^2y\approx xyx\approx yx^2\},\\
&\mathbf D_2=\var\{\Phi,\,\sigma_1,\,\sigma_2\},\\
&\mathbf M=\var\{\Phi,\,xyzxy\approx yxzxy,\,\sigma_2\},\\
&\mathbf N=\var\{\Phi,\,\sigma_2\}.
\end{align*}
\fi
\begin{align*}
&\mathbf C=\var\{x^2\approx x^3,\,xy\approx yx\},\enskip\mathbf D_1=\var\{x^2\approx x^3,\,x^2y\approx xyx\approx yx^2\},\\
&\mathbf D_2=\var\{\Phi,\,\sigma_1,\,\sigma_2\},\enskip\mathbf M=\var\{\Phi,\,xyzxy\approx yxzxy,\,\sigma_2\},\enskip\mathbf N=\var\{\Phi,\,\sigma_2\}.
\end{align*}
If $\mathbf V$ is a monoid variety then we denote by $\overleftarrow{\mathbf V}$ the variety \emph{dual to} $\mathbf V$, i.e., the variety consisting of monoids antiisomorphic to monoids from $\mathbf V$.
The main result of the paper is the following
\begin{theorem}
\label{main result}
\begin{itemize}
\item[\textup{(i)}] The variety $\mathbf N\vee \overleftarrow{\mathbf M}$ covers the variety $\mathbf N$.
\item[\textup{(ii)}] The interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$ of the lattice $L(\mathbf N\vee \overleftarrow{\mathbf M})$ contains continuum many subvarieties and does not satisfy the ascending chain condition and the descending chain condition.
\end{itemize}
\end{theorem}
The proof of Theorem~\ref{main result} implies that the lattice $L(\mathbf N\vee\overleftarrow{\mathbf M})$ ''modulo'' the interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$ has the form shown in Fig.~\ref{L(N vee dual M)}.
\begin{figure}[htb]
\unitlength=1mm
\linethickness{0.4pt}
\begin{center}
\begin{picture}(55,85)
\put(35,5){\circle*{1.33}}
\put(35,15){\circle*{1.33}}
\put(35,25){\circle*{1.33}}
\put(35,35){\circle*{1.33}}
\put(35,45){\circle*{1.33}}
\put(45,55){\circle*{1.33}}
\put(25,55){\circle*{1.33}}
\put(35,65){\circle*{1.33}}
\put(15,65){\circle*{1.33}}
\put(25,75){\circle*{1.33}}
\qbezier(25,75)(35,75)(35,65)
\qbezier(25,75)(25,65)(35,65)
\put(35,5){\line(0,1){40}}
\put(35,45){\line(-1,1){20}}
\put(35,45){\line(1,1){10}}
\put(25,55){\line(1,1){10}}
\put(25,55){\line(1,1){10}}
\put(15,65){\line(1,1){10}}
\put(45,55){\line(-1,1){10}}
\put(35,2){\makebox(0,0)[cc]{\textbf T}}
\put(37,15){\makebox(0,0)[lc]{\textbf{SL}}}
\put(37,25){\makebox(0,0)[lc]{$\mathbf C$}}
\put(37,35){\makebox(0,0)[lc]{$\mathbf D_1$}}
\put(37,45){\makebox(0,0)[lc]{$\mathbf D_2$}}
\put(23,55){\makebox(0,0)[rc]{$\mathbf M$}}
\put(13,65){\makebox(0,0)[rc]{$\mathbf N$}}
\put(47,55){\makebox(0,0)[lc]{$\overleftarrow{\mathbf M}$}}
\put(37,65){\makebox(0,0)[lc]{$\mathbf M\vee\overleftarrow{\mathbf M}$}}
\put(25,79){\makebox(0,0)[cc]{$\mathbf N\vee\overleftarrow{\mathbf M}$}}
\end{picture}
\end{center}
\caption{The lattice $L(\mathbf N\vee \dualM)$}
\label{L(N vee dual M)}
\end{figure}
We note also that one of the main goals of the work~\cite{Jackson-Lee-18} is to construct several examples of finitely generated monoid varieties with continuum many subvarieties. It is verified in Erratum to~\cite{Jackson-05} that $\overleftarrow{\mathbf M}$ and $\overleftarrow{\mathbf N}$ are finitely generated. Therefore, the variety $\mathbf N\vee\overleftarrow{\mathbf M}$ is finitely generated too. So, Theorem~\ref{main result} gives a new example of finitely generated variety of monoids with continuum many subvarieties. Besides that, this theorem provides some more new examples of finitely generated varieties of monoids with continuum many subvarieties (see~Corollary~\ref{D_2 join G} below). Since the variety $\mathbf N$ is finitely generated, this variety is locally finite. Moreover, $\mathbf N$ is finitely based and has only finite many subvarieties, i.e., it is a Cross variety. It follows that the covering of a Cross monoid variety can have a continuum cardinality subvariety lattice that violates the ascending chain condition and the descending chain condition. Since the cover $\mathbf M\vee\overleftarrow{\mathbf M}$ of the Cross varieties $\mathbf M$ and $\overleftarrow{\mathbf M}$ is non-finitely based~\cite{Jackson-05}, the class of Cross monoid varieties is not closed with respect to the formation of joins and of covers.
The article consists of three sections. Section~\ref{preliminaries} contains definitions, notation and auxiliary results, while Section~\ref{proof of theorem} is devoted to the proof of Theorem~\ref{main result}.
\section{Preliminaries}
\label{preliminaries}
\subsection{A useful construction}
\label{useful construction}
The following notion was introduced by Perkins~\cite{Perkins-69} and often appeared in the literature (see~\cite{Gusev-Vernikov-18,Jackson-05,Jackson-Lee-18,Jackson-Sapir-00}, for instance). Let $W$ be a set of possibly empty words. We denote by $\overline W$ the set of all subwords of words from $W$ and by $I\bigl(\,\overline W\,\bigr)$ the set $F^1 \setminus \overline W$. It is clear that $I\bigl(\,\overline W\,\bigr)$ is an ideal of $F^1$. Then $S(W)$ denotes the Rees quotient monoid $F^1/I\bigl(\,\overline W\,\bigr)$. If $W=\{{\bf w}_1,{\bf w}_2,\dots,{\bf w}_k\}$ then we will write $S\bigl({\bf w}_1,{\bf w}_2,\dots,{\bf w}_k\bigr)$ rather than $S\bigl(\{{\bf w}_1,{\bf w}_2,\dots,{\bf w}_k\}\bigr)$. A word \textbf w is called an \emph{isoterm} for a class of semigroups if no semigroup in the class satisfies any non-trivial identity of the form $\mathbf w\approx\mathbf w'$.
\begin{lemma}
\label{S(W) in V}
Let $\mathbf V$ be a monoid variety and $W$ a set of possibly empty words. Then $S(W)$ lies in $\mathbf V$ if and only if each word in $W$ is an isoterm for $\mathbf V$.
\end{lemma}
\begin{proof}
It is easy to verify that it suffices to consider the case when $W$ consists of one word (see the paragraph after Lemma~3.3 in~\cite{Jackson-05}). Then necessity is obvious, while sufficiency is proved in~\cite[Lemma~5.3]{Jackson-Sapir-00}.
\end{proof}
The following statement is dual to Proposition~1 in Erratum to~\cite{Jackson-05}.
\begin{lemma}
\label{generator of M}
The variety $\mathbf M$ is generated by monoid $S(xysxty)$.\qed
\end{lemma}
\subsection{Word problems for the varieties $\mathbf M$ and $\mathbf N$}
\label{word problems}
We introduce a series of new notions and notation. The set of all simple [multiple] letters in a word \textbf w is denoted by $\simple(\mathbf w)$ [respectively $\mul(\mathbf w)$]. The \emph{content} of a word \textbf w, i.e., the set of all letters occurring in $\bf w$, is denoted by $\con({\bf w})$. For a word $\mathbf w$ and letters $x_1,x_2,\dots,x_k\in \con(\mathbf w)$, let $\mathbf w(x_1,x_2,\dots,x_k)$ denotes the word obtained from $\mathbf w$ by retaining the letters $x_1,x_2,\dots,x_k$.
Let $\mathbf w$ be a word and $\simple(\mathbf w)=\{t_1,t_2,\dots, t_m\}$. We can assume without loss of generality that $\mathbf w(t_1,t_2,\dots, t_m)=t_1t_2\cdots t_m$. Then $\mathbf w = \mathbf w_0 t_1 \mathbf w_1 \cdots t_m \mathbf w_m$ where $\mathbf w_0,\mathbf w_1,\dots,\mathbf w_m$ are possibly empty words and $t_0=\lambda$. The words $\mathbf w_0$, $\mathbf w_1$, \dots, $\mathbf w_m$ are called \emph{blocks} of $\mathbf w$, while $t_0,t_1,\dots,t_m$ are said to be \emph{dividers} of $\mathbf w$. The representation of the word \textbf w as a product of alternating dividers and blocks, starting with the divider $t_0$ and ending with the block $\mathbf w_m$ is called a \emph{decomposition} of the word \textbf w. A block of the word $\mathbf w$ is called a $k$-\emph{block} if this block consists of $k$th occurrences of letters in $\mathbf w$. If every block of the word $\mathbf w$ is either $1$-block or $2$-block then we say that the word $\mathbf w$ is \emph{reduced}. Recall that a word \textbf w is called \emph{linear} if every letter from $\con(\mathbf w)$ is simple in $\mathbf w$. We note that if $\mathbf w$ is a reduced word and $x$ is a multiple letter in $\mathbf w$ then $x$ cannot occur twice in the same block of $\mathbf w$. In other words, the following is true.
\begin{remark}
\label{blocks of a reduced word}
Every block of a reduced word is a linear word.
\end{remark}
Further, let $\mathbf w$ be a reduced word. Let us consider an arbitrary $1$-block $\mathbf w_i$ of $\mathbf w$. The maximal subwords of this block consisting of the letters whose second occurrences in $\mathbf w$ lie in the same $2$-block of $\mathbf w$ are called \emph{subblocks of $1$-block} $\mathbf w_i$ of the word $\mathbf w$. The representation of $1$-block as a product of subblocks is called a \emph{decomposition} of this $1$-block. The notions of $2$-blocks and $2$-decompositions of the word $\mathbf w$ are defined dually. The representation of the reduced word $\mathbf w$ as a product of alternating dividers and decompositions of blocks is called \emph{full decomposition} of $\mathbf w$. Below we underline dividers to distinguish them from blocks and we divide subblocks of the same block by the symbol ''$|$''. We illustrate the introduced notions by the following
\begin{example}
\label{example blocks}
Put $\mathbf w=abcdxcbyezaed$. Clearly, $\simple(\mathbf w)=\{x,y,z\}$. Therefore, the letters $x$, $y$ and $z$ are the dividers of $\mathbf w$, while the words $abcd$, $cb$, $e$ and $aed$ are the blocks of $\mathbf w$. Evidently, the blocks $abcd$ and $e$ consist of the first occurrences of letters in $\mathbf w$, while the blocks $cb$ and $aed$ consist of the second occurrences of letters in $\mathbf w$. Consequently, $abcd$ and $e$ are the $1$-blocks of the word $\mathbf w$, while $cb$ and $aed$ are the $2$-blocks of this word. So, the word $\mathbf w$ is reduced. The second occurrences of the letters $a$, $b$ and the letters $c$, $d$ lie in the different $2$-blocks, while the second occurrences of the letters $b$ and $c$ lie in the same $2$-blocks. Therefore, the decomposition of $1$-block $abcd$ has the form $a\,|\,bc\,|\,d$. The decomposition of $2$-block $cb$ consist of one subblock because the first occurrences of letters $b$ and $c$ lie in the same $1$-block. Finally, the decomposition of $2$-block $aed$ equals $a\,|\,e\,|\,d$ because the first occurrences of the letters $a$, $d$ and the letter $e$ lie in different $1$-blocks. Thus, the full decomposition of the word $\mathbf w$ has the form $a\,|\,bc\,|\,d\underline x\,cb\,\underline y\,e\,\underline z\,a\,|\,e\,|\,d$.
\end{example}
An identity $\mathbf u\approx \mathbf v$ is called \emph{reduced} if the words $\mathbf u$ and $\mathbf v$ are reduced. The words $\mathbf u$ and $\mathbf v$ are said to be \emph{equivalent} if their decompositions are
\begin{align}
\label{decomposition of u}
&\mathbf u_0 t_1 \mathbf u_1 \cdots t_m \mathbf u_m,\\
\label{decomposition of v}
&\mathbf v_0 t_1 \mathbf v_1 \cdots t_m \mathbf v_m
\end{align}
respectively and $\con(\mathbf u_i)=\con(\mathbf v_i)$ for every $0\le i\le m$. Let $\mathbf u$ and $\mathbf v$ be equivalent reduced words. Suppose that~\eqref{decomposition of u} and~\eqref{decomposition of v} are the decompositions of these words respectively. In this case, the blocks $\mathbf u_i$ and $\mathbf v_i$ are said to be \emph{corresponding} to each other. Let's say that the corresponding blocks $\mathbf u_i$ and $\mathbf v_i$ are \emph{equivalent} if their decompositions are
\begin{align}
\label{decomposition of u_i}
&\mathbf u_{i1}\mathbf u_{i2}\dots \mathbf u_{ik_i},\\
\label{decomposition of v_i}
&\mathbf v_{i1}\mathbf v_{i2}\dots \mathbf v_{ik_i}
\end{align}
respectively and $\con(\mathbf u_{ij})=\con(\mathbf v_{ij})$ for every $1\le j\le k_i$. In this case, the subblocks $\mathbf u_{ij}$ and $\mathbf v_{ij}$ are said to be \emph{corresponding} to each other. The equivalent words $\mathbf u$ and $\mathbf v$ are said to be $1$-\emph{equivalent} if every two corresponding $1$-blocks of these words are equivalent each other.
\begin{lemma}
\label{word problem M}
A reduced identity $\mathbf u\approx \mathbf v$ holds in the variety $\mathbf M$ if and only if the words $\mathbf u$ and $\mathbf v$ are $1$-equivalent.
\end{lemma}
\begin{proof}
\emph{Necessity}. Suppose the reduced identity $\mathbf u\approx \mathbf v$ holds in the variety $\mathbf M$.
First, we will prove that $\mathbf u$ and $\mathbf v$ are equivalent. It is proved in~\cite[Proposition 2.13]{Gusev-Vernikov-18} that an identity $\mathbf u\approx\mathbf v$ holds in the variety $\mathbf D_1$ if and only if $\simple(\mathbf u)=\simple(\mathbf v)$, $\mul(\mathbf u)=\mul(\mathbf v)$ and the simple letters appear in the words $\mathbf u$ and $\mathbf v$ in the same order. This fact and inclusion $\mathbf D_1\subset \mathbf M$ imply that if~\eqref{decomposition of u} is the decomposition of the word $\mathbf u$ then the decomposition of the word $\mathbf v$ has the form~\eqref{decomposition of v}. Let $z\in\mul(\mathbf u)$. Suppose that, the first occurrence of $z$ in $\mathbf u$ lies in the block $\mathbf u_p$, while the second occurrence of $z$ in $\mathbf u$ lies in the block $\mathbf u_q$ for some $0\le p,q\le m$. Remark~\ref{blocks of a reduced word} implies that $p<q$. The word $ztz$ is an isoterm for the variety $\mathbf M$ by Lemmas~\ref{S(W) in V} and~\ref{generator of M}. Consequently, $\mathbf u(z,t_r)=zt_rz=\mathbf v(z,t_r)$ for every $p< r\le q$. Moreover, $\mathbf v(z,t_s)\ne zt_sz$ and $\mathbf v(z,t_\ell)\ne zt_\ell z$ for all $1\le s \le p$ and $q<\ell\le m$. Therefore, the first occurrence of $z$ in $\mathbf v$ lies in the block $\mathbf v_p$, while the second occurrence of $z$ in $\mathbf v$ lies in the block $\mathbf v_q$. This implies that the words $\mathbf u$ and $\mathbf v$ are equivalent.
Suppose now that the words $\mathbf u$ and $\mathbf v$ are not $1$-equivalent. Then there exist a $1$-block $\mathbf u_i$ of $\mathbf u$ and letters $x,y\in\con(\mathbf u_i)$ such that the second occurrences of $x$ and $y$ lie in different $2$-blocks of $\mathbf u$, say $\mathbf u_j$ and $\mathbf u_k$ respectively, while the first occurrence of $x$ precedes the first occurrence of $y$ in $\mathbf u$, but the first occurrence of $y$ precedes the first occurrence of $x$ in $\mathbf v$. In view of Remark~\ref{blocks of a reduced word}, we have that $i<j$. Besides that, we can assume without loss of generality that $j<k$. Since the words $\mathbf u$ and $\mathbf v$ are equivalent, $x\in\con(\mathbf v_i)\cap\con(\mathbf v_j)$ and $y\in\con(\mathbf v_i)\cap\con(\mathbf v_k)$. Then $\mathbf M$ satisfi es the identity $\mathbf u(x,y,t_{i+1},t_{j+1})\approx \mathbf v(x,y,t_{i+1},t_{j+1})$, where
$$
\mathbf u(x,y,t_{i+1},t_{j+1})=xyt_{i+1}xt_{j+1}y\text{ and } \mathbf v(x,y,t_{i+1},t_{j+1})=yxt_{i+1}xt_{j+1}y,
$$
so that $xysxty$ is not an isoterm for $\mathbf M$. But this is impossible by Lemmas~\ref{S(W) in V} and~\ref{generator of M}.
\smallskip
\emph{Sufficiency}. Let~\eqref{decomposition of u} be the decomposition of $\mathbf u$. Since the words $\mathbf u$ and $\mathbf v$ are $1$-equivalent, the decomposition of $\mathbf v$ has the form~\eqref{decomposition of v}. We are going to verify that the word $xysxty$ is an isoterm for the variety $\var\{\mathbf u\approx \mathbf v\}$. Arguing by contradiction, we suppose that this variety satisfies a non-trivial identity $xysxty\approx \mathbf w$. We can assume without any loss that $xysxty=\mathbf a\xi(\mathbf u)\mathbf b$ and $\mathbf w=\mathbf a\xi(\mathbf v)\mathbf b$ for some words $\mathbf a$, $\mathbf b$ and some endomorphism $\xi$ of $F^1$. Since the identity $xysxty\approx \mathbf w$ is non-trivial and the words $\mathbf u$ and $\mathbf v$ are equivalent, there are letters $c,d\in\mul(\mathbf u)$ such that $x\in\con(\xi(c))$ and $y\in\con(\xi(d))$. Clearly, $c\ne d$, whence $\xi(c)=x$ and $\xi(d)=y$. This implies that $\mathbf a=\mathbf b=\lambda$ and $s=\xi(t_\ell)$, $t=\xi(t_r)$ for some $1\le \ell<r\le m$. Then, since the word $\mathbf u$ is reduced, $c\in\con(\mathbf u_i)\cap\con(\mathbf u_j)$ and $d\in\con(\mathbf u_p)\cap\con(\mathbf u_q)$ for some $i,j,p$ and $q$ such that $i\le p<\ell\le j<r\le q$. Further, we have that $c\in\con(\mathbf v_i)\cap\con(\mathbf v_j)$ and $d\in\con(\mathbf v_p)\cap\con(\mathbf v_q)$ because the words $\mathbf u$ and $\mathbf v$ are equivalent. If $i<p$ then the first occurrence of $c$ precedes the first occurrence of $d$ in $\mathbf v$. We obtain a contradiction with the fact that $\mathbf w\ne xysxty$. If $i=p$ then the first occurrence of $c$ precedes the first occurrence of $d$ in $\mathbf v$ again because the $\mathbf u$ and $\mathbf v$ are $1$-equivalent and the second occurrences of the letters $c$ and $d$ lie in different blocks of the words $\mathbf u$ and $\mathbf v$. So, the word $xysxty$ is an isoterm for $\var\{\mathbf u\approx \mathbf v\}$. In view of Lemmas~\ref{S(W) in V} and~\ref{generator of M}, the identity $\mathbf u\approx \mathbf v$ holds in $\mathbf M$.
\end{proof}
If \textbf u and \textbf v are words and $\varepsilon$ is an identity then we will write $\mathbf u\stackrel{\varepsilon}\approx\mathbf v$ in the case when the identity $\mathbf u\approx\mathbf v$ follows from $\varepsilon$.
\begin{lemma}
\label{word problem N}
A reduced identity $\mathbf u\approx \mathbf v$ holds in the variety $\mathbf N$ if and only if the words $\mathbf u$ and $\mathbf v$ are equivalent and corresponding $1$-blocks of the words $\mathbf u$ and $\mathbf v$ equal each other.
\end{lemma}
\begin{proof}
\emph{Necessity}. Suppose the reduced identity $\mathbf u\approx \mathbf v$ holds in the variety $\mathbf N$. Lemma~\ref{word problem M} and inclusion $\mathbf M\subset \mathbf N$ imply that the words $\mathbf u$ and $\mathbf v$ are $1$-equivalent. In particular, the words $\mathbf u$ and $\mathbf v$ are equivalent. Then if~\eqref{decomposition of u} is the decomposition of the word $\mathbf u$ then the decomposition of the word $\mathbf v$ has the form~\eqref{decomposition of v}. Suppose that there exist two corresponding $1$-blocks $\mathbf u_i$ and $\mathbf v_i$ such that $\mathbf u_i\ne\mathbf v_i$. Clearly, $i<m$. Since the $1$-blocks $\mathbf u_i$ and $\mathbf v_i$ are $1$-equivalent, there is a subblock $\mathbf u'$ of the block $\mathbf u_i$ which does not coincide with the corresponding subblock $\mathbf v'$ of the block $\mathbf v_i$. In view of Remark~\ref{blocks of a reduced word}, the subblocks $\mathbf u'$ and $\mathbf v'$ are linear words. At the same time, $\con(\mathbf u')=\con(\mathbf v')$ because the blocks $\mathbf u_i$ and $\mathbf v_i$ are $1$-equivalent. Therefore, there exist letters $x$ and $y$ such that $x$ precedes $y$ in $\mathbf u'$, while $y$ precedes $x$ in $\mathbf v'$. Then $\mathbf N$ satisfies the identity
$$
xyt_{i+1}\mathbf w=\mathbf u(x,y,t_{i+1})\approx \mathbf v(x,y,t_{i+1})=yxt_{i+1}\mathbf w',
$$
where $\mathbf w,\mathbf w'\in\{xy,yx\}$. Then $\mathbf N$ satisfies the identities
$$
xyt_{i+1}xy\stackrel{\sigma_2}\approx xyt_{i+1}\mathbf w\approx yxt_{i+1}\mathbf w'\stackrel{\sigma_2}\approx yxt_{i+1}xy.
$$
We obtain a contradiction with the fact that the varieties $\mathbf M$ and $\mathbf N$ are different.
\smallskip
\emph{Sufficiency}. Let~\eqref{decomposition of u} be the decomposition of $\mathbf u$. Since the word $\mathbf u$ and $\mathbf v$ are equivalent, the decomposition of $\mathbf v$ has the form~\eqref{decomposition of v}. Consider arbitrary corresponding $2$-blocks $\mathbf u_i$ and $\mathbf v_i$. The words $\mathbf u_i$ and $\mathbf v_i$ are linear (see Remark~\ref{blocks of a reduced word}) and depend on the same letters (since $\mathbf u$ and $\mathbf v$ are equivalent). The identity $\sigma_2$ allows us to swap the second occurrences of two multiple letters whenever these occurrences are adjacent to each other. Thus, if we replace the $2$-block $\mathbf u_i$ by $\mathbf v_i$ in $\mathbf u$ then the word we obtain should be equal to $\mathbf u$ in $\mathbf N$. By the hypothesis, the corresponding $1$-blocks of the words $\mathbf u$ and $\mathbf v$ equal each other. Therefore, the identity
$$
\mathbf u= t_0\mathbf u_0 t_1 \mathbf u_1 \cdots t_m \mathbf u_m\stackrel{\sigma_2}\approx t_0\mathbf v_0 t_1 \mathbf v_1 \cdots t_m \mathbf v_m=\mathbf v,
$$
holds in the variety $\mathbf N$.
\end{proof}
\subsection{Some words and their properties}
\label{some words}
We introduce some new notation. As usual, the symbol $\mathbb N$ stands for the set of all natural numbers. For all $n,k\in \mathbb N$, we put
$$
\mathcal M_n^k=\{1\}\times\underbrace{\mathbb N_n\times \mathbb N_n \times \cdots \times \mathbb N_n}_{k-1 \text{ copies}}.
$$
where $\mathbb N_n=\{1,2,\dots, n\}$. If $\gamma=(1,i_1,i_2,\dots,i_{k-1})\in \mathcal M_n^k$ and $1\le j\le n$ then we put
$$
\gamma+j=(1,i_1,i_2,\dots,i_{k-1},j)\in \mathcal M_n^{k+1}.
$$
The usual lexicographical order is defined on the set $\mathcal M_n^k$. So, the expression
$$
\prod_{\gamma\in \mathcal M_n^k} \mathbf w_{\gamma}
$$
mean an abbreviated notation of the product of the words $\mathbf w_{\gamma}$ in ascending order $\gamma$. Put
\begin{align*}
\mathbf c_n=\prod_{\gamma\in \mathcal M_n^n} \biggl(\prod_{j=1}^n s_{\gamma}^{(j)}x_{\gamma}^{(j)}\biggr)
&\text{ and }
\mathbf d_n^{(k)}=s_k\cdot\biggl(\prod_{\gamma\in \mathcal M_n^k} \biggl(\prod_{j=1}^n x_{\gamma}^{(j)}\biggl(\prod_{\ell=1}^n x_{\gamma+j}^{(\ell)}\biggr)\biggr)\biggr),\\[-3pt]
\mathbf e_m=s_mx_mt_my_m &\text{ and }\mathbf f_m=s_mx_mx_{m+1}y_{m+1}y_m
\end{align*}
for all $n,k\ge 1$ and $m\ge 0$. Further, for any natural $n$ put
\begin{align*}
&\mathbf a_n=xy\cdot\biggl(\prod_{i=1}^n \mathbf d_{2n}^{(2i-1)}\biggr)\cdot \mathbf c_{2n}\cdot\biggl(\prod_{i=n-1}^1 \mathbf d_{2n}^{(2i)}\biggr)\cdot sx\cdot\biggl(\prod_{i=1}^{2n} x_1^{(i)}\biggr)\cdot y,\\[-3pt]
&\mathbf a_n'=yx\cdot\biggl(\prod_{i=1}^n \mathbf d_{2n}^{(2i-1)}\biggr)\cdot \mathbf c_{2n}\cdot\biggl(\prod_{i=n-1}^1 \mathbf d_{2n}^{(2i)}\biggr)\cdot sx\cdot\biggl(\prod_{i=1}^{2n} x_1^{(i)}\biggr)\cdot y,\\[-3pt]
&\mathbf b_n=x_0y_0\biggl(\prod_{i=1}^n \mathbf f_{2i-1}\biggr)\cdot \mathbf e_{2n}\cdot\biggl(\prod_{i=n-1}^0 \mathbf f_{2i}\biggr),\\[-3pt]
&\mathbf b_n'=y_0x_0\biggl(\prod_{i=1}^n \mathbf f_{2i-1}\biggr)\cdot \mathbf e_{2n}\cdot\biggl(\prod_{i=n-1}^0 \mathbf f_{2i}\biggr).
\end{align*}
We note that the words $\mathbf a_n$, $\mathbf a_n'$, $\mathbf b_n$ and $\mathbf b_n'$ are reduced. The following simple fact can be easily verified directly.
\begin{remark}
\label{rm s-decompositions of a_n,a_n'}
The expression
\begin{equation}
\label{s-decompositions of a_n,a_n'}
\begin{aligned}
&\chi(xy)\cdot\biggl(\prod_{i=1}^{n-1}\underline{s_i} \biggl(\prod_{\gamma\in \mathcal M_{2n}^{2i-1}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\,\biggl|\,\prod_{\ell=1}^{2n} x_{\gamma+j}^{(\ell)}\,\biggr|\,\biggr)\biggr)\biggr)\\[-3pt]
&\cdot\biggl(\underline{s_{2n-1}} \prod_{\gamma\in \mathcal M_{2n}^{2n-1}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\biggl(\prod_{\ell=1}^{2n}\,|\,x_{\gamma+j}^{(\ell)}\,|\,\biggr)\biggr)\biggr)\biggl(\prod_{\gamma\in \mathcal M_{2n}^{2n}} \biggl(\prod_{j=1}^{2n} \underline{s_{\gamma}^{(j)}}x_{\gamma}^{(j)}\biggr)\biggr)\\[-3pt]
&\cdot\biggl(\prod_{i=n-1}^1 \underline{s_i}\biggl(\prod_{\gamma\in \mathcal M_{2n}^{2i}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\,\biggl|\,\prod_{\ell=1}^{2n} x_{\gamma+j}^{(\ell)}\,\biggr|\,\biggr)\biggr)\biggr)\cdot \underline{s}x\cdot\,\biggl|\,\prod_{i=1}^{2n} x_1^{(i)}\biggr|\,\cdot y
\end{aligned}
\end{equation}
is the full decomposition of $\mathbf a_n$ whenever $\chi(xy)=xy$, and the full decomposition of $\mathbf a_n'$ whenever $\chi(xy)=yx$.
\end{remark}
The following two observations play an important role below.
\begin{remark}
\label{subwords of word equivalent to b_n}
Suppose that
\begin{equation}
\label{equalities for b_n}
\{\zeta_\ell(x_{2\ell+1}),\zeta_\ell(y_{2\ell+1})\}=\{x_{2\ell+1},y_{2\ell+1}\}
\end{equation}
for all $0\le \ell\le n-1$. Then every subword of length $>1$ of the word
\begin{equation}
\label{word equivalent to b_n}
x_0y_0\biggl(\prod_{i=1}^n \mathbf f_{2i-1}\biggr)\cdot \mathbf e_{2n}\cdot\biggl(\prod_{i=n-1}^0 s_{2i}x_{2i}\zeta_i(x_{2i+1})\zeta_i(y_{2i+1})y_{2i}\biggr)
\end{equation}
has exactly one occurrence in this word.
\end{remark}
\begin{remark}
\label{subwords of word equivalent to a_n}
Suppose that
\begin{equation}
\label{equalities for a_n}
\{\zeta_\alpha(x_\alpha^{(i)})\mid 1\le i\le 2n\}=\{x_\alpha^{(i)}\mid 1\le i\le 2n\}
\end{equation}
for all odd $1\le\ell\le 2n-1$ and for all $\alpha\in \mathcal M_{2n}^\ell$. Then every subword of length $>1$ of the word
\begin{equation}
\label{word equivalent to a_n}
\begin{aligned}
&xy\cdot\biggl(\prod_{i=1}^{n-1}\underline{s_i} \biggl(\prod_{\gamma\in \mathcal M_{2n}^{2i-1}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\,\biggl|\,\prod_{\ell=1}^{2n} x_{\gamma+j}^{(\ell)}\,\biggr|\,\biggr)\biggr)\biggr)\\[-3pt]
&\cdot\biggl(\underline{s_{2n-1}}\prod_{\gamma\in \mathcal M_n^{2n-1}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\biggl(\prod_{\ell=1}^{2n}\,|\,x_{\gamma+j}^{(\ell)}\,|\,\biggr)\biggr)\biggr)\cdot\biggl(\prod_{\gamma\in \mathcal M_{2n}^{2n}} \biggl(\prod_{j=1}^{2n} \underline{s_{\gamma}^{(j)}}x_{\gamma}^{(j)}\biggr)\biggr)\\[-3pt]
&\cdot\biggl(\prod_{i=n-1}^1\underline{s_i}\biggl(\prod_{\gamma\in \mathcal M_{2n}^{2i}} \biggl(\prod_{j=1}^{2n} x_{\gamma}^{(j)}\,\biggl|\,\prod_{\ell=1}^{2n} \zeta_{\gamma+j}(x_{\gamma+j}^{(\ell)})\,\biggr|\,\biggr)\biggr)\biggr)\cdot \underline{s}x\cdot\,\biggl|\,\prod_{i=1}^{2n} \zeta_1(x_1^{(i)})\,\biggr|\,\cdot y
\end{aligned}
\end{equation}
has exactly one occurrence in this word.
\end{remark}
Remarks~\ref{subwords of word equivalent to b_n} and~\ref{subwords of word equivalent to a_n} follow from the directly verifiable fact that if $\mathbf w$ is one of the words~\eqref{word equivalent to b_n} or~\eqref{word equivalent to a_n} and $ab$ is a subword of the word $\mathbf w$ then this subwords has exactly one occurrence in this word.
\smallskip
\begin{lemma}
\label{substitution a_k in b_n}
Let $n$ be a natural number, $\xi$ be an endomorphism of $F^1$ and $\mathbf w\approx \mathbf w'$ be a non-trivial identity. Suppose that the word $\mathbf w$ coincides with the word~\eqref{word equivalent to b_n} where $\zeta_1,\zeta_2,\dots, \zeta_{n-1}$ are endomorphisms of $F^1$ such that the equality~\eqref{equalities for b_n} is true for all $0\le\ell\le n-1$. Then if $\mathbf w=\mathbf u\xi(\mathbf a_k)\mathbf v$ and $\mathbf w'=\mathbf u\xi(\mathbf a_k')\mathbf v$ for some words $\mathbf u$ and $\mathbf v$ and some $k\ge n$ then $n=k$.
\end{lemma}
\begin{proof}
We note that $\xi(x)\ne\lambda$ and $\xi(y)\ne\lambda$ because the identity $\mathbf w\approx \mathbf w'$ is non-trivial. It follows that the length of the word $\xi(xy)$ is more than $2$. In view of Remark~\ref{subwords of word equivalent to b_n}, the words $\xi(x)$ and $\xi(y)$ are letters. Remark~\ref{rm s-decompositions of a_n,a_n'} implies that the words $\mathbf a_k$ and $\mathbf a_k'$ are $1$-equivalent. According to Lemma~\ref{word problem M} the identity $\mathbf a_k\approx \mathbf a_k'$ holds in $\mathbf M$. Consequently, $\mathbf M$ satisfies the identity $\mathbf w\approx\mathbf w'$. Since the first occurrences of the letters $\xi(x)$ and $\xi(y)$ occur in the words $\mathbf w$ and $\mathbf w'$ in the opposite order, Lemma~\ref{word problem M} implies that some subblock of some $1$-block of the word $\mathbf w$ contains the subword $\xi(xy)$. It is easy to see that the full decomposition of the word $\mathbf w$ has the form
\begin{equation}
\label{decomposition of e-b_n}
\begin{aligned}
x_0y_0\cdot\biggl(\prod_{i=1}^{n-1} \underline{s_{2i-1}}x_{2i-1}\,|\,x_{2i}y_{2i}\,|\,y_{2i-1}\biggr)\cdot&\underline{s_{2n-1}}x_{2n-1}\,|\,x_{2n}\,|\,y_{2n}\,|\,y_{2n-1}\\[-3pt]
\cdot\,\underline{s_{2n}}x_{2n}\underline{t_{2n}}y_{2n}\cdot\biggl(\prod_{i=n-1}^0 \underline{s_{2i}}x_{2i}&\,|\,\zeta_i(x_{2i+1})\zeta_i(y_{2i+1})\,|\,y_{2i}\biggr).
\end{aligned}
\end{equation}
Then $\xi(xy)=x_{2p}y_{2p}$ for some $0\le p< n$, whence $\xi(x)=x_{2p}$ and $\xi(y)=y_{2p}$. So,
$$
\zeta_p(x_{2p+1})\zeta_p(y_{2p+1})=\prod_{i=1}^{2k} \xi(x_1^{(i)}).
$$
In view of Remark~\ref{subwords of word equivalent to b_n} and the equality~\eqref{equalities for b_n}, there are $c_1<d_1$ such that $\xi(x_1^{(c_1)})=\zeta_p(x_{2p+1})$ and $\xi(x_1^{(d_1)})=\zeta_p(y_{2p+1})$. Since the first occurrence of $x_1^{(c_1)}$ precedes the first occurrence of $x_1^{(d_1)}$ in $\mathbf a_k$, we have that the first occurrence of $\xi(x_1^{(c_1)})$ precedes the first occurrence of $\xi(x_1^{(d_1)})$ in $\mathbf w$, whence $\xi(x_1^{(c_1)})=x_{2p+1}$ and $\xi(x_1^{(d_1)})=y_{2p+1}$.
Further, we will prove by induction that for all $1\le q\le 2n-2p$ the equalities
\begin{equation}
\label{xi(x_2p+q) and xi(y_2p+q)}
x_{2p+q}=\xi(x_{\alpha_q}^{c_q}) \text{ and } y_{2p+q}=\xi(x_{\beta_q}^{d_q})
\end{equation}
are true for some letters $x_{\alpha_q}^{c_q}$ and $x_{\beta_q}^{d_q}$ such that $\alpha_q,\beta_q\in \mathcal M_{2k}^q$ and the first occurrence of $x_{\alpha_q}^{c_q}$ precedes the first occurrence of $x_{\beta_q}^{d_q}$ in $\mathbf w$, i.e.,
\begin{equation}
\label{condition for alpha_q beta_q}
\alpha_q\le\beta_q \text{ and if }\alpha_q=\beta_q \text{ then }c_q< d_q.
\end{equation}
The induction base is considered in the previous paragraph. Suppose that for all $1\le r< q\le 2n-2p$ there exist letters $x_{\alpha_r}^{c_r}$ and $x_{\beta_r}^{d_r}$ such that $x_{2p+r}=\xi(x_{\alpha_r}^{c_r})$ and $y_{2p+r}=\xi(x_{\beta_r}^{d_r})$, $\alpha_r,\beta_r\in \mathcal M_{2k}^r$, $\alpha_r\le\beta_r$ and if $\alpha_r=\beta_r$ then $c_r\le d_r$. We need to check that there are letters $x_{\alpha_q}^{c_q}$ and $x_{\beta_q}^{d_q}$ such that $\alpha_q,\beta_q\in \mathcal M_{2k}^q$ and the claims~\eqref{xi(x_2p+q) and xi(y_2p+q)} and~\eqref{condition for alpha_q beta_q} are true.
If $a,b\in\mul(\mathbf w)$ and $i\in\{1,2\}$ then $\mathbf w_i[a,b]$ denotes the subword of the word $\mathbf w$ located between $i$th occurrences of $a$ and $b$.
Suppose that $q$ is odd. Then
$$
\zeta_{\frac{2p+q-1}{2}}(x_{2p+q})\zeta_{\frac{2p+q-1}{2}}(y_{2p+q})=\xi(\mathbf w_2[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}]).
$$
We note that if $i=2$ and $\alpha_{q-1}=\beta_{q-1}$ then
\begin{equation}
\label{w_i[] short}
\mathbf w_i[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}]=\biggl(\prod_{\ell=1}^{2k} x_{\alpha_{q-1}+c_{q-1}}^{(\ell)}\biggr)\cdot\biggl(\prod_{j=c_{q-1}+1}^{d_{q-1}-1} x_{\alpha_{q-1}}^{(j)}\biggl(\prod_{\ell=1}^{2k} x_{\alpha_{q-1}+j}^{(\ell)}\biggr)\biggr),
\end{equation}
while if $i=2$ and $\alpha_{q-1}<\beta_{q-1}$ then
\begin{equation}
\label{w_i[] long}
\begin{aligned}
&\mathbf w_i[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}]=\biggl(\prod_{\ell=1}^{2k} x_{\alpha_{q-1}+c_{q-1}}^{(\ell)}\biggr)\cdot\biggl(\prod_{j=c_{q-1}+1}^{2k} x_{\alpha_{q-1}}^{(j)}\biggl(\prod_{\ell=1}^{2k} x_{\alpha_{q-1}+j}^{(\ell)}\biggr)\biggr)\\[-3pt]
&\cdot\biggl(\prod_{\alpha_{q-1}<\gamma<\beta_{q-1}}\biggl(\prod_{j=1}^{2k} x_{\gamma}^{(j)}\biggl(\prod_{\ell=1}^{2k} x_{\gamma+j}^{(\ell)}\biggr)\biggr)\biggr)\cdot\biggl(\prod_{j=1}^{d_{q-1}-1} x_{\beta_{q-1}}^{(j)}\biggl(\prod_{\ell=1}^{2k} x_{\beta_{q-1}+j}^{(\ell)}\biggr)\biggr).
\end{aligned}
\end{equation}
By the induction assumption, the endomorphism $\xi$ maps all letters located between the first occurrences of $x_{\alpha_{q-1}}^{(c_{q-1})}$ and $x_{\beta_{q-1}}^{(d_{q-1})}$ in $\mathbf w$ into the empty word, i.e.,
$$
\xi(\mathbf w_1[x_{\alpha_{q-1}}^{(c_{q-1})},x_{\beta_{q-1}}^{(d_{q-1})}])=\lambda.
$$
Then, taking into account the equalities~\eqref{w_i[] short} and~\eqref{w_i[] long}, we have that if $\alpha_{q-1}=\beta_{q-1}$ then the equality
\begin{equation}
\label{xi(w_i[]) short}
\xi(\mathbf w_i[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}])=\prod_{j=c_{q-1}}^{d_{q-1}-1}\biggl(\prod_{\ell=1}^{2k} \xi(x_{\alpha_{q-1}+j}^{(\ell)})\biggr)
\end{equation}
with $i=2$ is true, while if $\alpha_{q-1}<\beta_{q-1}$ then the equality
\begin{equation}
\label{xi(w_i[]) long}
\begin{aligned}
&\xi(\mathbf w_i[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}])=\biggl(\prod_{j=c_{q-1}}^{2k} \biggl(\prod_{\ell=1}^{2k} \xi(x_{\alpha_{q-1}+j}^{(\ell)})\biggr)\biggr)\\
&\cdot\biggl(\prod_{\alpha_{q-1}<\gamma<\beta_{q-1}}\biggl(\prod_{j=1}^{2k} \biggl(\prod_{\ell=1}^{2k} \xi(x_{\gamma+j}^{(\ell)})\biggr)\biggr)\biggr)\cdot
\biggl(\prod_{j=1}^{d_{q-1}-1} \biggl(\prod_{\ell=1}^{2k} \xi(x_{\beta_{q-1}+j}^{(\ell)})\biggr)\biggr)
\end{aligned}
\end{equation}
with $i=2$ is true. This fact and Remark~\ref{subwords of word equivalent to b_n} imply that there exist $\alpha_q,\beta_q\in \mathcal M_{2k}^q$ and $1\le c_q,d_q\le 2k$ such that the claim~\eqref{condition for alpha_q beta_q} is true, $\zeta_{\frac{2p+q-1}{2}}(x_{2p+q})=\xi(x_{\alpha_q}^{c_q})$ and $\zeta_{\frac{2p+q-1}{2}}(y_{2p+q})=\xi(x_{\beta_q}^{d_q})$. Since the first occurrence of $x_{\alpha_q}^{c_q}$ precedes the first occurrence of $x_{\beta_q}^{d_q}$ in $\mathbf a_k$, we obtain that the first occurrence of $\xi(x_{\alpha_q}^{c_q})$ precedes the first occurrence of $\xi(x_{\beta_q}^{d_q})$ in $\mathbf w$, whence $\xi(x_{\alpha_q}^{c_q})=\zeta_{\frac{2p+q-1}{2}}(x_{2p+q})=x_{2p+q}$ and $\xi(x_{\beta_q}^{d_q})=\zeta_{\frac{2p+q-1}{2}}(y_{2p+q})=y_{2p+q}$. So, we have proved the equality~\eqref{xi(x_2p+q) and xi(y_2p+q)} for all odd $q$.
Suppose now that $q$ is even. Then $x_{2p+q}y_{2p+q}=\xi(\mathbf w_1[x_{\alpha_{q-1}}^{c_{q-1}},x_{\beta_{q-1}}^{d_{q-1}}])$. Note that if $\alpha_{q-1}=\beta_{q-1}$ then the equality~\eqref{w_i[] short} is true whenever $i=1$ and if $\alpha_{q-1}<\beta_{q-1}$ then the equality~\eqref{w_i[] long} with $i=1$ is true. By the induction assumption, the endomorphism $\xi$ maps all letters located between the second occurrences of $x_{\alpha_{q-1}}^{(c_{q-1})}$ and $x_{\beta_{q-1}}^{(d_{q-1})}$ in $\mathbf w$ into the empty word, i.e. $\xi(\mathbf w_2[x_{\alpha_{q-1}}^{(c_{q-1})},x_{\beta_{q-1}}^{(d_{q-1})}])=\lambda$. Then, taking into account the equalities~\eqref{w_i[] short} and~\eqref{w_i[] long}, we have that if $\alpha_{q-1}=\beta_{q-1}$ then the equality~\eqref{xi(w_i[]) short} with $i=1$ is true and if $\alpha_{q-1}<\beta_{q-1}$ then the equality~\eqref{xi(w_i[]) long} with $i=1$ is true. This fact and Remark~\ref{subwords of word equivalent to b_n} imply that there exist $\alpha_q,\beta_q\in \mathcal M_{2k}^q$ and $1\le c_q,d_q\le 2k$ such that the claim~\eqref{condition for alpha_q beta_q} is true, $x_{2p+q}=\xi(x_{\alpha_q}^{c_q})$ and $y_{2p+q}=\xi(x_{\beta_q}^{d_q})$.
Thus, we have shown that, for any $1\le q\le 2n-2p$, there are letters $x_{\alpha_q}^{c_q}$ and $x_{\beta_q}^{d_q}$ such that $\alpha_q,\beta_q\in \mathcal M_{2k}^q$ and the claims~\eqref{xi(x_2p+q) and xi(y_2p+q)} and~\eqref{condition for alpha_q beta_q} are true. In particular, $x_{2n}=\xi(x_{\alpha_{2n-2p}}^{c_{2n-2p}})$ and $y_{2n}=\xi(x_{\beta_{2n-2p}}^{d_{2n-2p}})$. It follows that $\xi(\mathbf w_2[x_{\alpha_{2n-2p}}^{c_{2n-2p}},x_{\beta_{2n-2p}}^{d_{2n-2p}}])=t_{2n}$. If $k>n$ then $\con(\mathbf w_2[x_{\alpha_{2n-2p}}^{c_{2n-2p}},x_{\beta_{2n-2p}}^{d_{2n-2p}}])\subseteq\mul(\mathbf a_k)$. This contradicts the fact that $t_{2n}\in\simple(\mathbf w)$.
\end{proof}
\section{Proof of the main result}
\label{proof of theorem}
(i) We are going to verify that the lattice $L(\mathbf N\vee\overleftarrow{\mathbf M})$ ''modulo'' the interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$ has the form shown in Fig.~\ref{L(N vee dual M)}. In view of~\cite[Proposition~5.2]{Gusev-Vernikov-18} and~\cite[Fig.~1]{Jackson-05}, the lattices $L(\mathbf N)$ and $L(\mathbf M\vee\overleftarrow{\mathbf M})$ have the form shown in Fig.~\ref{L(N vee dual M)}. Let $\mathbf V$ be a proper subvariety of the variety $\mathbf N\vee\overleftarrow{\mathbf M}$ which is not contained in $\mathbf N$ and $\mathbf M\vee\overleftarrow{\mathbf M}$. We need to check that $\mathbf V$ belongs to the interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$.
A variety of monoids is said to be \emph{completely regular} if it consists of \emph{completely regular monoids}~(i.e., unions of groups). If the variety $\mathbf V$ is completely regular then it is a variety of \emph{bands} (i.e. idempotent monoids) because it satisfies the identity
\begin{equation}
\label{xx=xxx}
x^2\approx x^3.
\end{equation}
Evidently, every variety of bands which satisfies the identity
\begin{equation}
\label{xxy=yxx}
x^2y\approx yx^2,
\end{equation}
is commutative. Therefore, $\mathbf V$ is one of the varieties $\mathbf T$ or $\mathbf{SL}$, a contradiction. So, $\mathbf V$ is non-completely regular. Suppose that $\mathbf D_2 \nsubseteq \mathbf V$. Then it follows from~\cite[Lemma 2.15]{Gusev-Vernikov-18} that $\mathbf V$ satisfies the identity
\begin{equation}
\label{xyx=x^qyx^r}
xyx\approx x^qyx^r
\end{equation}
where either $q>1$ or $r>1$. If $q>1$ then $\mathbf V$ satisfies the identities
$$
xyx\stackrel{\eqref{xyx=x^qyx^r}}\approx x^qyx^r\stackrel{\eqref{xx=xxx}}\approx x^2yx^r\stackrel{\eqref{xxy=yxx}}\approx yx^{2+r}\stackrel{\eqref{xx=xxx}}\approx yx^2\stackrel{\eqref{xxy=yxx}}\approx x^2y,
$$
whence $\mathbf V\subseteq \mathbf D_1$, a contradiction. If $r>1$ then the identities
$$
xyx\stackrel{\eqref{xyx=x^qyx^r}}\approx x^qyx^r\stackrel{\eqref{xx=xxx}}\approx x^qyx^2\stackrel{\eqref{xxy=yxx}}\approx x^{2+q}y\stackrel{\eqref{xx=xxx}}\approx x^2y\stackrel{\eqref{xxy=yxx}}\approx yx^2
$$
hold in $\mathbf V$. We obtain a contradiction again. Thus, $\mathbf D_2 \subseteq \mathbf V$.
If $\mathbf M\nsubseteq\mathbf V$ then it follows from~\cite[Lemma~4.9(i)]{Gusev-Vernikov-18} that the variety $\mathbf V$ satisfies the identity $\sigma_1$. Therefore, $\mathbf V\subseteq \overleftarrow{\mathbf N}$. The lattice $L(\overleftarrow{\mathbf N})$ is isomorphic to the lattice $L(\mathbf N)$. Besides that, $\mathbf V\ne \overleftarrow{\mathbf N}$. This implies the wrong inclusion $\mathbf V\subseteq \overleftarrow{\mathbf M}\subset \mathbf M\vee\overleftarrow{\mathbf M}$. By symmetry, if $\overleftarrow{\mathbf M}\nsubseteq\mathbf V$ then $\mathbf V\subseteq\mathbf N$. This inclusion is also impossible. Thus, we have shown that $\mathbf M\vee\overleftarrow{\mathbf M}\subseteq \mathbf V$. Therefore, $\mathbf V\in[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$.
(ii) We denote an identity basis of the variety $\mathbf N\vee\overleftarrow{\mathbf M}$ by $\Sigma$. Let $\mathrm K$ be a subset of $\mathbb N$. Put $\Sigma_{\mathrm K}=\{\mathbf a_n\approx \mathbf a_n'\mid n \in \mathrm K\}$. We are going to verify that different subsets of the form $\Sigma_{\mathrm K}$ define different subvarieties within the variety $\mathbf N\vee\overleftarrow{\mathbf M}$. Arguing by contradiction, suppose that there are $n$ and $\mathrm K\subseteq \mathbb N$ such that $n\notin \mathrm K$ and the identity $\mathbf a_n\approx \mathbf a_n'$ follows from the identity system $\Sigma\cup \Sigma_{\mathrm K}$. Then there exists a sequence of words $\mathbf a_n= \mathbf w_0,\mathbf w_1,\dots,\mathbf w_m= \mathbf a_n'$ such that, for any $i\in\{0,1,\dots,m-1\}$ there exist words $\mathbf u_i,\mathbf v_i\in F^1$, an endomorphism $\xi_i$ of $F^1$ and an identity $\mathbf p_i\approx \mathbf q_i\in \Sigma\cup \Sigma_{\mathrm K}$ such that $\mathbf w_i= \mathbf u_i\xi_i(\mathbf p_i)\mathbf v_i$ and $\mathbf w_{i+1}= \mathbf u_i\xi_i(\mathbf q_i)\mathbf v_i$. We can assume without loss of generality that $\mathbf w_i\ne\mathbf w_{i+1}$ for all $i\in\{0,1,\dots,m-1\}$. The words $\mathbf a_n$ and $\mathbf a_n'$ are $1$-equivalent by Remark~\ref{rm s-decompositions of a_n,a_n'}. But the $1$-block $xy$ of $\mathbf a_n$ does not coincide with the corresponding $1$-block $yx$ of $\mathbf a_n'$. Then Lemma~\ref{word problem N} implies that the variety $\mathbf N$ violates the identity $\mathbf a_n\approx \mathbf a_n'$, whence there is a number $r\in\{0,1,\dots,m-1\}$ such that $\mathbf p_r\approx \mathbf q_r$ equals one of the identities $\mathbf a_k\approx \mathbf a_k'$ or $\mathbf a_k'\approx \mathbf a_k$ for some $k\ne n$. Let $r$ be the least number with such a property. Then the identity $\mathbf a_n\approx \mathbf w_r$ holds in the variety $\mathbf N\vee\overleftarrow{\mathbf M}$. In view of Remark~\ref{rm s-decompositions of a_n,a_n'}, the full decomposition of the word $\mathbf a_n$ has the form~\eqref{s-decompositions of a_n,a_n'} with $\chi(xy)=xy$. Then Lemma~\ref{word problem N} and the dual to Lemma~\ref{word problem M} imply that the word $\mathbf w_r$ coincides with~\eqref{word equivalent to a_n} where the equality~\eqref{equalities for a_n} is true for all odd $1\le\ell\le 2n-1$ and for all $\alpha\in \mathcal M_{2n}^\ell$.
We note that $\xi_r(x)\ne\lambda$ and $\xi_r(y)\ne\lambda$ because the identity $\mathbf w_r\approx \mathbf w_{r+1}$ is non-trivial. This implies that the length of the word $\xi_r(xy)$ is more than $2$. In view of Remark~\ref{subwords of word equivalent to a_n}, the words $\xi_r(x)$ and $\xi_r(y)$ are letters. Since the first occurrences of the letters $\xi_r(x)$ and $\xi_r(y)$ occur in the words $\mathbf w_r$ and $\mathbf w_{r+1}$ in the opposite order, Lemma~\ref{word problem M} implies that some subblock of some $1$-block of the word $\mathbf w_r$ contains the subword $\xi_r(xy)$ whenever $\mathbf p_r=\mathbf a_k$, and the subword $\xi_r(yx)$ whenever $\mathbf p_r=\mathbf a_k'$. Recall that the full decomposition of the word $\mathbf w_r$ has the form~\eqref{word equivalent to a_n}. Therefore, if the identity $\mathbf p_r\approx \mathbf q_r$ equals $\mathbf a_k'\approx \mathbf a_k$ then either $\xi_r(yx)=xy$ or $\xi_r(yx)=x_\gamma^{(p)}x_\gamma^{(p+1)}$ for some $1\le p< 2n$, $1\le h<n$ and $\gamma\in \mathcal M_{2n}^{2h}$. Then either $\xi_r(y)=x$ and $\xi_r(x)=y$ or $\xi_r(y)=x_\gamma^{(p)}$ and $\xi_r(x)=x_\gamma^{(p+1)}$. Since the second occurrence of $x$ precedes the second occurrence of $y$ in $\mathbf a_k'$, we have that the second occurrence of $\xi(x)$ precedes the second occurrence of $\xi(y)$ in $\mathbf w_r$. But this is impossible, because the second occurrence of $y$ is preceded by the second occurrence of $x$ in $\mathbf w_r$, while the second occurrence of $x_\gamma^{(p+1)}$ is preceded by the second occurrence of $x_\gamma^{(p)}$ in $\mathbf w_r$. So, the identity $\mathbf p_r\approx \mathbf q_r$ cannot coincide with the identity $\mathbf a_k'\approx \mathbf a_k$ and, therefore, $\mathbf p_r\approx \mathbf q_r$ equals $\mathbf a_k\approx \mathbf a_k'$.
Suppose that $k<n$. Since the full decomposition of the word $\mathbf w_r$ has the form~\eqref{word equivalent to a_n}, either $\xi_r(xy)=xy$ or $\xi_r(xy)=x_\gamma^{(p)}x_\gamma^{(p+1)}$ for some $1\le p< 2n$, $1\le h<n$ and $\gamma\in \mathcal M_{2n}^{2h}$. If $\xi(xy)=xy$ then
$$
\prod_{i=1}^{2k} \xi_r(x_1^{(i)})=\prod_{i=1}^{2n} \zeta_1(x_1^{(i)}).
$$
Since $k<n$, there is $1\le i\le 2k$ such that the length of the word $\xi(x_1^{(i)})$ is more than $1$. This contradicts the claim~\eqref{equalities for a_n} and Remark~\ref{subwords of word equivalent to a_n}. If $\xi_r(xy)=x_\gamma^{(p)}x_\gamma^{(p+1)}$ for some $1\le p< 2n$, $1\le h<n$ and $\gamma\in \mathcal M_{2n}^{2h}$ then
$$
\prod_{i=1}^{2k} \xi_r(x_1^{(i)})=\prod_{i=1}^{2n} \zeta_{\gamma+p}(x_{\gamma+p}^{(i)}).
$$
Taking into account the claim~\eqref{equalities for a_n}, we get a contradiction with Remark~\ref{subwords of word equivalent to a_n}.
Suppose now that $k>n$. We need some more notation. The smallest element of the set $\mathcal M_s^t$ is denoted by $\gamma_s^t$. Further, we denote by $\eta$ the endomorphism which is defined by the following equalities:
\begin{align*}
\eta(x)=x_0,\ \eta(y)=y_0,&\ \eta(s)=s_0,\ \eta(s_{q'})=s_{q'},\\
\eta(x_{\gamma_{2n}^q}^{(1)})=x_q,\ \eta(x_{\gamma_{2n}^q}^{(2)})=y_q,&
\ \eta(s_{\gamma_{2n}^{2n}}^{(1)})=s_{2n},\ \eta(s_{\gamma_{2n}^{2n}}^{(2)})=t_{2n},\\
\eta(z)&=\lambda,
\end{align*}
where $1\le q\le 2n$, $1\le q'\le 2n-1$ and $z$ is an arbitrary letter which differs from $x$, $y$, $s$, $s_{q'}$, $x_{\gamma_{2n}^q}^{(1)}$, $x_{\gamma_{2n}^q}^{(2)}$, $s_{\gamma_{2n}^{2n}}^{(1)}$ and $s_{\gamma_{2n}^{2n}}^{(2)}$.
We note that the word $\eta(\mathbf w_r)$ equals the word~\eqref{word equivalent to b_n} for some endomorphisms $\zeta_1,\zeta_2,\dots, \zeta_{n-1}$ of $F^1$ such that the equality~\eqref{equalities for b_n} is true for all $0\le\ell\le n-1$. Obviously, $\eta(\mathbf w_r)=\eta(\mathbf u_r)\eta(\xi_r(\mathbf p_r))\eta(\mathbf v_r)$ and $\eta(\mathbf w_{r+1})=\eta(\mathbf u_r)\eta(\xi_r(\mathbf q_r))\eta(\mathbf v_r)$. Note that the identity $\eta(\mathbf w_r)\approx \eta(\mathbf w_{r+1})$ is non-trivial because the first occurrences of the letters $\eta(\xi_r(x))$ and $\eta(\xi_r(y))$ occur in the words $\eta(\xi_r(\mathbf p_r))$ and $\eta(\xi_r(\mathbf q_r))$ in the opposite order. Then we have a contradiction with Lemma~\ref{substitution a_k in b_n} and inequality $n<k$. So, we have proved that different subsets of the form $\Sigma_{\mathrm K}$ define different subvarieties within the variety $\mathbf N\vee\overleftarrow{\mathbf M}$. All these subvarieties belongs to the interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$ by Lemma~\ref{word problem M}, the dual to Lemma~\ref{word problem M} and Remark~\ref{rm s-decompositions of a_n,a_n'}. This implies that the lattice of all subsets of $\mathbb N$ order-embeds into the interval $[\mathbf M\vee\overleftarrow{\mathbf M},\mathbf N\vee\overleftarrow{\mathbf M}]$. It is well known that the lattice of all subsets of $\mathbb N$ is uncountable and violates the ascending chain condition and the descending chain condition. Theorem~\ref{main result} is proved.\qed
\smallskip
The following statement provide some more new examples of finitely generated varieties of monoids with continuum many subvarieties.
\begin{corollary}
\label{D_2 join G}
If $G$ is a finite non-Abelian group then the monoid $S(xtx)\times G$ generates a variety with continuum many subvarieties.
\end{corollary}
\begin{proof}
Let $\mathbf G$ denotes the monoid variety generated by $G$. We note that the monoid $S(xtx)$ generates the variety $\mathbf D_2$~\cite{Jackson-Sapir-00}. It follows that $\mathbf D_2\vee \mathbf G$ is generated by $S(xtx)\times G$. The word $xtx$ is an isoterm for $\mathbf D_2\vee \mathbf G$ by Lemma~\ref{S(W) in V}. Then~\cite[Fact~3.1(ii)]{Sapir-15} implies that if the variety $\mathbf D_2\vee \mathbf G$ satisfies a non-trivial identity $xysxty\approx \mathbf w$ then $\mathbf w=yxsxty$. But every group that satisfies the identity $\sigma_1$ is Abelian one. Hence $xysxty$ is an isoterm for $\mathbf D_2\vee \mathbf G$. According to Lemma~\ref{S(W) in V}, $S(xysxty)\in \mathbf D_2\vee \mathbf G$. Analogously, $S(xsytxy)\in \mathbf D_2\vee \mathbf G$. The word $xtx$ is an isoterm for the variety $\mathbf D_2$. Therefore, if this variety satisfies a non-trivial identity $xytxy\approx \mathbf v$ then $\mathbf v\in\{xytyx,yxtxy,yxtyx\}$. But every group that satisfies one of the identities $xytxy\approx xytyx$, $xytxy\approx yxtxy$ or $xytxy\approx yxtyx$ is Abelian one. Hence $xytxy$ is an isoterm for $\mathbf D_2\vee \mathbf G$. According to Lemma~\ref{S(W) in V}, $S(xytxy)\in \mathbf D_2\vee \mathbf G$. Analogously, $S(xytyx)\in \mathbf D_2\vee \mathbf G$.
In view of the dual to Lemma~\ref{generator of M}, $\overleftarrow{\mathbf M}$ is generated by $S(xsytxy)$, whence $\overleftarrow{\mathbf M}\subseteq \mathbf D_2\vee \mathbf G$. It follows from the dual to Example~1 in Erratum to~\cite{Jackson-05} that $\mathbf N$ is generated by the monoid $S(xysxty)$ and some quotient monoid of $S(xytxy,xytyx)$. It follows that $\mathbf N\subseteq \mathbf D_2\vee \mathbf G$. Therefore, $\mathbf N\vee\overleftarrow{\mathbf M}\subseteq \mathbf D_2\vee \mathbf G$. Theorem~\ref{main result} implies that $\mathbf D_2\vee \mathbf G$ contains continuum many subvarieties.
\end{proof}
In conclusion, we note that Fig.~\ref{L(N vee dual M)} and Theorem~\ref{main result}(ii) imply that the lattice $L(\mathbf N\vee\overleftarrow{\mathbf M})$ is non-modular. The following question seems to be interesting.
\begin{question}
\label{question non-trivial}
Does the lattice $L(\mathbf N\vee\overleftarrow{\mathbf M})$ satisfy any non-trivial identity?
\end{question}
\subsection*{Acknowledgments.} The author is sincerely grateful to Professor Vernikov for his attention and assistance in the writing of the article and to the anonymous referee for several useful remarks.
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Good.
7 comments:
Lovely card Kim, great colours and love the quilted effect. Hugs, Claire x
Beautiful. Love the design.
Have a creative day.
Hugs Nana
♥ My Crafting Channel ♥
♥ 2 Creative Chicks ♥
♥ Silhouette Challenges ♥
Love how you combined quilts and Christmas!
Beautiful card Kim, love the quilting and the cute image.
Pat xx
Kim this is sooo cute, I love your fabulous quilting design and the gorgeous papers! Thanks for stopping by with your sweet comment! Take care of yourself and I will pop by too!
Big Hugs, Tammy
YAY she's back!
Great card hun, love the quilting & the papers are perfect!
Hugs Shell xx
Ah Kim this is so cute, I love it, Hpope alls well with you, Much Love Hazelxox
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TITLE: Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
QUESTION [7 upvotes]: Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding places in the lower-triangle will be filled with $1$, and $2k < n^2-n$). All other elements are independent uniform random variables over $[0,1]$.
Is there known bound (lower and upper) for the largest eigenvalue of such random matrices?
If there is not, any suggestion of possible method (I can think of using Gershgorin circle) or reference to related materials is very much appreciated.
Gershgorin circle could help with the upper bound. For example, if we assume all those $1$s are in the same row, then we should be able to find the probabilistic bound for this case with Irwin–Hall distribution; but I currently have trouble dealing with the "randomly scattered" $1$s.
I am not familiar with the random matrix theory. I am not sure if there is anything from it can help this.
REPLY [3 votes]: The diagonal elements just shift the spectrum (and the top eigenvalue) by $1$. So we may assume they are $0$.
You are essentially dealing with a symmetric matrix whose entries above the diagonal are iid, sum of a Bernoulli $\{0,1\}$ of parameter (=mean) $q=2k/n^2$ and of a uniform random variable on $[0,1]$. The mean is therefore $p=q+(1-q)/2$ and the variance is $\sigma^2=q+(1-q)/3-p^2$. Since, for any value of q, the mean is bounded below and so is the variance, you are dealing with the perturbation by $\sqrt{n}$ times a Wigner matrix of a matrix of rank $1$ and norm $pn$. Thus, your top eigenvalue will concentrate around $pn$ ($\pm O(\sqrt{n}))$ by estimates on the top eigenvalue of a Wigner matrix and
Weyl's inequalities.
I earlier referred to https://arxiv.org/pdf/math/0605624v1.pdf, but this is a very different scaling, so I scraped that answer.
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What.
Where did Broun get that idea? It doesn't seem sane. What kind of extremist is this representative of you folks? The rest of us in the United States might classify him as the real extremist.
Rev. James Yeaw
Sun City, Ariz.
Athens, GA ? Athens Banner-Herald © 2015. All Rights Reserved. | Terms of Service | Privacy Policy / About Our Ads | Content Removal Policy
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Cokie Roberts, longtime ABC News, NPR reporter, political analyst, dies at 75
Cokie Roberts was both reporter and commentator and was widely respected by fellow journalists and by those she covered.Photo: Heidi Gutman / Walt Disney Television Via Getty Images
Cokie Roberts was both reporter and commentator and was widely respected by fellow journalists and by those she covered.
Caption
Cokie Roberts was both reporter and commentator and was widely respected by fellow journalists and by those she covered.
Cokie Roberts was both reporter and commentator and was widely respected by fellow journalists and by those she covered.
Cokie Roberts, the pioneering broadcast journalist known to millions for her work with ABC News and National Public Radio, died Tuesday. She was 75.
ABC News, in a posting on its website Tuesday, said the cause was breast cancer, which was first diagnosed in 2002.
Roberts started her radio career at CBS, then in 1978 began working for NPR covering Capitol Hill. She joined ABC in 1988. Her three decades at the network included anchoring, with Sam Donaldson, the news program “This Week” from 1996 to 2002.
“Cokie’s kindness, generosity, sharp intellect and thoughtful take on the big issues of the day made ABC a better place and all of us better journalists,” James Goldston, president of ABC News, said in a statement.
Roberts was both reporter and commentator during her career and was widely respected both by her fellow journalists and by those she covered.
Danielle Kurtzleben, an NPR reporter, praised Roberts as an example for younger generations of journalists.
“I’m proud as hell — proud as hell — to work at a news organization that has ‘Founding Mothers’ whom we all look up to,” she said on Twitter. “God bless Cokie Roberts.”
If Roberts brought deep knowledge and keen insight to her work, that was in part because she was a child of politicians and first walked the halls of Congress as a young girl.
Her father was Hale Boggs, a longtime Democratic representative from Louisiana who in the early 1970s was House majority leader. After he died in a plane crash in 1972, his wife and Roberts’ mother, Lindy Boggs, was elected to fill his seat. She served until 1991 and later became U.S. ambassador to the Holy See.
It was a background that gave Roberts a deep respect for the institutions of government that she covered.
Mary Martha Corinne Morrison Claiborne Boggs was born Dec. 27, 1943, in New Orleans. She said that her brother, Tommy, invented her nickname because he couldn’t say “Corinne.”
Although her father had considerable influence on her, so did her mother, who was active in furthering her father’s career, and other women she came to know like Lady Bird Johnson.
Many of her eight books explored the role of women in shaping the country, including “We Are Our Mothers’ Daughters” (1998) and “Ladies of Liberty: The Women Who Shaped Our Nation” (2008).
In 1966 she married Steven Roberts, then a correspondent for The New York Times.
She became a radio correspondent for CBS, then in 1978 joined NPR. With Nina Totenberg and Linda Wertheimer, she began to change the journalistic landscape.
Neil Genzlinger is a New York Times writer.
| 241,475
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TITLE: Proof that sequence converges
QUESTION [0 upvotes]: How can I prove that $ (a_n) = \frac{n^3 -1}{2n^3-n} $ converges?
I've calculated the limit and got a result of a 1/2.
Now I need to prove that this limit exists. So, I tried to use the definition and find an $M$ that $n > M \implies |a_n - L| < \epsilon$ for $ \epsilon > 0 $, but I couldn't reach in a result.
Is there any strategy to prove that this sequence converges?
REPLY [1 votes]: If you have proved the algebraic limit laws for sequences, then this isn't too bad. $$\begin{align*}\lim_{n\to\infty}\frac{n^3-1}{2n^3-n}&=\lim_{n\to\infty}\frac{1-\frac{1}{n^3}}{2-\frac{1}{n^2}}\\&=\frac{\lim\limits_{n\to\infty}(1-\frac{1}{n^3})}{\lim\limits_{n\to\infty}(2-\frac{1}{n^2})}\\&=\frac{1}{2}\end{align*}$$ This works because both the top and bottom limits exist and the bottom limit is non-zero for all $n\in\mathbb{N}$.
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Development and Relations Committee
The SURA Board of Trustees has appointed a Development & Relations Committee Steering Group which represents the larger full committee:
- Fred King, West Virginia University, Chair
- Leo Chalupa, The George Washington University, DTRA/STEP Subcommittee Chair
- Dennis Manos, College of William & Mary, Tech Center Subcommittee Chair
- Prakash Nagarkatti, University of South Carolina, SRNL Subcommittee Chair
- Jerry Draayer, SURA President & CEO
- Greg Kubiak, SURA Chief Public Affairs Officer, Staff Liaison
Responsibilities of the Committee:
- Identify and promote opportunities to expand existing collaborations or initiate new collaborations that build strategic alliances among SURA members and outside partners.
- Advance development activities/initiatives that support SURA’s mission, including the review and approval of teaming agreements, partnerships, and other arrangements, as deemed appropriate to fulfill its charge.
- Create scenarios to advocate for multi-institutional collaborations that complement the interest of SURA’s membership.
- Advise and support commercialization of SURA technologies through government and/or industry partnerships.
- Oversee and support an effective external relations effort that promotes SURA’s mission and programs, and proactively advocates for the same as necessary and appropriate.
- Serve as SURA’s primary committee interface with the SURA Council of Presidents, helping to ensure alignment with shared institutional objectives.
- Solicit, screen, and select the recipient of the SURA Distinguished Scientist Award.
| 186,407
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Minnesota novelist Jon Hassler died last week. His funeral will be tomorrow at the Basilica of St. Mary here in Minneapolis. I found out that a member of my parish, raised Catholic but married to a Lutheran, will be singing at his funeral. It seems that her aunt was married to Mr. Hassler.
The first book I read by Jon Hassler was Grand Opening. I read it in my mid-twenties, I think, and was instantly enchanted. It's the story of a family that moves back to the father's small home town to open a grocery story. So they will have two grocery stores -- the "Catholic" one and the "Lutheran" one. I found Grand Opening to be a kinder, gentler Main Street. There were plenty of critiques of small town small-mindedness and prejudice, but there was an underlying affection that made those critiques much easier to take.
After Grand Opening, I moved backward to Hassler's first novel, Staggerford: one week in the life of a high school teacher. Then I read the novel that I think is still my favorite: Simon's Night, about an old man who checks himself into a nursing home after he makes some unfortunate mistakes in his kitchen. Simon's Night is both poignantly sad and funny, at the same time. I remember that the small Lyric Theatre here in my hometown adapted it for the stage.
The summer I married, I went to a 2-day writing workshop at Mount Carmel Lutheran Camp near Alexandria. Our workshop leader was Jon Hassler. Besides spending a lot of time journaling that weekend, I remember him talking about his Parkinson's disease, his gratitude for his wife, and making jokes about the Catholic at a Lutheran camp. (Why don't you get Garrison Keillor? he asked. We like you better, was the answer.)
I read a review of his life recently that said he was a "middle-brow" author. Probably correct. Also, they noted a book I haven't read, North of Hope, as his best. Now, that goes on my reading list.
He was a lovely example of an artist working out his faith in stories. His themes are universal: love, sin, redemption, sacrifice, loss. He made me wonder again, as I have off and on throughout my life, whether I too can work out my faith in stories.
Maybe, in the end, it is the only way we can work out our faith.
7 comments:
Thanks for a wonderful lifting up of an author I will have to check out. I do think, as you say, we work out faith in our stories, the ones we write, or preach and the ones we share with a friend over coffee when we scratch our heads and ask "why" or shed a tear, or rejoice.
I haven't read any of Hassler's work. I'll have to put him on the "want to read" list.
Thanks for writing this Diane. I think you're on to something.
That is a really nice post and may he rest in peace.
As Lindy says- you are onto something.
i'd say yes stories are foundational to our faith - they are the blocks we use, the art of choice in how we pass on, wrestle with, & celebrate the mystery... some are better storytellers. some are better listeners. some better writers. some better at sensing the untold stories. all gifts. but the same Spirit.
Another author to add to my reading list...thank you for your thoughtful post.
Grand Opening and Stagerford are classics. You are so right, this man from Plainview will be missed; not only in these parts but I hope beyond.
He made sense of people and the ways we don't always make sense.
It's sad to loose a great story teller but it's good to have the stories around.
| 211,068
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Child Custody & Spousal Support Modifications
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Stress Analysis Assignment Help
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Stress analysis might be carried out through classical mathematical methods, analytic mathematical modelling or computational simulation, speculative screening, or a mix of approaches. In a stress analysis, Element professionals attach bondable single and multi-axis pressure gages to your item. The item is then subjected to a load and/or other ecological tensions, such as vibration, temperature level, or pressure. Mathematically, the state of stress at a point in a flexible body is identified by 6 independent stress parts and is defined by a second-order symmetric Cartesian tensor, likewise understood as the stress tensor. The worths of these stress parts alter with the orientation of the coordinate system in which each stress part is specified. Such details is beneficial in determining the stress concentrations in a stress analysis.
Advanced stress analysis is engineering disciplines which have lots of approaches for figuring out the stress and pressure in structures and products with regard to the forces filled on them. In sophisticated stress analysis, deflections and tensions are examined in the engineering structures under various loading conditions and likewise load vectors and tightness matrices are identified for the iso-parametric issues. Stress stress analysis (or stress analysis) is an engineering discipline that utilizes lots of approaches to figure out the tensions and pressures in products and structures subjected to forces. In continuum mechanics, stress is a physical amount that reveals the internal forces that surrounding particles of a constant product put in on each other, while stress is the step of the contortion of the product.
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Zane Hijazi
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Zane Hijazi, an American YouTuber, is famous for his hilarious videos and vines on his YouTube channel named, Zane Hijazi. With his growing popularity, his channel has reached around 2.8 million subscribers. Zane came to fame with his partner, Heath Husar, with the help of Vine videos.
Zane was born in Miami, Florida, USA on 18th November 1992. He is of American nationality, and his birth sign is Scorpio. Zane shared his childhood with his two younger sisters and a brother.
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Zane Hijazi assembles the total assets just like a YouTube Star. He has been acquiring the income from his YouTube channel name ‘Zane Hijazi.’ According to Social Blade, Zane has expected earnings of around $72K and $1.2 Million. He likewise is gathering the income from channel Zane and Heath, working together with Heath Hussar. With his works, Zane’s net worth is estimated at $3 million.
Zane’s most watched videos on YouTube include “SURPRISING MY LITTLE SISTER WITH NEW CAR!!” with 4.2 million views, “LIZA FOUND THIS LIVING IN HER CAR!!” with 6.2 million views and “CAUGHT THE PEOPLE WHO STOLE HEATH’S CAR” with 4.2 million views. The performer has shown up on different TV indicates like “The pizza guys” in 2016, FML in 2016 and substantially more.
Before YouTubing, Zane worked for a few organizations and worked in air terminals and inns from West Palm Beach to Miami Beach in Florida. He additionally acted in couples of motion pictures and TV arrangement like FML (2016), Average Joe (2012) and 1 Minute Horror (2015).
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The 25-year-old YouTuber, Zane is frequently observed tweeting about a sweetheart and furthermore make a video related on point sweetheart. On 7 April 2016, Zane posted the mocking photograph of himself acting like spying sweetheart. In the tweet, he expounded on the wry thing that each sweetheart would do when they question their better half.
Further on 12th May 2018, Zane posted an image of him with Jason Nash and Trisha Paytas and wrote in an inscription that he got separated from everyone else, which indicated that he isn’t dating anybody. However, his fans thought he was gay as he was spotted acting like a gay in his YouTube recordings.
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, loss of a job or health, transition in life (i.e. retiring, aging), and relationships ending. The gatherings are designed so that each session stands alone, so participants can attend any amount of sessions they choose. Download a detailed flyer about the seminars.
In each seminar, we will learn from experts and from each other, different ways to process and live with grief. We will share tools, techniques, and gifts of healing, and we’ll begin by sharing a meal together as we nourish our bodies and our hearts. Each workshop will cover all kinds of grief:
- Death of a loved one
- Becoming an empty nester
- Loss of a job or health
- Transitions in life (i.e. retiring, aging)
- Relationships ending
REGISTER TODAY! RSVP for the first workshop by September 12, 2019 to Suzanne Martinez, Suzanne.martinez@advocatehealth.com
Questions? Call: 847-842-4285
Location Sites of the Four-Part Series
Sunday, September 15, 2019 – Seasons of Grief
Location: Bethany Lutheran Church, 5-7 p.m.
76 W Crystal Lake Ave, Crystal Lake, IL Room: Luther Hall
Presenter: Christine SPUHLER, LCPC, Psychologist
Sunday, November 17, 2019 – Making it through the Holidays
Location: St. Barnabas Church, 5-7p.m.
8901 Cary Algonquin Rd, Cary, IL 60013 Room: Family Life Center
Presenter: Maggie MOORE, Widow Coach and Certified Grief Recovery Specialist
Sunday, January 19, 2020 – New Beginnings?!? Faltering Steps in Grief
Location: Bethany Lutheran Church, 5-7 p.m.
76 W Crystal Lake Ave, Crystal Lake, IL Room: Luther Hall
Presenter: TBA
Sunday, March 1, 2020 – Life… Even in Grief |Body • Mind • Spirit
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8901 Cary Algonquin Rd, Cary, IL 60013 Room: Family Life Center
Presenter: TBA
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!-Andre Gisson American. 1921-2003-->. travelled).
Allinson Gallery Index.
Paintings.
American Fine Art.
The still life painting is located in Connecticut. To order, to report broken links or to be placed on the email list, please contact Jane Allinson (jane@allinsongallery.com) or fax (860) 429 2825.
The other paintings are located at the RA GAllery in Texas. To order, please contact Rebecca Allinson (allinson.rebecca@gmail.com) Allinson.
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In Cepsa's Technological Development and Engineering department we work across the areas of research, development, and the construction of new projects to meet the needs of our company.
We construct and start up a natural fatty alcohols plant in Dumai on the island of Sumatra to produce the raw materials for detergents, cosmetics, and personal care products, among others.
Construction of a chemicals plant in Shanghai to position the company as the world's leading producer of cumene and the second in phenol and acetone.
We develop the installations and start up the BMS (Bir El Msana) oilfield in the Algerian desert.
We use IWH technology to increase the production capacity at the Ourhoud field, which boosts water treatment and its reinjection into the field
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Hotel Sarum
Buenos Aires, Distrito Federal, Argentina (AEP)
Hotel Specials
WAS
Package Specials
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Standard (Standard Room with double bed)
- One Double Bed
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\begin{document}
\title{Variations of projectivity for $C^*$-algebras}
\author{Don Hadwin}
\address{University of New Hampshire}
\email{operatorguy@gmail.com}
\author{Tatiana Shulman}
\address{Institute of Mathematics of the Polish Academy of Sciences, Poland }
\email{tshulman@impan.pl}
\subjclass[2010]{46L05}
\keywords{Projective $C^*$-algebras, RFD $C^*$-algebras, almost commuting matrices, tracial ultraproducts, order zero maps}
\maketitle
\begin{abstract} We consider various lifting problems for $C^*$-algebras. As an application of our results we show that any commuting family of
order zero maps from matrices to a von Neumann central sequence algebra can be lifted to a commuting family of order zero maps to the $C^*$-central sequence algebra.
\end{abstract}
\bigskip
\section{Introduction}
Many important properties of $C^*$-algebras are formulated in terms of liftings.
By a lifting property we mean the following. Suppose we are given a surjective $\ast$-homomorphism $\mathcal B \twoheadrightarrow \mathcal M$. We will say that a C*-algebra $\mathcal A$ has the lifting property corresponding to this surjection if for any $\ast$-homomorphism $\phi: \mathcal A \to \mathcal M$ there is a $\ast$-homomorphism $\psi: \mathcal A \to \mathcal B$ such that the following diagram commutes.
$$\xymatrix { & \mathcal B \ar@{->>}[d]^-{} \\
\mathcal A \ar[r]_-{\phi} \ar@{-->}[ur]^{\psi} & \mathcal M} $$
In other words any $\ast$-homomorphism from $\mathcal A$ to the $C^*$-algebra $\mathcal M$ "downstairs" "lifts" to a $\ast$-homomorphism to the $C^*$-algebra $\mathcal B$ "upstairs".
$C^*$-algebras which have the lifting property with respect to any surjection are called {\it projective}, they were introduced by B. Blackadar in \cite{Bl}.
Many problems that arise in C*-algebras reduce to the question of the existence of liftings in various special situations.
Here are some examples:
1) Problems about approximation of almost commuting matrices by commuting ones and, more generally, matricial weak semiprojectivity for $C^*$-algebras (\cite{LoringBook}, \cite{Lin}, \cite{FR}, \cite{ELP}), is expressed as the lifting property corresponding to the surjection $\prod_{n\in \mathbb N} M_{n} \twoheadrightarrow \prod_{n\in \mathbb N} M_{n}/\oplus_{n\in \mathbb N}M_n$ (here $M_n$ is the $C^*$-algebra of all n-by-n matrices).
2) Stability of $C^*$-algebraic relations under small Hilbert-Schmidt perturbations in matrices is expressed as the lifting property corresponding to the surjection $\prod_{n\in \mathbb N} M_n \twoheadrightarrow \prod_{n\in \mathbb N}^{\alpha} (M_n, tr_n)$ (here $\alpha$ is a non-trivial ultrafilter on $\mathbb N$ and the $C^*$-algebra $\prod_{n\in \mathbb N}^{\alpha} (M_n, tr_n)$ "downstairs" is the tracial ultraproduct of the matrix algebras) \cite{TracialStability}. Stability under small tracial perturbations in $II_1$-factors is expressed as the lifting property corresponding to the surjection $\prod_{n\in \mathbb N} N_n \twoheadrightarrow \prod_{n\in \mathbb N}^{\alpha} (N_n, \tau_n)$ (here $N_n$ is a $II_1$-factor with a faithful trace $\tau_n$) (\cite{TracialStability}). Similar problems for groups are discussed in \cite{DT2} and \cite{AP}.
3) The property of a $C^*$-algebra to be residually finite-dimensional (RFD) was proved in \cite{DonRFD} to be the lifting property corresponding to the surjection $\mathcal B \twoheadrightarrow B(H)$, where $\mathcal B \subseteq \prod M_n$ is defined as the $C^*$-algebra of all $\ast$-strongly convergent sequences of matrices and the surjection $\mathcal B \twoheadrightarrow B(H)$ is defined by sending each sequence to its $\ast$-strong limit. Here we identify $M_n$ with $B(l^2\{1, \ldots, n\}) $ naturally included in $ B(l^2\{\mathbb N\}) = B(H)$.
4) The famous Brown-Douglas-Fillmore theory deals with lifting of
injective $\ast$-homomorphisms from $C(X)$ to the Calkin algebra $\mathcal{C}(H)$ with respect to the surjection $B(H)\twoheadrightarrow \mathcal{C}(H)$.
5) In the classification program for $C^*$-algebras one sometimes has to deal with liftings of $\ast$-homomorphisms to a von Neumann central sequence algebra $N^{\omega}\cap
N^{\prime}$ to $\ast$-homomorphisms to the $C^*$-central sequence algebra $A_{\omega}\cap A^{\prime}$ (see for instance \cite{TWW}). More details on this and on the surjection $A_{\omega}\cap A^{\prime} \twoheadrightarrow N^{\omega}\cap
N^{\prime}$ are given in section 3.
We see that in the examples above the corresponding surjections sometimes have a von Neumann algebra "upstairs", sometimes "downstairs", sometimes at both places. This leads us to introducing the following more general notions.
We say that a $C^*$-algebra $\it A$ is {\it $C^*$-$W^*$-projective} if it has the lifting property with respect to any surjection $\mathcal B \twoheadrightarrow \mathcal M$ with $\mathcal M$ being a von Neumann algebra; in a similar way {\it $W^*$-$C^*$-projectivity} and {\it $W^*$-$W^*$-projectivity} are defined.
In this terminology the usual projectivity may be called $C^*$-$C^*$-projectivity.
Dealing with specific lifting problems, one has to look at liftability of projections, isometries, matrix units, various commutational relations, etc. So it is natural to explore whether and which of those basic relations have more general property of being $C^*$-$W^*$, $W^*$-$W^*$, $W^*$-$C^*$-projective, and we do it in this paper. Main focus is given to commutational relations, that is to the $C^*$-$W^*$, $W^*$-$W^*$ and $W^*$-$C^*$-projectivity of commutative $C^*$-algebras, but we consider basic non-commutative relations here as well. Note that for the usual projectivity a characterization of when a separable commutative $C^*$-algebra is projective is obtained in \cite{ChDr} and is the following: $C(K)$ is projective if and only if $K$ is a compact absolute retract of covering dimension not larger than 1.
In section 2 we give necessary definitions and discuss a relation between unital and non-unital cases.
In section 3 we study $C^*$-$W^*$-projectivity. The main result of the section is a characterization of when a separable unital commutative $C^*$-algebra is $C^*$-$W^*$-projective: $C(K)$ is $C^*$-$W^*$-projective if and only if $K$ is connected and locally path-connected (Theorem \ref{C*-W*Comm}).
Thus for commutative $C^*$-algebras $C^*$-$W^*$-projectivity is very different from the usual projectivity.
We also give restrictions on a $C^*$-algebra to be $C^*$-$W^*$-projective, namely it has to be RFD and cannot have non-trivial projections (Propositions \ref{NoProjections} and \ref{RFD}); furthermore we prove that tensoring a separable non-unital commutative $C^*$-$W^*$-projective $C^*$-algebra with matrices preserves $C^*$-$W^*$-projectivity (Theorem \ref{MatricesOverCommAlgebras}). These results are applied to certain lifting problems for order zero maps (completely positive maps preserving orthogonality).
A commonly used tool in classification of $C^{*}$-algebras is the fact that an
order zero map from the matrix algebra $M_{n}$ to any quotient $C^{*}$-algebra
lifts (the so-called projectivity of order zero maps). In particular a
possibility to lift an order zero map from $M_{n}$ to a von Neumann central sequence algebra $N^{\omega}\cap N^{\prime}$ to
an order zero map to the $C^*$-central sequence algebra $A_{\omega}\cap A^{\prime}$ is a key ingredient to
obtain uniformly tracially large order zero maps (\cite{TWW}). As an application of our results we prove
a stronger statement: one can lift any commuting family of order zero maps
$M_{n} \to N^{\omega}\cap N^{\prime}$ to a commuting family of order zero maps $M_{n}\to A_{\omega}\cap A^{\prime}$ (Theorem \ref{OrderZero}).
In section 4 we study $W^*$-$C^*$-projectivity. This seems to be the most
intractable case. We don't have a characterization of $W^*$-$C^*$-projectivity for commutative $C^*$-algebras, we only have a sufficient condition (Corollary \ref{totallydisconnected}) and, in case when the spectrum is a Peano continuum, a necessary condition (Proposition \ref{Peano}). We prove basic non-commutative results such as lifting projections and partial isometries, consider $W^*$-$C^*$-projectivity of matrix algebras, Toeplitz algebra, Cuntz algebras and discuss a relation with extension groups Ext. Techniques developed in this section are applied in section 5.
In section 5 we study $W^*$-$W^*$-projectivity. The main result here is that all separable subhomogeneous $C^*$-algebras are $W^*$-$W^*$-projective (Theorem \ref{subhomogeneous}). In particular all separable commutative $C^*$-algebras are $W^*$-$W^*$-projective. We discuss also a relation between $W^*$-$W^*$-projectivity and property RFD. It is easy to show that if a $C^*$-algebra $\mathcal{A}$ is separable nuclear $W^*$-$W^*$- projective and has a faithful trace, then it must be RFD. Moreover if Connes' embedding problem has an affirmative answer, then every unital
$W^*$-$W^*$-projective C*-algebra with a faithful trace is RFD. The converse to this statement is not true. Indeed in
\cite{TracialStability} we constructed a nuclear $C^{*}$-algebra which is RFD (hence has a faithful trace) but is
not matricially tracially stable (that is not stable under small Hilbert-Schmidt perturbations in matrices) and hence is not $W^*$-$W^*$-projective.
In this paper we give an example which is not only nuclear but even AF (Theorem \ref{AF}). Our arguments
of why it is not matricially tracially stable are much simpler than the ones
in \cite{TracialStability}.
\bigskip
{\bf Acknowledgements} The first author was supported by a
Collaboration Grant from the Simons Foundation. The research of the
second author was supported by the Polish National Science Centre grant
under the contract number DEC- 2012/06/A/ST1/00256, by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS" and Polish Government grant 3542/H2020/2016/2, and from the Eric Nordgren
Research Fellowship Fund at the University of New Hampshire.
\section{Definitions}
\begin{definition}
Suppose $\mathcal{X}$ and $\mathcal{Y}$ are classes of unital $C^*$-algebras that are
closed under isomorphism. We say that a unital $C^*$-algebra $\mathcal{A}$ is $\mathcal{X}
$\emph{-}$\mathcal{Y}$\emph{ projective} if, for every $\mathcal{B}\in \mathcal{X}$, $\mathcal{M}
\in \mathcal{Y}$ and unital surjective $\ast$-homomorphisms $\pi:\mathcal{B}
\rightarrow \mathcal{M}$ and every unital $\ast$-homomorphism $\phi:\mathcal{A}\rightarrow \mathcal{M}$, there is a unital $\ast$-homomorphism $\psi:\mathcal{A}\rightarrow \mathcal{B}$
such that $\pi\circ \psi = \phi.$
\end{definition}
$$\xymatrix { & \mathcal B \ar@{->>}[d]^-{\pi} \\
\mathcal A \ar[r]_-{\phi} \ar@{-->}[ur]^{\psi} & \mathcal M} $$
The same conditions with all the words "unital" taken away define $\mathcal{X}
$\emph{-}$\mathcal{Y}$\emph{ projectivity in the non-unital category}.
We use the term \emph{$C^*$-$W^*$-projective} when $\mathcal{X}$ is the class of all unital $C^*$-algebras and
$\mathcal{Y}$ is the class of all von Neumann algebras. We use the term \emph{$C^*$-$W^*$-projective in the non-unital category} when $\mathcal{X}$ is the class of all $C^*$-algebras and
$\mathcal{Y}$ is the class of all von Neumann algebras.
The terms
\emph{W*}-\emph{C*}\emph{-projective(in the non-unital category)}, \emph{W*-W*-projective (in the non-unital category)}, \emph{C*-C*-projective (in the non-unital category)},
are defined similarly. The usual notion of projectivity defined by B. Blackadar \cite{Bl} is the $C^*$-$C^*$-projectivity in the non-unital category.
The term $RR0$\emph{-projectivity} is used when
$\mathcal{X}=\mathcal{Y}$ is the class of unital real rank zero $C^*$-algebras.
Thus in the introduction in the formulation of some of our results we in fact should of added "in the non-unital category", which we did not do to not confuse the readers too much.
We will work mostly with the unital category, but with some exceptions. Namely in section 3 dealing with order zero maps one has to consider the non-unital case, and in sections 4 and 5
proving stability of the class of $W^*$-$W^*$ and $W^*$-$C^*$-projective $C^*$-algebras under tensoring with matrices and taking direct sums, one has to deal with the non-unital category.
In fact the relation between the unital and non-unital cases is simple. For a $C^*$-algebra $\mathcal A$, let $\tilde{ \mathcal A} = \mathcal A^+$ if $\mathcal A$ is non-unital and $\tilde{ \mathcal A} = \mathcal A \oplus \mathbb C$ if $\mathcal A$ is unital.
\begin{proposition} Let $\mathcal A$ be a $C^*$-algebra. Then $\mathcal A$ is W*-C*-projective in the non-unital category (W*-W*, C*-W*, C*-C*-projective in the non-unital category respectively) if and only if
$\tilde{\mathcal A}$ is W*-C*-projective (W*-W*, C*-W*, C*-C*-projective respectively).
\end{proposition}
\section{$C^*$-$W^*$-projectivity}
The following result puts a severe restriction on being C*-W* projective. In
particular, if $C\left( K\right) $ is C*-W* projective, then $K$ must be connected.
\begin{proposition}
\label{NoProjections} Let $\mathcal{A}$ be a unital $C^{*}$-algebra. If
$\mathcal{A}$ is C*-W* projective, then $\mathcal{A}$ is $\ast$-isomorphic to
a unital C*-subalgebra of the unitization of the cone of $\mathcal{A}^{**}$.
In particular, $\mathcal{A}$ has no non-trivial projections.
\end{proposition}
\begin{proof}
Let $\mathcal{B}$ be the unitization of the cone of $\mathcal{A}^{**}$. We
know that there is a unital $\ast$-homomorphism $\pi:\mathcal{B\rightarrow
A}^{**}$ and there is a faithful unital $\ast$-homomorphism $\rho
:\mathcal{A}\rightarrow \mathcal{A}^{**}$. If $\mathcal{A}$ is C*-W*
projective, then there must be a unital $\ast$-homomorphism $\tau
:\mathcal{A}\rightarrow \mathcal{B}$ such that $\rho=\pi \circ \tau$. Since
$\rho$ is faithful, $\tau$ is an embedding. However, $\mathcal{B}$ has no
nontrivial projections, so $\mathcal{A}$ has no nontrivial projections.
\end{proof}
\begin{proposition}\label{RFD}
Let $\mathcal{A}$ be a separable $C^{*}$-algebra. If $\mathcal{A}$ is
$C^{*}-W^{*}$-projective in either unital or non-unital category, then
$\mathcal{A}$ is RFD.
\end{proposition}
\begin{proof} Let $H = l^2(\mathbb N)$. We will identify the algebra $M_n$ of $n$-by-$n$ matrices with $$B(l^2\{1, \ldots, n\}) \subseteq B(H).$$ Let $\mathcal B \subseteq \prod M_n$ be the $C^*$-algebra of all $\ast$-strongly convergent sequences and let $I$ be the ideal of all sequences $\ast$-strongly convergent to zero. Then one can identify $\mathcal B /I$ with $B(H)$ by sending each sequence to its $\ast$-strong limit.
In \cite{DonRFD} the first-named author answered a question of Loring by proving the following: a separable $C^*$-algebra $\mathcal A$ is RFD if and only if each $\ast$-homomorhism from $\mathcal A$ to $\mathcal B /I$ lifts to a $\ast$-homomorphism from $\mathcal A$ to $\mathcal B$. Since $\mathcal B/I = B(H)$ is a von Neumann algebra, the result follows.
\end{proof}
We next characterize $C^*$-$W^*$- projectivity and $C^*$-$AW^*$-projectivity for separable commutative $C^{*}$-algebras.
Recall that a $C^{\ast}$-algebra is $AW^{\ast}$ \cite{K} if every set of
projections has a least upper bound and every maximal abelian selfadjoint
subalgebra is the C*-algebra generated by its projections.
\begin{theorem}\label{C*-W*Comm}
Let $K$ be a compact metric space. Then the following are equivalent:
\begin{enumerate}
\item $C\left( K\right) $ is C*-W* projective,
\item $C\left( K\right) $ is C*-AW* projective,
\item $C\left( K\right) $ is C*-$\mathcal{Y}$ projective, where
$\mathcal{Y}$ is the class of all unital C*-algebras in which every
commutative separable C*-subalgebra is contained in a commutative C*-algebra
generated by projections.
\item $K$ is a continuous image of $\left[ 0,1\right] $,
\item $K$ is connected and locally path-connected.
\item Every continuous function from a closed subset of $\left[ 0,1\right] $
into $K$ can be extended to a continuous function from $\left[ 0,1\right] $
into $K$.
\item Every continuous function from a closed subset of the Cantor set into
$K$ can be extended to a continuous function from $\left[ 0,1\right] $ into
$K$.
\end{enumerate}
\end{theorem}
\begin{proof}
The implications $\left( 3\right) \Rightarrow \left( 2\right)
\Rightarrow \left( 1\right) $ and $\left( 6\right) \Rightarrow \left(
7\right) $ are clear.
$\left( 4\right) \Leftrightarrow \left( 5\right) $. This is the
Hahn-Mazurkiewicz theorem (\cite{TopologyBook}, Th. 3-30).
$\left( 1\right) \Rightarrow \left( 4\right) .$ Suppose $C\left( K\right)
$ is C*-W* projective. Then there is a compact Hausdorff space $X$ such that
$C\left( K\right) ^{**}=C\left( X\right) $. We know that there is an
embedding $\varphi:X\rightarrow
{\displaystyle \prod_{i\in I}}
[0, 1] $ (product topology). Hence there is a surjective $\ast
$-homomorphism from $C\left(
{\displaystyle \prod_{i\in I}}
[0, 1] \right) $ onto $C\left( K\right) ^{**}$. Since the
canonical embedding from $C\left( K\right) $ to $C\left( K\right) ^{**}$
is an injective $\ast$-homomorphism, we know from that $\left( 1\right) $
there is an injective unital $\ast$-homomorphism from $C\left( K\right) $ to
$C\left(
{\displaystyle \prod_{i\in I}}
[0, 1] \right) $. Hence there is a continuous surjective map
$\beta:
{\displaystyle \prod_{i\in I}}
[0, 1] \rightarrow K$. Suppose $D$ is a countable dense subset of
$K$ and $E$ is the set of all elements of $
{\displaystyle \prod_{i\in I}}
[0, 1] $ with finite support, i.e., only finitely many nonzero
coordinates. If $x\in D,$ then there is a countable set $E_{x}$ of $E$ such
that $x\in \overline{\beta \left( E_{x}\right) }$. Hence
\[
\beta \overline{\left( \cup_{x\in D}E_{x}\right) }=K,
\]
since $\beta \overline{\left( \cup_{x\in D}E_{x}\right) }$ is compact and
contains $D$. It follows that there is a countable subset $J\subseteq I$ such
that
\[
\overline{\left( \cup_{x\in D}E_{x}\right) }\subseteq
{\displaystyle \prod_{i\in J}}
[0, 1] \times
{\displaystyle \prod_{i\in I\backslash J}}
\left \{ 0\right \} ;
\]
whence,
\[
\beta \left(
{\displaystyle \prod_{i\in J}}
[0, 1] \times
{\displaystyle \prod_{i\in I\backslash J}}
\left \{ 0\right \} \right) =K.
\]
Hence $K$ is a continuous image of $
{\displaystyle \prod_{i\in J}}
[0, 1] ,$ which in turn a continuous image of $\left[
0,1\right] .$ Thus $K$ is a continuous image of $\left[ 0,1\right] $.
$\left( 5\right) \Longrightarrow \left( 6\right) $. Suppose $K$ is
connected and locally path connected and $E$ is any closed subset of $\left[
0,1\right] ,$ and $f:E\rightarrow K$ is continuous. For $x,y\in K,$ let
$\Delta \left( x,y\right) $ be the infimum of the diameters of every path in
$K$ from $x$ to $y$. We first note that
\[
\lim_{d\left( x,y\right) \rightarrow0}\Delta \left( x,y\right) =0.
\]
If this is not true, then there is an $\varepsilon>0$ and sequences $\left \{
x_{n}\right \} $ and $\left \{ y_{n}\right \} $ in $K$ such that $d\left(
x_{n},y_{n}\right) \rightarrow0$ and $\Delta \left( x_{n},y_{n}\right)
\geq \varepsilon$ for every $n\in \mathbb{N}$. Since $K$ is compact we can
replace $\left \{ x_{n}\right \} $ and $\left \{ y_{n}\right \} $ with
subsequences that converge to $x$ and $y$, respectively. Since $d\left(
x,y\right) =\lim_{n\rightarrow \infty}d\left( x_{n},y_{n}\right) =0$, we see
that $x=y.$ Since $X$ is locally connected, there is path connected
neighborhood $U$ of $x$ such that $U$ is contained in the ball centered at $x$
with radius $\varepsilon/3$. There must be an $n$ such that $x_{n},y_{n}\in
U,$ and there must be a path $\gamma$ in $U$ from $x_{n}$ to $y_{n}$. Since
$\gamma$ is in the ball centered at $x$ with radius $\varepsilon/3$, the
diameter of $\gamma$ is at most $2\varepsilon/3$, which implies $\Delta \left(
x_{n},y_{n}\right) <\varepsilon$, a contradiction.
We can clearly add $0$ and $1$ to $E$ and extend $f$ so that it is still
continuous. Hence we can assume that $0,1\in E$. We can write
\[
\left[ 0,1\right] \backslash E=\bigcup_{n\in I}\left( a_{n},b_{n}\right)
\]
where $\left \{ \left( a_{n},b_{n}\right) :n\in I\right \} $ is a disjoint
set of open intervals with $I\subseteq \mathbb{N}$. For each $n\in I$ we chose
a path $\gamma_{n}:\left[ a_{n},b_{n}\right] \rightarrow K$ from $f\left(
a_{n}\right) $ to $f\left( b_{n}\right) $ so that the diameter of
$\gamma_{n}\ $is less than $2\Delta \left( f\left( a_{n}\right) ,f\left(
b_{n}\right) \right) $.
$\left( 7\right) \Longrightarrow \left( 3\right) $. Suppose $\left(
7\right) $ is true. Suppose $\mathcal{B}$ is a unital C*-algebra,
$\mathcal{M}\in \mathcal{Y}$ and $\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a
unital surjective $\ast$-homomorphism. Let $\rho:C\left( K\right)
\rightarrow \mathcal{M}$ be a unital $\ast$-homomorphism. Since $K$ is
metrizable, $C\left( K\right) $ is separable, and since $\mathcal{M}
\in \mathcal{Y}$, there a countable commuting family $\left \{ p_{1}
,p_{2},\ldots \right \} $ of projections in $\mathcal{M}$ such that
$\mathcal{\rho}\left( C\left( K\right) \right) \mathcal{\subseteq C}
^{\ast}\left( p_{1},p_{2},\ldots \right) $. Let $E$ be the maximal ideal
space of $\mathcal{C}^{\ast}\left( p_{1},p_{2},\ldots \right) $. Since
$\mathcal{C}^{\ast}\left( p_{1},p_{2},\ldots \right) $ is generated by
countable many projections, $E$ is a totally disconnected compact metric space
and is therefore homeomorphic to a subset of the Cantor set. Hence there is an
$a=a^{\ast}\in \mathcal{C}^{\ast}\left( p_{1},p_{2},\ldots \right) $ such that
$\mathcal{C}^{\ast}\left( p_{1},p_{2},\ldots \right) =C^{\ast}\left(
a\right) $ and $\sigma \left( a\right) $ (homeomorphic to $E$) is a subset
of the Cantor set. Let $\Gamma:C^{\ast}\left( a\right) \rightarrow C\left(
\sigma \left( a\right) \right) $ be the Gelfand map. Then $\Gamma \circ
\rho:C\left( K\right) \rightarrow C\left( \sigma \left( a\right) \right)
$ is a unital $\ast$-homomorphism, so there is a continuous function
$\psi:\sigma \left( a\right) \rightarrow K$ so that
\[
\Gamma \left( \rho \left( f\right) \right) =f\circ \psi
\]
for every $f\in C\left( K\right) .$ By applying $\Gamma^{-1}$, we get
\[
\rho \left( f\right) =\left( f\circ \psi \right) \left( a\right)
\]
for every $f\in C\left( K\right) .$
By $\left( 7\right) $, we can assume (by extending) that $\psi:\left[
0,1\right] \rightarrow K$. We can find $A\in \mathcal{B}$ with $0\leq A\leq1$
such that $\pi \left( A\right) =a.$ We now define $\nu:C\left( K\right)
\rightarrow \mathcal{B}$ by
\[
\nu \left( f\right) =\left( f\circ \psi \right) \left( A\right) .
\]
Then $\nu$ is a unital $\ast$-homomorphism and, for every $f\in C\left(
K\right) $, we have
\[
\pi \left( \nu \left( f\right) \right) =\pi \left( \left( f\circ
\psi \right) \left( A\right) \right) =\left( f\circ \psi \right) \left(
a\right) =\rho \left( f\right) .
\]
Hence, $\rho=\pi \circ \nu$. This proves $\left( 3\right) $.
\end{proof}
\bigskip
Let $K$ be a compact metric space and let $x_0$ be any point in $K$. Let us denote by $C_{0}(K\setminus\{x_0\})$ the $C^*$-algebra of all continuous functions on $K$ vanishing at $x_0$.
\begin{theorem}
\label{MatricesOverCommAlgebras} Let $K$ be a continuous image of $\left[
0,1\right] $, $n\in \mathbb{N}$. Then the $C^{*}$-algebra $C_{0}(K\setminus\{x_0\}) \otimes M_{n}$
is $C^{*}$-$W^{*}$-projective in the non-unital category.
\end{theorem}
\begin{proof}
Since $C_0(K\setminus\{x_0\})^{**}$ is a commutative von Neumann algebra, $$C_0(K\setminus\{x_0\})^{**} \cong C(X),$$ for some extremally disconnected space $X$.
Let $i_{\ast}: C_0(K\setminus\{x_0\}) \to C(X)$ be the canonical embedding into the bidual. It is induced by some surjective continuous map $i: X \twoheadrightarrow K$.
Since $K$ is a continuous image of $\left[ 0,1\right] $, there is a surjective continuous map $\alpha: [0, 1]\twoheadrightarrow K$.
By the universal property of extremally disconnected spaces (Gleason's theorem), $i$ factorizes through a continuous map $\beta: X \to [0, 1]$.
$$\xymatrix {[0, 1] \ar@{->>}[d]^-{\alpha} & \\
K & X \ar@{->>}[l]_-{i} \ar@/_/[ul]^{\beta} }$$
Let $$\alpha_{\ast}: C_0(K\setminus\{x_0\}) \to C_0(0, 1], \; \; \beta_{\ast}: C_0(0, 1]\to C_0(X) \subset C(X)$$ be the $\ast$-homomorphisms induced by $\alpha$ and $\beta$.
Let $\mathcal M$ be a von Neumann algebra, $\mathcal B$ a $C^*$-algebra, $q: \mathcal B\to \mathcal M$ a surjective $\ast$-homomorphism. Let $\phi: C_0(K\setminus\{x_0\}) \otimes M_n \to \mathcal M$ be a $\ast$-homomorphism. By the universal property of double duals we can extend $\phi$ to a $\ast$-homomorphism $\tilde \phi: C(X) \otimes M_n \cong (C_0(K\setminus\{x_0\})\otimes M_n)^{**} \to \mathcal M$.
$$\xymatrix { B \ar[dd]^{q} & & &\\ && C_0(0, 1]\otimes M_n \ar[dr]_{\beta_{\ast}\otimes id_{M_n}} \ar@{.>}[ull]_{\psi}& \\ \mathcal M & C_0(K\setminus \!\{x_0\})\!\otimes \! M_n \ar[l]_{\phi\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \ar@{^{(}->}[rr]^{i_{\ast}\otimes id_{M_n}} \ar[ur]_{\alpha_{\ast}\otimes id_{M_n}} & & {C(X)\otimes M_n}
\ar@/^/[lll]^{\tilde \phi}
}$$
Since $C_0(0, 1]\otimes M_n$ is projective (\cite{LoringBook}, Th. 10.2.1), there is a $\ast$-homomorphism $\psi: C_0(0, 1]\otimes M_n \to \mathcal B$ such that
$$q\circ \psi = \tilde \phi \circ (\beta_{\ast}\otimes id_{M_n}).$$ Then $$q\circ \psi \circ (\alpha_{\ast}\otimes id_{M_n}) = \tilde \phi \circ (\beta_{\ast}\otimes id_{M_n}) \circ (\alpha_{\ast}\otimes id_{M_n}) =\tilde \phi \circ (i_{\ast}\otimes id_{M_n}) = \phi.$$ Thus $\psi \circ (\alpha_{\ast}\otimes id_{M_n})$ is a lift of $\phi$.
\end{proof}
We don't know if the previous result can be generalized to get the
following: if $A$ is $C^{*}$-$W^{*}$-projective in the non-unital category, then
so is $A\otimes M_{n}$.
\medskip
Recent developments in the classification of $C^{*}$-algebras show the
importance of the analysis of central sequence algebras. Let
$\omega$ be a free ultrafilter on $\mathbb{N}$. With any $C^*$-algebra $A$ with a faithful tracial state $\tau$ one can associate its $C^{*}$-central sequence algebra $A_{\omega}\cap A^{\prime}$ and its $W^{*}$-central sequence algebra $N^{\omega}\cap
N^{\prime}$, where $N$ is the weak closure of $A$ under the
GNS representation $\pi_{\tau}$ of $A$. There is a natural $\ast$-homomorphism $\gamma: A_{\omega}\cap A^{\prime} \to N^{\omega}\cap
N^{\prime}$; it was proved in \cite{Sato} and \cite{KR} that $\gamma$ is surjective.
A commonly used tool in classification of $C^{*}$-algebras is the fact that an
order zero map from the matrix algebra $M_{n}$ to any quotient $C^{*}$-algebra
lifts (the so-called projectivity of order zero maps). In particular a
possibility to lift an order zero map $M_{n} \to N^{\omega}\cap N^{\prime}$ to
an order zero map $M_{n}\to A_{\omega}\cap A^{\prime}$ is a key ingredient to
obtain uniformly tracially large order zero maps (\cite{TWW}). Below we prove
a stronger statement: one can lift any commuting family of order zero maps
$M_{n} \to N^{\omega}\cap N^{\prime}$ to a commuting family of order zero maps $M_{n}\to A_{\omega}\cap A^{\prime}$.
\begin{lemma}
\label{universal} Let $d\in \mathbb{N}$, $k\in \mathbb{N }\cup \{ \infty \}$. The
$C^{*}$-algebra $C_{0}([0, 1]^{k}) \otimes M_{kd}$ is isomorphic to the
universal $C^{*}$-algebra with generators $e_{ij}^{n, l}$, $l\le k$, $i, j\le
d$, $n\in \mathbb{N}$ and relations
\[
0\le e_{ii}^{n, l} \le1
\]
\[
\left( e_{ij}^{n, l}\right) ^{\ast} = e_{ji}^{n, l},
\]
\[
e_{ij}^{m, l}e_{ks}^{n, l} = \delta_{jk}e_{is}^{m,l}, \; m< n
\]
\[
e_{ij}^{n, l}e_{ks}^{n, l} = \delta_{jk}e_{ii}^{n, l}e_{is}^{n, l},
\]
\[
[e_{ij}^{n, l}, e_{i^{\prime}j^{\prime}}^{n^{\prime}, l^{\prime}}] =
[e_{ij}^{n, l}, \left( e_{i^{\prime}j^{\prime}}^{n^{\prime}, l^{\prime}
}\right) ^{*}] =0 \; \text{when} \; l\neq l^{\prime}.
\]
\end{lemma}
\noindent The proof of the lemma is analogous to the proof of Lemma 6.2.4 in
\cite{LoringBook}, so we don't write it.
\begin{theorem}\label{OrderZero}
Any commuting family of order zero maps from $M_{n}$ to $N^{\omega}\cap
N^{\prime}$ lifts to a commuting family of order zero maps from $M_{n}$ to
$A_{\omega}\cap A^{\prime}$.
\end{theorem}
\begin{proof} As was proved in \cite{WZ}, with any order zero map $\phi:B\to D$ one can associate a $\ast$-homomorphism $f: CB\to D$ and vice versa. Here $CB = C_0(0, 1]\otimes B$ is the cone over $B$.
Moreover it follows from the construction in \cite{WZ} that order zero maps $\phi_i$'s have commuting ranges if and only if the corresponding $f_i$'s have commuting ranges. It also follows from the construction that if for an order zero map $\phi: A \to B/I$ the corresponding $f: CA \to B/I$ lifts to a $\ast$-homomorphism $\tilde f: CA \to B$, then
$\phi$ lifts to the order zero map $\tilde \phi$ corresponding to $\tilde f$. Thus we need to prove that
any family of $\ast$-homomorphisms $f_i: CM_n \to N^{\omega}\cap N'$ with pairwisely commuting ranges lifts to a family of $\ast$-homomorphisms $\tilde f_i: CM_n \to A_{\omega}\cap A'$ with pairwisely commuting ranges. It follows from Lemma \ref{universal} that it is equivalent to lifting of one $\ast$-homomorphism from $C_0([0, 1]^k) \otimes M_{kd}$ to $ N^{\omega}\cap N'$, where $k$ is the number of $\ast$-homomorphisms in the family. Since $[0, 1]^k$ is a continuous image of $[0, 1]$, the statement follows from Theorem \ref{MatricesOverCommAlgebras}.
\end{proof}
\section{$W^*$-$C^*$-projectivity and $RR0$-projectivity}
Recall that a $C^*$-algebra has {\it real rank zero } (RR0) if each its self-adjoint element can be approximated by self-adjoint elements with finite spectra.
Since a $\ast$-homomorphic image of a real rank zero C*-algebra is real rank
zero, and since every von Neumann algebra has real rank zero \cite{BP}, it follows that every
$RR0$-projective C*-algebra is W*-C* projective.
We first prove that in the $W^{*}$-$C^{*}$-case we can lift projections. This was proved in the RR0-case by L. G. Brown and G. Pederson
\cite{BP}. We include the brief W*-C* proof because it is much shorter.
\begin{proposition}
\label{proj}$\mathbb{C}\oplus \mathbb{C}$ is W*-C*-projective.
\end{proposition}
\begin{proof}
Suppose $\mathcal{B}$ is a von Neumann algebra,
$\mathcal{M}$ is a C*-algebra, and $\pi:\mathcal{B}\rightarrow \mathcal{M}$ is
a unital $\ast$-homomorphism. Since $\mathbb{C}\oplus \mathbb{C}$ is the unital universal $C^*$-algebra of one projection, suppose $q\in \mathcal{M}$ is a projection. By [\cite{LoringBook}, Lemma 10.1.12], we can lift $q$ to $b_1$ and $1-q$ to $b_2$ in $\mathcal B$ such that $0\le b_j\le 1$, $\pi(b_1) = q, \pi(b_2) = 1-q$ and $b_{1}b_{2}=b_{2}b_{1}=0.$ Let
$p\in \mathcal{B}$ be the range projection for $b_{1}$. Then $pb_{2}=b_{2}p=0.$
Thus $b_{1}\leq p$ and $b_{2}\leq1-p$. Hence $q\leq \pi \left( p\right) $ and
$1-q\leq1-\pi \left( p\right) $. Hence, $\pi \left( p\right) =q$.
\end{proof}
\begin{definition}
Suppose $B,\mathcal{M}$ are unital C*-algebras and $\pi:\mathcal{B}
\rightarrow \mathcal{M}$ is a surjective $\ast$-homomorphism, and suppose
$\mathcal{S}$ is a C*-subalgebra of $\mathcal{M}$. We say that $\gamma$\emph{
is a }$\ast$\emph{-cross section for }$\pi$\emph{ on }$S$ if $\gamma
:\mathcal{S}\rightarrow \mathcal{B}$ is a $\ast$-homomorphism and $\pi
\circ \gamma$ is the identity on $\mathcal{S}$. Clearly, every such $\gamma$ is injective.
\end{definition}
\begin{theorem}
\label{projlift}Suppose $\mathcal{B}$ is a real rank zero C*-algebra,
$\mathcal{M}$ is a C*-algebra, and $\pi:\mathcal{B}\rightarrow \mathcal{M}$ is
a unital surjective $\ast$-homomorphism. Suppose $\left \{ p_{1},p_{2}
,\ldots \right \} \subseteq \mathcal{M}$ is a commuting family of projections.
Then there is an unital $\ast$-cross section $\gamma$ for $\pi$ on $C^{\ast
}\left( p_{1},p_{2},\ldots \right) .$ Moreover, if $\gamma_{n}:C^{\ast
}\left( p_{1},p_{2},\ldots,p_{n}\right) \rightarrow \mathcal{M}$ is a unital
$\ast$-cross section for $\pi$ on $C^{\ast}\left( p_{1},p_{2},\ldots
,p_{n}\right) ,$ then there is a unital $\ast$-cross section $\gamma_{n+1}$
for $\pi$ on $C^{\ast}\left( p_{1},p_{2},\ldots,p_{n+1}\right) $ whose
restriction to $C^{\ast}\left( p_{1},\ldots,p_{n}\right) $ is $\gamma_{n}$.
\end{theorem}
\begin{proof}
We know that if $0\neq p\neq1$ is a projection in $\mathcal{M}$, there is a
projection $P\in \mathcal{B}$ with $\pi \left( P\right) =p.$ Clearly $0\neq
P\neq1$. Suppose $\gamma_{n}:C^{\ast}\left( p_{1},p_{2},\ldots,p_{n}\right)
\rightarrow \mathcal{M}$ is a unital $\ast$-cross section for $\pi$ on
$C^{\ast}\left( p_{1},p_{2},\ldots,p_{n}\right) $. We know $C^{\ast}\left(
p_{1},p_{2},\ldots,p_{n}\right) $ is generated by an orthogonal family of
projections $\left \{ q_{1},\ldots,q_{m}\right \} $ whose sum is $1$, and, for
$1\leq k\leq m$, let $Q_{k}=\gamma_{n}\left( q_{k}\right) $. Now $p_{n+1}$
commutes with $\left \{ q_{1},\ldots,q_{m}\right \} $, so $C^{\ast}\left(
p_{1},\ldots,p_{n+1}\right) $ is generated by the orthogonal family
$$\cup_{k=1}^{m}\left \{ q_{k}p_{n+1}q_{k},q_{k}-q_{k}p_{n+1}q_{k}\right \} .$$
If $q_{k}p_{n+1}q_{k}=0$ or $q_{k}-q_{k}p_{n+1}q_{k}=0$, there is nothing new
to lift. If $q_{k}p_{n+1}q_{k}\neq0$ and $q_{k}-q_{k}p_{n+1}q_{k}\neq0$, we
need to find a lifting of $q_{k}p_{n+1}q_{k}$ in $Q_{k}\mathcal{B}Q_{k}$.
However, if was proved in \cite{BP} that $Q_{k}\mathcal{B}Q_{k}$ has real rank $0$. Thus such a lifting is
possible.
\end{proof}
As a corollary we get a sufficient condition for $C(K)$ to be $RR0$-projective.
\begin{corollary}
\label{totallydisconnected} If $K$ is a totally disconnected compact metric
space, then $C\left( K\right) $ is RR0-projective, and hence W*-C* projective.
\end{corollary}
The following corollary uses the fact that if $\gamma:\mathcal{A}
\rightarrow \mathcal{B}$ is a unital $\ast$-homomorphism and $\mathcal{B}$ is a
von Neumann algebra, then there is an extension to a weak*-weak* continuous
$\ast$-homomorphism $\hat{\gamma}:\mathcal{A}^{**}\rightarrow \mathcal{B}$.
\begin{corollary}
\label{commW*-C*proj} Let $K$ be a compact metric space. The following are equivalent.
\begin{enumerate}
\item $C\left( K\right) $ is W*-C* projective
\item Whenever $\mathcal{B}$ is a von Neumann algebra, $\mathcal{M}$ is a
C*-algebra, $\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a surjective $\ast
$-homomorphism, and $\rho:C\left( K\right) \rightarrow \mathcal{M}$ is a
unital $\ast$-homomorphism, there is a commutative C*-subalgebra $\mathcal{D}$
of $\mathcal{M}$ that contains $\rho(C(K))$ such that the maximal ideal space
of $\mathcal{D}$ is totally disconnected,
\item Whenever $\mathcal{B}$ is a von Neumann algebra, $\mathcal{M}$ is a
C*-algebra, $\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a surjective $\ast
$-homomorphism, and $\rho:C\left( K\right) \rightarrow \mathcal{M}$ is a
unital $\ast$-homomorphism, $\rho$ extends to a $\ast$-homomorphism $\hat
{\rho}:C\left( K\right) ^{**}\rightarrow \mathcal{M}$.
\end{enumerate}
\end{corollary}
\bigskip
\begin{remark}
Without the separablility assumption on $C\left( K\right) $ (i.e., the
metrizability of $K$), it is not generally true that $C\left( K\right) $ is
W*-C*-projective whenever $K$ is compact and totally disconnected. For
example, let $\mathcal{A}$ be the universal C*-algebra generated by a mutually
orthogonal family $\left \{ P_{t}:t\in \left[ 0,1\right] \right \} $ of
projections. The maximal ideal space of $\mathcal{A}$ is the one-point
compactification $K$ of the discrete space $\left[ 0,1\right] $. If we let
$\mathcal{B}=B\left( \ell^{2}\right) $ and $\mathcal{M}=B\left( \ell
^{2}\right) /\mathcal{K}\left( \ell^{2}\right) $ and let $\pi
:\mathcal{B}\rightarrow \mathcal{M}$ be the quotient map, then there is an
injective unital $\ast$homomorphism $\rho:\mathcal{A}\rightarrow \mathcal{M}$,
but there is no injective unital $\ast$-homomorphism from $\mathcal{A}$ to
$\mathcal{B}$ since $B\left( \ell^{2}\right) $ does not contain an
uncountable orthogonal family of nonzero projections. Hence $C\left(
K\right) $ is not W*-C* projective although $K$ is totally disconnected. This
shows that an attempt at a transfinite inductive version of the proof of
Theorem \ref{projlift} is doomed to failure. This also shows that being W*-C*
projective is not closed under arbitrary direct limits, since $\mathcal{A}$ is
the direct limit of the family $\left \{ C^{\ast}\left( \left \{ P_{t}:t\in
E\right \} \right) :E\subseteq \left[ 0,1\right] \text{ is countable}
\right \} $. We doubt that $\rho$ can be extended to a $\ast$-homomorphism
from $C\left( K\right) ^{**}$ to $\mathcal{M}$, so the equivalence of
$\left( 1\right) $ and $\left( 3\right) $ in Corollary \ref{commW*-C*proj}
may conceivably be true.
\end{remark}
\medskip
In the case when $K$ is a Peano continuum (that is a non-empty compact connected metric space which is locally connected at each point) there is a necessary condition for $C(K)$ to be $W^*$-$C^*$-projective.
\begin{proposition}\label{Peano} Suppose $K$ is a Peano continuum and $C(K)$ is $W^*$-$C^*$-projective. Then $dim_{\text{cov}}(K) \le 1$ (here $dim_{\text{cov}}$ is the covering dimension).
\end{proposition}
\begin{proof} Suppose $dim_{\text{cov}} K > 1$. Then by [\cite{ChDr}, Prop. 3.1] $K$ contains a circle, $S^1$. Let $j: C(K) \to C(S^1)$ be the restriction map. Let $\pi: B(H) \to B(H)/K(H)$ be the canonical surjection. Let $T\in B(H)$ be the unilateral shift. Define a $\ast$-homomorphism $\rho: C(S^1) \to B(H)/K(H)$ by sending the identity function $z$ to $\pi(T)$. We claim that $\rho\circ j$ is not liftable. Indeed suppose it lifts to a $\ast$-homomorphism $\gamma: C(K) \to B(H)$. Let $f\in C(K)$ be any preimage of $z\in C(S^1)$ under the map $j$. Since $f$ is normal, $\gamma(f)\in B(H)$ is a normal preimage of $\pi(T)$. Since any preimage of $\pi(T)$ has Fredholm index $-1$ and since any normal Fredholm operator has Fredholm index zero, we come to a contradiction.
\end{proof}
\medskip
Below we give a few non-commutative examples and non-examples of $W^*$-$C^*$-projective $C^*$-algebras.
The following lemma shows that Murray-von Neumann equivalent projections can
be lifted to Murray von Neumann equivalent projections in the W*-C* case. More
simply, it states that partial isometries can be lifted.
\begin{lemma}
\label{PI} Let $\mathcal{B}$ be a unital C*-algebra, $\mathcal{M}$ be a von
Neumann algebra and $\pi:\mathcal{M}\rightarrow \mathcal{B}$ be a surjective
$\ast$-homomorphism. Let $v\in \mathcal B$ be a partial isometry with $v^{\ast}v=p$ and
$vv^{\ast}=q$. Let $X\in \mathcal{M}$, $\left \Vert X\right \Vert \leq1$, and let
$P,Q$ be projections in $\mathcal{M}$ such that $\pi \left( P\right)
=p,\pi \left( Q\right) =q$ and $\pi \left( X\right) =v$. If $V$ is the
parial isometry in the polar decomposition of $PXQ$, then $\pi \left(
V\right) =v$.
\end{lemma}
\begin{proof}
Let $Y=PXQ$. Then the range projection $P_{1}$ of $Y$ is less than or equal to
$P$ and the range projection of $Y^{\ast}$ is less than or equal to $Q$.
Moreover, $\left( YY^{\ast}\right) ^{1/2}\leq P_{1}\leq P$. Also,
\[
p=\pi \left( \left( Y^{\ast}Y\right) ^{1/2}\right) \leq \pi \left(
P_{1}\right) \leq \pi \left( P\right) =p\text{.}
\]
If $Y=\left( YY^{\ast}\right) ^{1/2}V$ is the polar decomposition, then
\[
v=p\pi \left( V\right) =\pi \left( P_{1}V\right) =\pi \left( V\right) .
\]
\end{proof}
\begin{corollary}\label{Toeplitz}
Let $\mathcal{T}$ be the Toeplitz algebra. Then $\mathcal{T }\oplus \mathbb{C}$
is W*-C* projective.
\end{corollary}
\begin{proof}
$\mathcal T \oplus \mathbb C$ is the universal unital $C^*$-algebra generated by $v$ with the relation that $v$ is a
partial isometry.
\end{proof}
\begin{corollary}
\label{M_n} $\mathcal{M}_{n}\left( \mathbb{C}\right) \oplus \mathbb C$ is W*-C* projective.
\end{corollary}
\begin{proof} It would be equivalent to prove that $\mathcal{M}_{n}\left( \mathbb{C}\right)$ is W*-C* projective
in the non-unital category.
We will use induction on $n$. The case when $n=1$ amounts to lifting a projection.
Assume the theorem is true for $n$.
Suppose $\mathcal{M}$ is a
von Neumann algebra, $\mathcal{B}$ is a C*-algebra, $\pi:\mathcal{M}\rightarrow \mathcal{B}$ is a surjective $\ast$-homomorphism.
Suppose $\rho:\mathcal{M}_{n}\left( \mathbb{C}\right) \rightarrow
\mathcal{B}$ is a $\ast$-homomorphism.
It follows from the induction assumption that there is a
$\ast$-homomorphism $$\gamma_{0}:C^{\ast}\left( \left \{ e_{1k}:2\leq k\leq
n\right \} \right) \rightarrow \mathcal{M}$$ so that $\gamma_{0}\left(
e_{jk}\right) =E_{jk}\ $and $\left( \pi \circ \gamma_{0}\right) \left(
e_{1k}\right) =\rho \left( e_{1k}\right) $ for $2\leq k\leq n$. We then have
$E_{1k}E_{1k}^{\ast}=\gamma_{0}\left( e_{11}\right) $ for $2\leq k\leq n$.
Choose $X\in \mathcal{M}$ so that $\pi \left( X\right) =\rho \left(
e_{1,n+1}\right) .$ If we replace $X$ with $E_{11}X\left( 1-\sum_{1\leq
k\leq n}E_{kk}\right) ,$ and let $V$ be the partial isometry in the polar
decomposition of $E_{11}X\left( 1-\sum_{1\leq k\leq n}E_{kk}\right) $, we
have from Lemma \ref{PI} that $\pi \left( V\right) =\rho \left(
e_{1,n+1}\right) ,$ and $VV^{\ast}\leq E_{11}$ and $\pi \left( VV^{\ast
}\right) =\rho \left( e_{11}\right) $. If we replace $E_{1k}$ with
$F_{1k}=VV^{\ast}E_{1k}$ for $2\leq k\leq n$, and define $F_{1,n+1}=V$, we
obtain a representation $\gamma:\mathcal{M}_{n+1}\left( \mathbb{C}\right)
\rightarrow \mathcal{M}$ with $\gamma \left( e_{1k}\right) =F_{1k}$ for $1\leq
k\leq n+1$ such that $\pi \circ \gamma=\rho$.
\end{proof}
\begin{lemma}
\label{directsums} Let $C^*$-algebras $A$ and $ D$ be unital $W^{*}$-$ C^{*}$-projective ($W^{*}$-$W^{*}$-projective respectively) in the non-unital category. Then $A \oplus D$
is $W^{*}$-$C^{*}$-projective ($W^{*}$-$W^{*}$-projective respectively) in the
non-unital category.
\end{lemma}
\begin{proof} Suppose $\mathcal{B}$ and $\mathcal{M}$ are von Neumann algebras ($\mathcal{B}$ is a von Neumann algebra in the $W^*-W^*$-case respectively) and
$\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a surjective $\ast
$-homomorphism. Let $\phi: A\oplus D \to \mathcal M$ be a $\ast$-homomorphism. Let $$p= \phi(1_A\oplus 0), \; q= \phi(0\oplus 1_D).$$ Define $\phi_A: A\to pMp$ and $\phi_D: D \to qMq$ by $\phi_A(a) = \phi(a\oplus 0)$ and $\phi_D(d) = \phi(0\oplus d)$ respectively. It follows from Corollary \ref{totallydisconnected} that we can lift $p, q$ to projections $P, Q$ in $B$ which are orthogonal to each other. Let $\psi_A: A\to PBP$ and $\psi_D: D\to QBQ$ be lifts of $\phi_A$ and $\phi_D$.
Define lift $\psi$ of $\phi$ by $$\psi(a\oplus d) = \psi_A(a) + \psi_D(d).$$
\end{proof}
Combining this lemma with Corollary \ref{M_n} we obtain the following result.
\medskip
\begin{corollary}
\label{fd}If $\mathcal{A}$ is a finite-dimensional C*-algebra, then
$\mathcal{A}\oplus \mathbb{C}$ is W*-C* projective.
\end{corollary}
\begin{remark}
The result in Corollary \ref{fd} cannot be extended to AF-algebras even in the
W*-W* case. Indeed the tracial ultraproduct $
{\displaystyle \prod \limits_{n\in \mathbb{N}}^{\alpha}}
\left( \mathcal{M}_{2^{n}}\left( \mathbb{C}\right) ,\tau_{2^{n}}\right) $
with respect to a free ultrafilter $\alpha$, where $\tau_{2^{n}}$ is the
normalized trace on $\mathcal{M}_{2^{n}}\left( \mathbb{C}\right) $, is a von
Neumann algebra. Thus $\pi:
{\displaystyle \prod_{n\in \mathbb{N}}}
\mathcal{M}_{2^{n}}\left( \mathbb{C}\right) \rightarrow
{\displaystyle \prod \limits_{n\in \mathbb{N}}^{\alpha}}
\left( \mathcal{M}_{2^{n}}\left( \mathbb{C}\right) ,\tau_{2^{n}}\right) $
is a unital surjective $\ast$-homomorphism and the domain and range are both
von Neumann algebras. If $\mathcal{A}$ is the CAR algebra, then it is clear
that there is an embedding $\rho:\mathcal{A}\rightarrow
{\displaystyle \prod \limits_{n\in \mathbb{N}}^{\alpha}}
\left( \mathcal{M}_{2^{n}}\left( \mathbb{C}\right) ,\tau_{2^{n}}\right) $.
However, $\mathcal{A}$ is simple and infinite-dimensional, so there is no
embedding from $ \mathcal{A} \oplus \mathbb C$ into $
{\displaystyle \prod_{n\in \mathbb{N}}}
\mathcal{M}_{2^{n}}\left( \mathbb{C}\right) $ such that $\rho=\pi \circ \tau$.
\end{remark}
A trace $\psi$ on a unital MF-algebra $\mathcal A$ is called an {\it MF-trace} if there is a free ultrafilter $\alpha$ on $\mathbb N$ and a unital
$\ast$-homomorphism $\pi: \mathcal A \to \prod^{\alpha} M_k(\mathbb C)$ to the $C^*$-ultraproduct of matrices, such that $\psi = \tau_{\alpha} \circ \pi$, where $\tau_{\alpha}(\{A_k\}_{\alpha} = \lim_{k\to \alpha} \tau_k(A_k)$ [\cite{all}, Prop.4].
The ideas in the preceding remark easily extend to the following result.
\begin{proposition}
If $\mathcal{A}$ is a unital MF C*-algebra and is W*-C*-projective, then $\mathcal{A}$ must be RFD.
If the MF-traces are a faithful
set on $\mathcal{A}$, i.e., $\tau \left( a^{\ast}a\right) =0$ for every MF
trace implies $a=0$, and if $\mathcal{A}$ is W*-W* projective, then
$\mathcal{A}$ must be RFD.
\end{proposition}
The following result shows that without adding $\mathbb{C}$ as a direct
summand Corollaries \ref{Toeplitz} and \ref{M_n} no longer hold.
\begin{proposition}\label{M_nNonUnital}
$\mathcal{T}$ and $M_{n}(\mathbb{C})$ are not $W^{*}$-$C^{*}$-projective.
\end{proposition}
\begin{proof}
The Toeplitz algebra is not $W^*$-$C^*$-projective, since an isometry in Calkin algebra need not lift to an isometry in $B(H)$.
$M_n(\mathbb C)$ is not $W^*-C^*$-projective, because $M_n(\mathbb C)$ is a quotient of
$M_n(\mathbb C) \oplus \mathbb C $ and since $M_n(\mathbb C)$ does not admit any unital $\ast$-homomorphisms to $\mathbb C$, the identity
map on $M_n(\mathbb C)$ is not liftable.
\end{proof}
\begin{proposition} Suppose $\mathcal{A}$ is a separable unital C*-algebra.
\begin{enumerate}
\item If $Ext\left( \mathcal{A}\right) $ is not trivial, then $\mathcal{A}$
is not W*-C* projective
\item If $Ext_{w}\left( \mathcal{A}\right) $ is not trivial, then
$ \mathbb C \oplus \mathcal{A}$ is not W*-C* projective.
\end{enumerate}
\end{proposition}
\begin{proof} $\left( 1\right) $ This is obvious.
$\left( 2\right) $ Suppose $Ext_{w}\left( \mathcal{A}\right) $ is not
trivial. Then there is an injective unital $\ast$-homomorphism $\rho:\mathcal{A}
\rightarrow B\left( \ell^{2}\right)
/\mathcal{K}\left( \ell^{2}\right) $ that is not weakly equivalent to the
trivial element in $Ext_{w}\left( \mathcal{A}\right) $. Assume, via
contradiction that there is a nonunital $\ast$-homomorphism $\gamma
:\mathcal{A\rightarrow B}\left( \ell^{2}\right) $ such that $\pi \circ
\gamma=\rho$. Then $$\pi \left( 1-\gamma \left( 1\right) \right)
=1-\rho \left( 1\right) =0.$$ Thus $1-\gamma \left( 1\right) $ is a
finite-rank projection, and if $\gamma_{0}\left( A\right) =\gamma \left(
A\right) |_{\gamma \left( 1\right) \left( \ell^{2}\right) }$, we have
$\gamma=0\oplus \gamma_{0}$ relative to $\ell^{2}=\ker \gamma \left( 1\right)
\oplus \gamma \left( 1\right) \left( \ell^{2}\right) .$ Since $\rho=\pi
\circ \gamma$ is injective, $\gamma_{0}$ must be injective. Choose an isometry
$V$ in $B\left( \ell^{2}\right) $ whose range is $\gamma \left( 1\right)
\left( \ell^{2}\right) .$ Then $V^{\ast}\gamma \left( \cdot \right) V$ is
unitarily equivalent to $\gamma_{0}$. Thus $\pi \left( V\right) $ is unitary
in $B\left( \ell^{2}\right) /\mathcal{K}\left( \ell^{2}\right) $ and
$\pi(V^{\ast})\rho \left( \cdot \right) \pi(V)$ lifts to $V^{\ast}\gamma \left( \cdot \right) V = U^{\ast}\gamma_{0}\left(
\cdot \right) U$ for some unitary $U.$ This means $\rho$ is weakly equivalent
to the trivial element in $Ext_{w}\left( \mathcal{A}\right) $, a
contradiction.
\end{proof}
\begin{corollary}
If $n\geq2$, the Cuntz algebra $\mathcal{O}_{n}$ is not W*-C* projective. If
$n\geq3$, $\mathbb{C}\oplus O_{n}$ is not W*-C* projective.
\end{corollary}
\begin{proof} By [\cite{Davidson}, Th. V.6.5] $Ext(O_n) \cong \mathbb Z$ and by [\cite{Davidson}, Th. V. 6.6] $Ext_w(O_n) \cong \mathbb Z_{n-1}$, when $n\ge 2$.
\end{proof}
\begin{remark}
By Corollary \ref{M_n} and Proposition \ref{M_nNonUnital}, if $n\geq2$, $\mathcal{M}_{n}\left( \mathbb{C}\right) $ is not W*-C*
projective, but $\mathbb{C}\oplus \mathcal{M}_{n}\left( \mathbb{C}\right) $
is W*-C* projective, and this happily coincides with the fact that
$Ext\left( \mathcal{M}_{n}\left( \mathbb{C}\right) \right) $ is not
trivial and $Ext_{w}\left( \mathcal{M}_{n}\left( \mathbb{C}\right) \right)
$ is trivial.
\end{remark}
The following is a consequence of the proof of a result of T. Loring and the
second-named author [\cite{LoringShulmanAlgebraicElements}, Th. 9]. It
generalizes Olsen's structure theorem for polynomially
compact operators \cite{Olsen}.
\begin{theorem}
Let $R\geq0$ and $p\in \mathbb{C}\left[ x\right] $. The universal C*-algebra
generated by $a$ such that $\left \Vert a\right \Vert \leq R$ and $p\left(
a\right) =0$ is $RR0$-projective and hence $W^{*}$-$C^{*}$-projective.
\end{theorem}
\section{$W^*$-$W^*$-projectivity}
We begin with the separable unital commutative C*-algebras.
\begin{theorem}
\label{commutative} Every separable unital commutative C*-algebra is RR0-AW*- projective. In particular every separable unital commutative C*-algebra is W*-W*-projective.
\end{theorem}
\begin{proof}
Suppose $\mathcal{B}$ is a real rank zero C*-algebra, $\mathcal{M}$ is an
AW*-algebra and
$\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a surjective unital $\ast
$-homomorphism, and suppose that $\mathcal{A}$ is a separable unital commutative C*-subalgebra
of $\mathcal{M}$. Since $\mathcal{M}$ is an AW*-algebra, every maximal abelian
selfadjoint C*-subalgebra of $\mathcal{M}$ is the C*-algebra generated by its
projections. Since $\mathcal{A}$ is contained in such a maximal algebra and
$\mathcal{A}$ is separable, it follows that there is a countable commuting
family $\left \{ p_{1},p_{2},\ldots \right \} $ of projections in $\mathcal{M}$
such that $\mathcal{A}\subset C^{\ast}\left( p_{1},p_{2},\ldots \right) $. By
Theorem \ref{projlift} there is a $\ast$-cross section $\gamma$ for $\pi$ on
$C^{\ast}\left( p_{1},p_{2},\ldots \right) .$ Clearly, the restriction of $g$
to $\mathcal{A}$ is a $\ast$-cross section of $\pi$ for $\mathcal{A}$.
\end{proof}
\begin{theorem}
\label{matrices over algebras} Let $A$ be a unital $C^{*}$-algebra. If $A$
is $W^{*}$-$C^{*}$-projective ($W^{*}$-$W^{*}$-projective respectively) in the
non-unital category, then for each $n\in \mathbb{N}$, $M_{n}(A)$ is
$W^{*}$-$C^{*}$-projective ($W^{*}$-$W^{*}$-projective respectively) in the
non-unital category.
\end{theorem}
\begin{proof} Our proof is a modification of Loring's proof of the
fact that the class of projective $C^*$-algebras is closed under tensoring with matrices
(\cite{LoringBook}).
Let $\phi: M_n\otimes A \to B/I$ be a $\ast$-homomorphism and $B$
(and $B/I$, for the $W^*$-$W^*$-projectivity case) be a von Neumann
algebra and let $\pi: B\to B/I$ denote the canonical surjection. We
need to prove that $\phi$ lifts. Define $j: M_n \to M_n\otimes A$ by
$$j(T) = T\otimes 1_A$$ and let $\phi_2 = \phi \circ j$. Since by
Corollary \ref{M_n} $M_n$ is $W^*$-$C^*$-projective in the non-unital category, $\phi_2$ lifts to $\psi:
M_n\to B$.
$$\xymatrix {M_n \ar[r]^-{\psi}\ar[d]^-{j} \ar@/_/[dr]^{\phi_2} & B\ar[d]^-{\pi} \\
M_n\otimes A \ar[r]_-{\phi}& B/I}$$ Let $(e_{ij})$ be a matrix unit
in $M_n$. Define a $\ast$-homomorphism $$i: M_n\otimes
\phi(e_{11}\otimes A) \to \phi(M_n\otimes A)$$ by $i(T\otimes
\phi(e_{11}\otimes a)) = \phi(T\otimes a).$ It is obviously
surjective. To see that it is injective, we will use the fact that
an ideal in a tensor product $C^*$-algebra is a tensor product of
ideals. Hence the kernel of $i$ is either $0$ or of the form
$M_n\otimes J$, where $J$ is an ideal in $\phi(e_{11}\otimes A)$.
Let $\phi(e_{11}\otimes a))\in J$. Then for each $T\in M_n$,
$\phi(T\otimes a) = T\otimes \phi(e_{11}\otimes a) =0$. In
particular, $\phi(e_{11}\otimes a) = 0$. Thus $\phi(e_{11}\otimes
a))=0$ and $J=0$. So $i$ is an isomorphism. Let $$p= \phi_2(e_{11}),
\;P = \psi(e_{11})$$ and let $i_1$ be the inclusion
$\phi(e_{11}\otimes A)\subseteq pB/Ip= PBP/PIP.$ Then the
composition $(id_{M_n}\otimes i_1)\circ i^{-1}\circ \phi: M_n\otimes A \to M_n\otimes
PBP/PIP$ is of the form $id_{M_n}\otimes \gamma$, where $\gamma:
A \to PBP/PIP$ is defined by $$\gamma(a) =
p\phi(e_{11}\otimes a)p.$$ Since $PBP$ (and $pB/Ip$, for the
$W^*$-$W^*$ -projectivity case) is a von Neumann algebra, by
$W^*$-$C^*$ ($W^*$-$W^*$) projectivity of $A$, it can be lifted to
$$id_{M_n}\otimes \tilde \gamma: M_n\otimes A \to M_n\otimes PBP.$$
$$\xymatrix { & & & M_n\otimes PBP \ar[d]^{id_{M_n}\otimes \pi|_{PBP}} \\
M_n\otimes A \ar[urrr]^-{id_{M_n}\otimes \tilde
\gamma}\ar[r]_-{\phi} & \phi(M_n \otimes A) \ar[r]_{i^{-1}} &
M_n\otimes \phi(e_{11}\otimes A) \ar[r]_{id_{M_n}\otimes i_1} & M_n\otimes pB/Ip}$$
Now we are going to embed $M_n\otimes PBP$ back into $B$ and $ M_n\otimes pB/Ip$ -- back into $B/I$. Define $$\tilde
\alpha: M_n\otimes PBP \to B \; \; \text{and} \; \; \alpha: M_n\otimes p
B/I p \to B/I$$ by $$\tilde \alpha(e_{ij}\otimes PbP) =
\psi(e_{i1})b\psi(e_{1j})$$ and $$\alpha(e_{ij}\otimes p\pi(b)p) =
\phi_2(e_{i1})\pi(b)\phi_2(e_{1j})$$ respectively, for each $b\in
B$. It is straightforward to check that $$\alpha \circ (id_{M_n}\otimes i_1) \circ
i^{-1} \circ \phi = \phi$$ and that the diagram
$$\xymatrix {M_n\otimes PBP \ar[r]^-{\tilde \alpha}\ar[d]^-{id_{M_n}\otimes \pi|_{PBP}} & B\ar[d]^-{\pi} \\
M_n\otimes pB/Ip \ar[r]_-{\alpha}& B/I}$$ commutes. It follows that
$\tilde \alpha \circ \left(id_{M_n}\otimes \tilde \gamma \right)$ is
a lift of $\phi$.
\end{proof}
\begin{remark}
We did not consider the $C^{*}$-$W^{*}$ case in the theorem because no unital
$C^{*}$-algebra is $C^{*}$-$W^{*}$-projective in the non-unital category.
Otherwise $A\oplus \mathbb{C}$ would be unital and $C^{*}$-$W^{*}$-projective
which would contradict to Proposition \ref{NoProjections} since $A\oplus
\mathbb{C}$ has a non-trivial projection.
\end{remark}
Recall that a $C^*$-algebra is {\it subhomogeneous} if there
is an upper bound for the dimensions of its irreducible representations.
\begin{theorem}
\label{subhomogeneous} Let $A$ be a separable subhomogeneous $C^{*}$-algebra. Then $A$
is $W^*$- $W^{*}$-projective in the non-unital category.
\end{theorem}
\begin{proof}
Suppose $\mathcal{B}$ and $\mathcal{M}$ are von Neumann algebras and
$\pi:\mathcal{B}\rightarrow \mathcal{M}$ is a surjective $\ast
$-homomorphism. Let $\phi: A \to \mathcal M$ be a $\ast$-homomorphism. If $A$ is non-unital, we can extend $\phi$ to a homomorphism from the unitization of $A$ to $\mathcal M$. It implies that it will be sufficient to prove the theorem in the assumption that $A$ is unital.
Since $\mathcal M \subseteq B(H)$, by the universal property of the second dual
there exists $\tilde \phi: A^{**}\to B(H)$ such that $\tilde \phi|_{A} = \phi$ and $\tilde \pi(A^{**}) = \pi(A)''.$ Hence $\tilde \phi$ is a $\ast$-homomorphism from $A^{**}$ to $M$. It can be easily deduced from some well-known properties of subhomogeneous algebras (see for instance \cite {TanyaOtogo}, Lemmas 2.3 and 2.4) that $A^{**}$ can be written as $$A^{**} = \oplus_{k=1}^n M_k(D_k),$$ where $D_k, k\le n,$ are abelian von Neumann algebras. Let $\pi_k: A^{**}\to M_k(D_k)$ be the projection on the k-th summand. Let $$F_k = \{b\in D_k\;|\; \exists a\in A \; \text{such that b is a matrix element
of}\; \pi_k(a)\}, $$ for each $k\le n$. Let $E_k$ denote the $C^*$-subalgebra of $D_k$ generated by $F_k$, for each $k\le n$. Then each $E_k$ is separable and $A\subseteq \oplus_{k=1}^n M_k(E_k) \subseteq A^{**}$.
By Theorem \ref{commutative}, Theorem \ref{matrices over algebras} and Lemma \ref{directsums}, $\tilde \phi|_{\oplus_{k=1}^n M_k(E_k)}$ lifts to some $\ast$-homomorphism $\psi: \oplus_{k=1}^n M_k(E_k) \to \mathcal B$. The restriction of $\psi$ onto $A$ is a lift of $\phi$.
\end{proof}
The following are easy observations.
\begin{proposition}
Suppose $\mathcal{A}$ is a separable unital $W^*$-$W^*$- projective C*-algebra.
1) If $\mathcal{A}$ is nuclear and has a faithful trace, then it must be RFD.
2) If Connes' embedding problem has an affirmative answer, then every unital
$W^*$-$W^*$-projective $C^*$-algebra with a faithful trace is RFD.
\end{proposition}
The opposite to the previous proposition is not true. Indeed in
\cite{TracialStability} we constructed a nuclear RFD $C^{*}$-algebra which is
not matricially tracially stable and hence is not $W^{*}-W^{*}$-projective.
Below we give an example which is not only nuclear but even AF. Our arguments
of why it is not matricially tracially stable are much simpler than the ones
in \cite{TracialStability}.
\begin{theorem}\label{AF}
There exists an AF RFD $C^{*}$-algebra which is not matricially tracially
stable and hence is not $W^{*}$-$W^{*}$-projective (in both unital and
non-unital categories).
\end{theorem}
\begin{proof}
Suppose $\mathcal{A}$ and $\mathcal{B}$ are separable unital AF-C*-algebras.
Suppose $\mathcal{A}=C^{\ast}\left( a_{1},a_{2},\ldots \right) $ and
$\mathcal{B=}C^{\ast}\left( b_{1},b_{2},\ldots \right) $ with each $a_{n}$
and $b_{n}$ selfadjoint. We can assume that $\sigma \left( a_{1}\right)
\subset \left[ 0,1\right] $ and $\sigma \left( b_{1}\right) \subset \left[
4,5\right] $. Then we can find, for each $n\in \mathbb{N}$, a
finite-dimesnional C*-subalgebra $\mathcal{A}_{n}$ of $\mathcal{A}$ and
elements $a_{1,n},\ldots,a_{n,n}\in \mathcal{A}_{n}$ such that $\left \Vert
a_{k}-a_{k,n}\right \Vert <1/n$ for $1\leq k\leq n$. Similarly, we can find,
for each $n\in \mathbb{N}$ a finite-dimensional C*-subalgebra $\mathcal{B}_{n}$
of $\mathcal{B}$ and elements $b_{1,n},\ldots,b_{n,n}\in \mathcal{B}_{n}$ such
that $\left \Vert b_{k}-b_{k,n}\right \Vert <1/n$ for $1\leq k\leq n$. We can
also assume that $\sigma \left( a_{1,n}\right) \subset \left[ -1,2\right] $
and $\sigma \left( b_{1,n}\right) \subset \left[ 3,6\right] $ for every
$n\in \mathbb{N}$. We can assume, for each $n\in \mathbb{N}$ that $\mathcal{A}
_{n},\mathcal{B}_{n}\subset \mathcal{M}_{s_{n}}\left( \mathbb{C}\right) $
(unital embeddings). For each $1\leq k\leq n<\infty$ define
\[
c_{k,n}=a_{k,n}^{\left( n\right) }\oplus b_{k,n}\in \mathcal{M}_{\left(
n+1\right) s_{n}}\left( \mathbb{C}\right) .
\]
Define $c_{k,n}=0$ when $1\leq n<k<\infty$.
Let $C_{k}=\sum_{n\in \mathbb{N}}^{\oplus}c_{k,n}\in
{\displaystyle \prod \limits_{n\in \mathbb{N}}}
\mathcal{M}_{\left( n+1\right) s_{n}}\left( \mathbb{C}\right) $ and define
the C*-algebra $C$ as the C*-algebra generated by $C_{1},C_{2},\ldots$ and
$\mathcal{J}=\sum_{n\in \mathbb{N}}^{\oplus}\mathcal{M}_{\left( n+1\right)
s_{n}}\left( \mathbb{C}\right) $. Clearly, $\mathcal{C}$ is RFD and
$$\mathcal{C}/\mathcal{J} \cong C^{\ast}\left( a_{1}\oplus
b_{1},a_{2}\oplus b_{2},...\right) \subseteq \mathcal{A}\oplus \mathcal{B}.$$
However, if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and $f=0$ on
$\left[ 0,1\right] $ and $f=1$ on $\left[ 2,3\right] $, we have $f\left(
a_{1}\oplus b_{1}\right) =0\oplus1.$ Thus $0\oplus1 \in C^{\ast}\left( a_{1}\oplus
b_{1},a_{2}\oplus b_{2},...\right)$ and hence $C^{\ast}\left( a_{1}\oplus
b_{1},a_{2}\oplus b_{2},...\right) =\mathcal{A}\oplus \mathcal{B}$. Since
$\mathcal{J}$ and $\mathcal{C}/\mathcal{J}$ are AF, $\mathcal{C}$ must be AF.
Now, to get an example we wanted, suppose $\mathcal{A}=\mathcal{B}=\mathcal{M}_{2^{\infty}}$ with
trace $\tau$. Let $\pi=\pi_{1}\oplus \pi_{2}:\mathcal{C}\rightarrow
\mathcal{A}\oplus \mathcal{B}$ be the map whose kernel is $\mathcal{J}$. Then
$\rho=\tau \circ \pi_{2}$ is a tracial state on $\mathcal{C}$. Note that since $\mathcal{J}\subset \mathcal C$, the
only irreducible finite-dimensional representations of $\mathcal{C}$ are
(unitarily equivalent to) the coordinate projections onto $\mathcal{M}
_{\left( n+1\right) s_{n}}\left( \mathbb{C}\right) $ and, for each of
these representations the trace of the image of $f\left( C_{1}\right)
=\sum_{n\in \mathbb{N}}^{\oplus}0^{\left( n\right) }\oplus1$ is at most
$1/2.$ However, $\rho \left( f\left( C_{1}\right) \right) =1$. Thus $\rho$
is not a weak*-limit of finite-dimensional traces. By [\cite{TracialStability}, Th. 3.10] $\mathcal C$ is not matricially tracially stable.
\end{proof}
| 10,519
|
Warm Spinach Salad
Not all salads need to be served cold or at room temperature. This one is perfect especially in the winter since it has smokey bacon (or spicy pancetta), the richness of dijon mustard, and of course the sweetness of the shallots. This is super easy too.
INGREDIENTS
- 4-5 Strips of Pre-cooked bacon or 3 ounces of Diced Pancetta
- 1 Bag of Fresh Spinach
- 1 Shallot diced/chopped
- 1 Tablespoon Dijon Mustard
- 2 Tablespoons Red Wine Vinegar
- 2 Tablespoons Spring Water
Heat the bacon or pancetta until it is brown and crispy. Add in the diced shallots and continue to cook for a few minutes. You want the shallots soft but not burnt. Deglaze with red wine vinegar and whisk in dijon mustard. Once smooth, add in your water to thin it out and cut some of the acidity and spiciness of the mustard until it is like a thick (but not too thick) of a dressing. Pour over your spinach, toss, and serve.
| 82,786
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\begin{document}
\title{Non-group gradings on simple Lie algebras}
\author{Alberto Elduque}
\address{Departamento de
Matem\'{a}ticas e Instituto Universitario de Matem\'aticas y
Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain}
\email{elduque@unizar.es}
\thanks{${}^3$
Supported by grants MTM2017-83506-C2-1-P (AEI/FEDER, UE) and E22\_17R
(Gobierno de Arag\'on, Grupo de referencia ``\'Algebra y
Geometr{\'\i}a'', cofunded by Feder 2014-2020
``Construyendo Europa desde Arag\'on'')}
\subjclass[2010]{Primary 17B70; Secondary 17B40}
\keywords{Set grading; group grading; pure grading; orthogonal Lie algebra;
Steiner system}
\date{July 21, 2021}
\begin{abstract}
A set grading on the split simple Lie algebra of type $D_{13}$, that cannot
be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern provided by the lines of the projective plane over $GF(3)$. This answers in the negative \cite[Question 1.11]{EKmon}.
Similar non-group gradings are obtained for types $D_{n}$ with $n\equiv 1\pmod{12}$,
by substituting the lines in the projective plane by blocks of suitable Steiner systems.
\end{abstract}
\maketitle
\section{Introduction}\label{se:intro}
The aim of this paper is the construction of a set grading on a simple Lie algebra
that cannot be realized as a group grading, thus answering in the negative
\cite[Question 1.11]{EKmon}.
\smallskip
The systematic study of gradings on Lie algebras was initiated in 1989 by Patera and Zassenhaus \cite{PZ}, although particular gradings have been used from the beginning of Lie theory.
A \emph{set grading} on a Lie algebra $\cL$ over a field $\FF$
is a decomposition into a direct sum
of vector subspaces: $\Gamma:\cL=\bigoplus_{s \in S}\cL_s$, where $\cL_s\neq 0$ for any $s$ in the grading set $S$, such that for any $s_1,s_2\in S$, either
$[\cL_{s_1},\cL_{s_2}]=0$ or there exists an element $s_3\in S$ such that
$[\cL_{s_1},\cL_{s_2}]\subseteq \cL_{s_3}$. The subspaces $\cL_s$ are called the
\emph{homogeneous components} of $\Gamma$.
Two such gradings $\Gamma:\cL=\bigoplus_{s \in S}\cL_s$ and
$\Gamma':\cL'=\bigoplus_{s' \in S'}\cL'_{s'}$ are said to be \emph{equivalent}
if there is an isomorphism of Lie algebras $\varphi:\cL\rightarrow\cL'$ such that
for any $s\in S$ there is an $s'\in S'$ such that $\varphi(\cL_s)=\cL'_{s'}$.
On the other hand, given a (semi)group $G$, a \emph{$G$-grading} on $\cL$ is
a decomposition as above $\Gamma:\cL=\bigoplus_{g\in G}\cL_g$, such that
$[\cL_{g_1},\cL_{g_2}]\subseteq \cL_{g_1g_2}$ for all $g_1,g_2\in G$. The \emph{support} of $\Gamma$ is the subset
$\supp(\Gamma)\bydef\{g\in G\mid \cL_g\neq 0\}$.
Given a set grading $\Gamma:\cL=\bigoplus_{s \in S}\cL_s$, it is said that
$\Gamma$ can be realized as a (semi)group grading, if there exists a (semi)group
$G$ and a one-to-one map $\iota: S\hookrightarrow G$ such that the subspaces
$\cL_{\iota(s)}\bydef\cL_s$, and $\cL_g\bydef 0$ if $g\not\in\iota(S)$, form a
$G$-grading of $\cL$.
Given the set grading $\Gamma:\cL=\bigoplus_{s \in S}\cL_s$, its
\emph{universal group} is the group defined by generators (the elements of $S$) and
relations as follows:
\[
U(\Gamma)\bydef\Bigl\langle S\mid s_1s_2=s_3\ \text{for all
$s_1,s_2,s_3\in S$ with $0\neq [\cL_{s_1},\cL_{s_2}]\subseteq\cL_{s_3}$}\Bigr\rangle.
\]
There is a natural map $\iota:S\rightarrow U(\Gamma)$ taking $s$ to its coset
modulo the relations, and \emph{$\Gamma$ can be realized as a group grading if and only
if $\iota$ is one-to-one}.
The \emph{diagonal group} of $\Gamma$ is the subgroup $\Diag(\Gamma)$
of the group $\Aut(\cL)$
of automorphisms of $\cL$, consisting of those automorphisms that act by
multiplication by a scalar on each homogeneous component. Any
$\varphi\in\Diag(\Gamma)$ gives a map $\chi:S\rightarrow \FF^\times$ by the equation $\varphi\vert_{\cL_{s}}=\chi(s)\id$. This map induces a group homomorphism
(a character with values in $\FF$), denoted by the same symbol,
$\chi:U(\Gamma)\rightarrow \FF^\times$. And conversely, any character $\chi$
determines a unique element in $\Diag(\Gamma)$. This shows that the diagonal group is isomorphic to the group of characters with values in
$\FF$ of the universal group. If $\FF$ is algebraically closed of characteristic $0$ and $U(\Gamma)$ is abelian, characters separate points, and hence if $\Gamma$
is a group grading, the homogeneous components are the common
eigenspaces for the action of $\Diag(\Gamma)$.
For the basic results on gradings, the reader is referred to
\cite[Chapter 1]{EKmon}.
\smallskip
In the seminal work \cite{PZ}, it was erroneously asserted that any set grading
on a finite-dimensional Lie algebra can be realized as a semigroup grading \cite[Theorem 1.(d)]{PZ}.
Counterexamples were given in \cite{Eld06} and \cite{Eld09}. In particular, in
\cite{Eld09} there appears a non-semigroup grading on the semisimple Lie algebra
$\frsl_2\oplus\frsl_2$. On the other hand, Cristina Draper proved that if $G$ is
a semigroup and $\Gamma:\cL=\bigoplus_{g\in G}\cL_g$ is a $G$-grading on a
simple Lie algebra $\cL$, then $\supp(\Gamma)$ generates an abelian subgroup
of $G$ so, in particular, $\Gamma$ can be realized as a group grading.
(See \cite[Proposition 1.12]{EKmon}.) As a direct consequence, \emph{the universal group
of any set grading on a finite-dimensional simple Lie algebra is always abelian}.
\smallskip
There appeared naturally the following question \cite[Question 1.11]{EKmon}:
\begin{center}
\emph{Can any set grading on a finite-dimensional simple Lie algebra over $\CC$\\
be realized as a group grading?}
\end{center}
The main goal of this paper is to answer this question in the negative.
\smallskip
The author is indebted to Alice Fialowski, who during the
``Workshop on Non-Associative Algebras and Applications'', held in Lancaster University in 2018,
showed him a
$\left(\ZZ/2\right)^n$-grading on the orthogonal Lie algebra
$\cL=\frso(2n)$ over $\CC$,
constructed by Andriy Panasyuk,
with some nice properties: one of the homogeneous component $\cL_g$,
with $g\neq e$, is a Cartan subalgebra,
while all the other nonzero homogeneous components are toral two-dimensional
subalgebras. With hindsight, this grading
turns out to be the one in Example \ref{ex:Dn}. This is an example of
what Hesselink \cite{Hes} called \emph{pure gradings}. Some properties of these
pure gradings, useful for our purposes, are developed in Section \ref{se:pure}.
In an attempt to understand this grading on $\frso(2n)$ and to define similar gradings,
the author used the $13$ lines in the projective plane over the field of
three elements to split the set of positive roots of $\frso(26)$ (say, over $\CC$) into the disjoint union of pairs of orthogonal roots, thus obtaining the grading in \eqref{eq:D13}.
Checking that this is indeed a set grading is not obvious and relies on a pure
group grading on $\frso(8)$ in Example \ref{ex:D4}. The next step
was to check whether it is a group grading, but surprisingly it turned out not to be the case (Theorem \ref{th:D13}). Moreover, the result can be extended to the
classical split simple Lie algebra of type $D_{13}$ over any arbitrary field
of characteristic not two (Theorem \ref{th:D13_2}). All this appears in Section
\ref{se:D13}.
The last Section \ref{se:Steiner} is devoted to obtain similar non-group gradings on
the classical split simple Lie algebras of type $D_{2n}$ with
$n\equiv 1\pmod{12}$, by
substituting the lines in the projective plane $\PP^2(\FF_3)$ by the blocks
of Steiner systems of type $S(2,4,n)$.
\smallskip
Throughout the work, all the algebras considered will be assumed to be
defined over a field $\FF$ and to be finite-dimensional. Up to Theorem \ref{th:D13},
the ground field $\FF$ will be assumed to be algebraically closed of characteristic $0$.
\smallskip
\section{Pure gradings on semisimple Lie algebras}\label{se:pure}
Let $\cL$ be a semisimple Lie algebra over an algebraically closed field $\FF$ of characteristic $0$. Following \cite{Hes}, a grading $\Gamma$
by the abelian group $G$: $\Gamma:\cL=\bigoplus_{g\in G}\cL_g$, is said to be
\emph{pure} if there exists a Cartan subalgebra $\cH$ of $\cL$ and an element
$g\in G$, $g\neq e$, such that $\cH$ is contained in $\cL_g$. (Here $e$ denotes the neutral element of $G$.)
This section follows the ideas in \cite{Hes}.
Let $\Phi$ be the set of roots relative to $\cH$, so that
$\cL=\cH\oplus\left(\bigoplus_{\alpha\in\Phi}\cL_\alpha\right)$ is the
root space decomposition of $\cL$, and fix a system of simple roots
$\Delta=\{\alpha_1,\ldots,\alpha_n\}$ relative to the Cartan subalgebra $\cH$. Denote by $\Phi^+$ the set of positive roots relative to $\Delta$.
As in \cite[\S 4]{Hes}, consider the torus $T\bydef\{\varphi\in\Aut(\cL)\mid \varphi\vert_\cH=\id\}$ and its
$2$-torsion part: $T_2\bydef\{\varphi\in T\mid \varphi^2=\id\}$. Consider too the
larger subgroup $T_e\bydef\{\varphi\in\Aut(\cL)\mid \varphi\vert_\cH=\pm\id\}$.
\begin{lemma}\label{le:phiH-id}
Let $\sigma$ be an automorphism of $\cL$ with $\sigma\vert_\cH=-\id$. Then the following conditions hold:
\begin{enumerate}
\item
The order of $\sigma$ is $2$: $\sigma^2=\id$, $\sigma$ permutes
$\cL_\alpha$ and $\cL_{-\alpha}$ for all $\alpha\in\Phi$, and a system of generators $e_i\in\cL_{\alpha_i}$, $f_i\in\cL_{-\alpha_i}$, $i=1,\ldots,n$, can be chosen
such that $[e_i,f_i]=h_i$,
$\sigma(e_i)=-f_i$, and $\sigma(f_i)=-e_i$, where $h_i\in\cH$ satisfies $\alpha_i(h_i)=2$.
\item The subgroup $T_e$ is the semidirect product of the torus $T$ and the cyclic
group of order $2$ generated by $\sigma$.
\item The centralizer of $\sigma$ in $T$ is $T_2$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows from \cite[Lemma 4.3]{Hes}. We include a proof for completeness.
From $\sigma\vert_\cH=-\id$ it follows at once that $\sigma$ permutes
$\cL_\alpha$ and $\cL_{-\alpha}$ for all $\alpha\in\Phi$. For any
$\alpha\in\Phi^+$, choose nonzero elements $e_\alpha\in\cL_\alpha$
and $f_\alpha\in\cL_{-\alpha}$ such that
$[e_\alpha,f_\alpha]=h_{\alpha}$ with $\alpha(h_{\alpha})=2$. Then we have
$\sigma(h_\alpha)=-h_\alpha$, $\sigma(e_\alpha)=\mu f_{\alpha}$,
and $\sigma(f_{\alpha})=\mu^{-1}e_\alpha$ for some
$0\neq \mu\in\FF$. Let
$\eta\in\FF$ be a square root of $-\mu^{-1}$. Then we get
$\sigma(\eta e_\alpha)=-\eta^{-1}f_\alpha$ and
$\sigma(\eta^{-1}f_\alpha)=-\eta e_\alpha$. This completes the proof of the first part.
For any $\varphi\in T_e$, either $\varphi\vert_\cH=\id$ and hence we have
$\varphi\in T$, or $\varphi\vert_\cH=-\id$ and hence $\varphi\sigma\in T$. Therefore, $T_e$ equals $T\cup T\sigma$. Besides, $T$ is the kernel of
the natural homomorphism $T_e\rightarrow \{\pm 1\}$, so it is a normal subgroup
of $T_e$. This proves (2).
For any $\varphi\in T$, there are nonzero scalars $\mu_i\in\FF$ such that
$\varphi(e_i)=\mu_ie_i$ and $\varphi(f_i)=\mu_i^{-1}f_i$ for all
$i=1,\ldots,n$.
Then we get
\[
\begin{split}
\sigma\varphi(e_i)&=\mu_i\sigma(e_i)=-\mu_if_i,\\
\varphi\sigma(e_i)&=-\varphi(f_i)=-\mu_i^{-1}f_i,
\end{split}
\]
and hence, if $\varphi$ and $\sigma$ commute, we get $\mu_i=\mu_i^{-1}$, so that
$\mu_i^2=1$ for all $i$ and, therefore, $\varphi^2=\id$.
\end{proof}
The automorphism $\sigma$ in item (1) above play a key role in the proof of
the existence of Chevalley bases. (See \cite[\S 25.2]{Humphreys}.)
\begin{proposition}\label{pr:pure}
Let $\cL$ be a semisimple Lie algebra over an algebraically closed field $\FF$ of characteristic $0$. Let $G$ be an abelian group and let $\Gamma$ be a pure
$G$-grading of $\cL$. Let $g\in G\setminus\{e\}$ such that $\cL_g$ contains
a Cartan subalgebra $\cH$ of $\cL$. With $T$ and $T_2$ as before, there is an automorphism $\sigma\in\Aut(\cL)$ with
$\sigma\vert_\cH=-\id$, such that
$\Diag(\Gamma)$ is the cartesian product of $\Diag(\Gamma)\cap T_2$ and the
subgroup generated by $\sigma$:
\begin{equation}\label{eq:DiagG}
\Diag(\Gamma)=\left(\Diag(\Gamma)\cap T_2\right)\times\langle\sigma\rangle .
\end{equation}
In particular, $\Diag(\Gamma)$ is a finite $2$-elementary group.
\end{proposition}
\begin{proof}
Without loss of generality, we may assume that $G$ is the universal group
$U(\Gamma)$. Then the group $\Diag(\Gamma)$ can be identified with the group of
characters of $G$.
Let $\chi$ be a character of $G$ with $\chi(g)\neq 1$. The automorphism $\sigma$ of $\Diag(\Gamma)$
defined by $\sigma(x)=\chi(h)x$ for any $h\in G$ and $x\in\cL_h$ satisfies that its restriction to $\cH$ is $\chi(g)\id$. This forces
$\chi(g)^{-1}\alpha$ to be a root for any root $\alpha$ relative to
$\cH$. As the only scalar multiples of a root $\alpha$ are $\pm\alpha$,
we get $\chi(g)=-1$, and hence $\sigma\vert_\cH=-\id$. This also shows
that $\chi(g)=\pm 1$ for any character
$\chi$ of $G$, and hence $\Diag(\Gamma)$ is contained in $T_e$.
We conclude that $\Diag(\Gamma)$ is contained in the centralizer of
$\sigma$ in
$T_e$, which is $T_2\times\langle\sigma\rangle$ by Lemma \ref{le:phiH-id}.(3).
The result follows.
\end{proof}
From now on, fix a Cartan subalgebra $\cH$ of $\cL$ and an automorphism
$\sigma\in\Aut(\cL)$ such that $\sigma\vert_\cH=-\id$. Note that $\sigma$ is unique up to conjugation because of Lemma \ref{le:phiH-id}.(1). Denote by $Q$ the root lattice $Q\bydef\ZZ\Phi=\ZZ\Delta$ (notation as above). The torus $T$ is naturally isomorphic to the group of characters of $Q$, and this isomorphism restricts to
a group isomorphism $T_2\simeq \Hom(Q/2Q,\{\pm 1\})$. Any
$\chi\in\Hom(Q/2Q,\{\pm 1\})$ corresponds to the automorphism $\tau_\chi$ whose
restriction to $\cL_\alpha$ is $\chi(\alpha +2Q)\id$ for all $\alpha\in\Phi$.
In particular, any element of $T_2$ acts a $\pm\id$ on
the two-dimensional subspace $\cL_\alpha+\cL_{-\alpha}$ for any $\alpha\in\Phi^+$.
This gives a bijection:
\begin{equation}\label{eq:T2}
\begin{split}
\{\text{subgroups of $T_2$}\}&\longrightarrow
\{\text{subgroups of $Q$ containing $2Q$}\}\\
S\quad&\mapsto\quad E\
\text{such that $E/2Q=\bigcap_{\tau_\chi\in S}\ker\chi$}\,.
\end{split}
\end{equation}
In the reverse direction, a subgroup $E$ with $2Q\leq E\leq Q$ corresponds to
the subgroup $T_E\bydef\{\tau_\chi\mid \chi(E/2Q)=1\}$.
For any positive root $\alpha\in\Phi^+$, pick a nonzero element
$x_\alpha\in\cL_\alpha$. Let $E$ be a subgroup with $2Q\leq E\leq Q$ and denote by
$\overline{Q}_E$ the
$2$-elementary group $Q/E\times \ZZ/2$. Define the $\overline{Q}_E$-grading
$\Gamma_E$ on $\cL$ as follows:
\begin{equation}\label{eq:GammaE}
\begin{split}
\cL_{(q+E,\bar 0)}
&=\bigoplus_{\alpha\in\Phi^+\cap(q+E)}\FF\left(x_\alpha+\sigma(x_\alpha)\right),\\
\cL_{(q+E,\bar 1)}
&=\begin{cases}
\bigoplus_{\alpha\in\Phi^+\cap(q+E)}\FF\left(x_\alpha-\sigma(x_\alpha)\right)
&\text{if $q+E\neq E$,}\\
\cH\oplus\left(\bigoplus_{\alpha\in\Phi^+\cap(q+E)}
\FF\left(x_\alpha-\sigma(x_\alpha)\right)\right)
&\text{if $q+E=E$.}
\end{cases}
\end{split}
\end{equation}
\begin{remark}\label{re:GammaE}
Note that the homogeneous spaces of $\Gamma_E$ are the common eigenspaces for
the action of
the group $T_E\times\langle\sigma\rangle$.
\end{remark}
The extreme cases are $\Gamma_Q$, which is a grading by $\ZZ/2$ with
$\cL\subo=\{x\in\cL\mid \sigma(x)=x\}$ and
$\cL\subuno=\{x\in\cL\mid \sigma(x)=-x\}$, and $\Gamma_{2Q}$, which is a grading
by $Q/2Q\times\ZZ/2\simeq \left(\ZZ/2\right)^{n+1}$.
For any $E$ as above, define the subgroup $E^\circ$ as follows:
\begin{equation}\label{eq:Ecirc}
E^\circ\bydef 2Q +\ZZ(\Phi^+\cap E)+\ZZ\{\alpha-\beta\mid \alpha,\beta\in\Phi^+\
\text{and}\ \alpha-\beta\in E\}.
\end{equation}
Note that $2Q\leq E^\circ\leq E\leq Q$.
\begin{remark}\label{re:Ecirc}
Another way to define $E^\circ$ is as the subgroup of $E$ generated by $2Q$,
by the elements $\alpha\in\Phi^+$ such that
$x_\alpha-\sigma(x_\alpha)$ is
in the homogeneous component of $\Gamma_E$ that contains
$\cH$, and by the elements
$\alpha-\beta$ for $\alpha,\beta\in\Phi^+$ such that
$x_\alpha-\sigma(x_\alpha)$
and $x_\beta-\sigma(x_\beta)$ are in the same homogeneous component of $\Gamma_E$.
\end{remark}
\begin{proposition}\label{pr:DiagGE}
Under the conditions above, the gradings $\Gamma_E$ and $\Gamma_{E^\circ}$ are equivalent and $\Diag(\Gamma_E)=\Diag(\Gamma_{E^\circ})$ equals
$T_{E^\circ}\times\langle\sigma\rangle$.
\end{proposition}
\begin{proof}
The homogeneous components of $\Gamma_E$ and $\Gamma_{E^\circ}$ coincide, so they are trivially equivalent.
Proposition \ref{pr:pure} gives $\Diag(\Gamma_E)=\left(\Diag(\Gamma_E)\cap T_2\right)\times \langle \sigma\rangle$. For any character $\chi\in\Hom(Q/2Q,\{\pm 1\})$,
let $\tau_\chi$ be the associated element in $T_2$. For any $\alpha\in\Phi^+$,
$\tau_\chi\left(x_\alpha\pm\sigma(x_\alpha)\right)
=\chi(\alpha+2Q)\left(x_\alpha\pm\sigma(x_\alpha)\right)$.
Then $\tau_\chi$ belongs to $\Diag(\Gamma_E)$ if and only if $\chi(\alpha+2Q)=1$
for any $\alpha\in\Phi^+\cap E$ (because $\tau_\chi\vert_\cH=\id$), and
$\chi(\alpha+2Q)=\chi(\beta+2Q)$ for any $\alpha,\beta\in\Phi^+$ with
$\alpha-\beta\in E$. In other words, $\tau_\chi$ lies in $\Diag(\Gamma_E)$ if
and only if $E^\circ/2Q$ is contained in $\ker\chi$, if and only if $\tau_\chi$ lies in $T_{E^\circ}$.
\end{proof}
Let us show now that $E^\circ$ may be strictly contained in $E$.
\begin{example}\label{ex:D6}
Let $\cL$ be the orthogonal Lie algebra $\frso(V,\bup)$ of a vector space $V$ of
dimension $12$, endowed with a nondegenerate symmetric bilinear form $\bup$.
Pick a basis $\{u_1,\ldots,u_6,v_1,\ldots,v_6\}$ of $V$ with
$\bup(u_i,u_j)=0=\bup(v_i,v_j)$ and $\bup(u_i,v_j)=\delta_{ij}$ (Kronecker delta) for all $i,j$.
The diagonal elements of $\frso(V,\bup)$ relative to this basis form a Cartan subalgebra $\cH$, and the weights of $V$ relative to $\cH$ are
$\pm\veps_1,\ldots,\pm\veps_6$, where
$h.u_i=\veps_i(h)u_i$, $h.v_i=-\veps_i(h)v_i$ for all $h\in\cH$ and $i=1,\ldots,6$. The set of roots is $\Phi=\{\pm\veps_i\pm\veps_j\mid i\neq j\}$.
Up to a scalar, the nondegenerate bilinear form on $\cH^*$ induced
by the Killing form is given by $(\veps_i\vert\veps_j)=\delta_{ij}$ (Kronecker delta).
As a system of simple roots
take $\Delta=\{\veps_1-\veps_2,\veps_2-\veps_3,\veps_3-\veps_4,\veps_4-\veps_5,
\veps_5-\veps_6,\veps_5+\veps_6\}$. The highest weight of one of the half-spin
modules es $\frac{1}{2}(\veps_1+\cdots+\veps_6)$. Denote by
$\Lambda'$ the set
of weights of this half spin module:
$\Lambda'=\bigl\{\frac12(\pm\veps_1\pm\cdots\pm\veps_6)\mid
\text{even number of $+$ signs}\bigr\}$. Write $W'=\ZZ\Lambda'$. Then $E=2W'$ satisfies
$2Q\leq E=
2Q+\ZZ(\veps_1+\cdots+\veps_6)\leq Q$.
It follows at once that for any $i<j$, $r<s$ and $(i,j)\neq (r,s)$, the element
$\frac12\bigl((\veps_i\pm\veps_j)-(\veps_r\pm\veps_s)\bigr)$ does not belong neither to $Q$, nor to $\frac12(\veps_1+\cdots\veps_6)+Q$. Hence we get that for any two different positive roots
$\alpha,\beta\in\Phi^+$, we get $\alpha-\beta\not\in E$. It is also clear that
$\alpha$ does not belong to $E$ for any $\alpha\in\Phi^+$, so \eqref{eq:Ecirc}
shows that $E^\circ$ equals $2Q$.
\end{example}
\begin{remark}\label{re:caution}
A word of caution is in order here. The notion of equivalence in
\cite{Hes} is more restrictive. It turns out that
$\Gamma_E$ and
$\Gamma_{E^\circ}$ are not equivalent in this more restrictive sense if $E^\circ$ is contained properly in $E$, as in Example \ref{ex:D6}, even though their
homogeneous components coincide, so they are trivially equivalent in our sense.
\end{remark}
\smallskip
The situation is different for $D_4$, and this will be crucial in our examples of non-group gradings in the next sections.
\begin{example}\label{ex:D4}
Let $\cL$ be the orthogonal Lie algebra $\frso(V,\bup)$ of a vector space $V$ of
dimension $8$, endowed with a nondegenerate symmetric bilinear form
$\bup$. As in Example \ref{ex:D6},
pick a basis $\{u_1,\ldots,u_4,v_1,\ldots,v_4\}$ of $V$ with
$\bup(u_i,u_j)=0=\bup(v_i,v_j)$ and $\bup(u_i,v_j)=\delta_{ij}$ for all $i,j$, and consider the corresponding Cartan subalgebra of diagonal elements relative to this basis, and the system of simple roots
$\Delta=\{\veps_1-\veps_2,\veps_2-\veps_3,\veps_3-\veps_4,
\veps_3+\veps_4\}$. Pick the highest weight
$\frac12(\veps_1+\veps_2+\veps_3+\veps_4)$
of one of the half-spin modules, and let $\Lambda'$ be its set of weights. Finally, write $W'=\ZZ\Lambda'$ and consider the subgroup
$E=2W'=2Q+\ZZ(\veps_1+\veps_2+\veps_3+\veps_4)$,
whose index $[E:2Q]$ is $2$.
For any $1\leq r<s\leq 4$, let $\bar r<\bar s$ be such that
$\{r,s,\bar r,\bar s\}=\{1,2,3,4\}$. Then the positive roots
$\veps_r+\veps_s$ and $\veps_{\bar r}+\veps_{\bar s}$
satisfy that its difference belongs to $E$, and the same happens with
$\veps_1+\veps_2+\veps_3+\veps_4=(\veps_r+\veps_s)-(\veps_{\bar r}+\veps_{\bar s})+2(\veps_{\bar r}+\veps_{\bar s})$. Therefore in this case we get $E=E^\circ$.
Proposition \ref{pr:DiagGE} shows that the diagonal group of the
grading $\Gamma_E$ is $T_E\times\langle\sigma\rangle$, which is
isomorphic to $\left(\ZZ/2\right)^4$.
\end{example}
Our last example is Panayuk's example mentioned in Section \ref{se:intro}.
\begin{example}\label{ex:Dn}
Let $\cL$ be the orthogonal Lie algebra $\frso(V,\bup)$ of a vector space $V$ of
dimension $2n$, endowed with a nondegenerate symmetric bilinear form $\bup$.
Pick a basis $\{u_1,\ldots,u_n,v_1,\ldots,v_n\}$ of $V$ with
$\bup(u_i,u_j)=0=\bup(v_i,v_j)$ and $\bup(u_i,v_j)=\delta_{ij}$ for all
$i,j$ as in Example \ref{ex:D6}, and use the notation there. Let
$W=\ZZ\veps_1\oplus\cdots\oplus\ZZ\veps_n$ be the $\ZZ$-span of the
weights of the natural module $V$. Note that $W=Q+\ZZ\veps_1$ and
$2\veps_1=(\veps_1-\veps_2)+(\veps_1+\veps_2)$ belongs to $Q$.
Hence the index $[W:Q]$ is $2$. Consider the subgroup $E=2W$, which lies between $2Q$ and $Q$, and the associated grading
$\Gamma_{2W}$ over $Q/(2W)\times \ZZ/2\simeq \left(\ZZ/2\right)^n$
in \eqref{eq:GammaE}.
The intersection $\Phi^+\cap 2W$ is empty, and the only pairs of positive roots that belong to the same coset modulo $2W$ are
$\veps_i-\veps_j$ and $\veps_i+\veps_j$ for $1\leq i< j\leq n$.
It follows that $E^\circ=E$ and that the homogeneous component
$\cL_{(E,\bar 1)}$ coincides with the Cartan subalgebra, while all the other nonzero homogeneous components have dimension two.
\end{example}
\begin{remark}\label{re:triality}
The same properties are valid for the grading in Example \ref{ex:D4}. But for $n=4$, the natural and half-spin modules are related by triality, so the grading in Example \ref{ex:D4} is equivalent to the one in
Example \ref{ex:Dn} for $n=4$. However, the specific $E$ in
Example \ref{ex:D4} will be crucial in the next section (proof of Proposition \ref{pr:set_grading}).
\end{remark}
\smallskip
\section{A non-group grading on $D_{13}$}\label{se:D13}
We may number the points in the projective plane $\PP^2(\FF_3)$
over the field $\FF_3$ of three elements from
$1$ to $13$, so that the lines in $\PP^2(\FF_3)$ are the ones consisting of the points:
\begin{equation}\label{eq:lines}
\begin{matrix}
\{1,2,3,4\}&\{2,5,8,11\}&\{3,5,9,13\}&\{4,5,10,12\}\\
\{1,5,6,7\}&\{2,6,9,12\}&\{3,6,10,11\}&\{4,6,8,13\}\\
\{1,8,9,10\}&\{2,7,10,13\}&\{3,7,8,12\}&\{4,7,9,11\}\\
\{1,11,12,13\}&&&
\end{matrix}
\end{equation}
(The reader can check that any two points lie in a unique line, and any two lines intersect in a unique point.) Denote by $\mathfrak{L}$ this set of lines.
As in Examples \ref{ex:D6} and \ref{ex:D4}, let $\cL$ be the orthogonal Lie algebra $\frso(V,\bup)$ of a vector space $V$ of
dimension $26$, endowed with a nondegenerate symmetric bilinear form $\bup$.
Pick a basis $\{u_1,\ldots,u_{13},v_1,\ldots,v_{13}\}$ of $V$ with
$\bup(u_i,u_j)=0=\bup(v_i,v_j)$ and $\bup(u_i,v_j)=\delta_{ij}$ for all $i,j$.
The diagonal elements of $\frso(V,\bup)$ relative to this basis form a Cartan subalgebra $\cH$, and the weights of $V$ relative to $\cH$ are
$\pm\veps_1,\ldots,\pm\veps_{13}$, where
$h.u_i=\veps_i(h)u_i$, $h.v_i=-\veps_i(h)v_i$ for all $h\in\cH$ and
$i=1,\ldots,13$. The set of roots is
$\Phi=\{\veps_i\pm\veps_j\mid i\neq j\}$. As a system of simple roots
take $\Delta=\{\veps_1-\veps_2,\veps_2-\veps_3,\veps_3-\veps_4,\dots,
\veps_{12}-\veps_{13},\veps_{12}+\veps_{13}\}$.
Fix an automorphism $\sigma$ of $\cL$ with $\sigma\vert_\cH=-\id$, and for any positive root $\alpha\in\Phi^+$ pick a nonzero element
$x_\alpha\in\cL_\alpha$.
For any line $\ell=\{i,j,k,l\}\in\frL$, with $i<j<k<l$ consider the set
$\frP_{\ell}$ whose elements are the following six subsets of pairs of orthogonal positive roots
\begin{equation}\label{eq:six}
\begin{matrix}
\{\veps_i+\veps_j,\veps_k+\veps_l\},&\
\{\veps_i+\veps_k,\veps_j+\veps_l\},&\
\{\veps_i+\veps_l,\veps_j+\veps_k\},\\
\{\veps_i-\veps_j,\veps_k-\veps_l\},&\
\{\veps_i-\veps_k,\veps_j-\veps_l\},&\
\{\veps_i-\veps_l,\veps_j-\veps_k\}.
\end{matrix}
\end{equation}
Denote by $\frP$ the union $\frP=\bigcup_{\ell\in\frL}\frP_{\ell}$. Its
elements are subsets consisting of a pair of orthogonal positive roots. Hence
$\frP$ contains $13\times 6=78$ elements, and each positive root appears in exactly one of the pairs in $\frP$. Then $\cL$ decomposes as
\begin{multline}\label{eq:D13}
\cL=\cH\oplus\Bigl(\bigoplus_{\{\alpha,\beta\}\in\frP}
\bigl(\FF(x_\alpha+\sigma(x_\alpha))
+\FF(x_\beta+\sigma(x_\beta))\bigr)\Bigr) \\\oplus
\Bigl(\bigoplus_{\{\alpha,\beta\}\in\frP}
\bigl(\FF(x_\alpha-\sigma(x_\alpha))
+\FF(x_\beta-\sigma(x_\beta))\bigr)\Bigr).
\end{multline}
Hence $\cL$ decomposes into the direct sum of the Cartan subalgebra and the direct sum of $13\times 6\times 2=156$ two-dimensional abelian subalgebras. The reader may recall here Example \ref{ex:Dn}, where a group-grading with these properties is highlighted.
\begin{proposition}\label{pr:set_grading}
The decomposition in Equation \eqref{eq:D13} is a set grading of
$\cL$.
\end{proposition}
\begin{proof}
For any $h\in\cH$ and $\alpha\in\Phi^+$,
$[h,x_\alpha\pm\sigma(x_\alpha)]
=\alpha(h)(x_\alpha\mp\sigma(x_\alpha))$. Also, for any two orthogonal positive roots $\alpha,\beta$,
$[x_\alpha\pm\sigma(x_\alpha),x_\beta\pm\sigma(x_\beta)]=0$, because $\alpha\pm\beta$ is not a root. We must check that the bracket
of two subspaces of the form $\FF(x_\alpha\pm\sigma(x_\alpha))+\FF(x_\beta\pm\sigma(x_\beta))$, with $\{\alpha,\beta\}\in\frP$, is contained in another subspace of this form. So let us take two such elements $\{\alpha,\beta\}\in\frP_{\ell_1}$ and $\{\alpha',\beta'\}\in\frP_{\ell_2}$. There are two different possibilities:
\begin{itemize}
\item
$\ell_1\cap\ell_2$ consists of a single point: $\ell_1=\{i,j,k,l\}$,
$\ell_2=\{i,p,q,r\}$ for distinct $i,j,k,l,p,q,r$. This shows that, up to reordering, $(\alpha\vert\alpha')\neq 0$, but $(\alpha\vert\beta')=0=
(\alpha'\vert\beta)$. This is because, up to reordering, the roots involved are of the form $\alpha=\pm\veps_i\pm\veps_j$,
$\beta=\pm\veps_k\pm\veps_l$, $\alpha'=\pm\veps_i\pm\veps_p$,
$\beta=\pm\veps_q\pm\veps_r$. Since $(\alpha\vert\alpha')\neq 0$,
either $\alpha+\alpha'$ or $\alpha-\alpha'$ is a root (but not both), while
$\alpha\pm\beta'$ and $\alpha'\pm\beta$ are not roots.
It follows that the bracket of $\FF(x_\alpha\pm\sigma(x_\alpha))+
\FF(x_\beta\pm\sigma(x_\beta))$ with
$\FF(x_{\alpha'}\pm\sigma(x_{\alpha'}))
+\FF(x_{\beta'}\pm\sigma(x_{\beta'}))$ equals
$\FF[x_\alpha\pm\sigma(x_\alpha),x_{\alpha'}\pm\sigma(x_{\alpha'})]$,
which equals
$\FF\left(x_{\alpha\pm\alpha'}\pm\sigma(x_{\alpha\pm\alpha'})\right)$, depending on whether $\alpha+\alpha'$ or $\alpha-\alpha'$ is a root.
\item
$\ell_1=\ell_2$. Without loss of generality we may assume that
$\ell=\{1,2,3,4\}$. Then the subalgebra generated by $x_\alpha$ and
$\sigma(x_\alpha)$ for $\alpha$ in the six subsets associated to $\ell$ generate a subalgebra of $\cL$ isomorphic to $D_4$. The decomposition induced on this subalgebra from the decomposition in \eqref{eq:D13} is exactly the group-grading $\Gamma_E$ in Example \ref{ex:D4}, and hence the bracket of $\FF(x_\alpha\pm\sigma(x_\alpha))+
\FF(x_\beta\pm\sigma(x_\beta))$ with
$\FF(x_{\alpha'}\pm\sigma(x_{\alpha'}))
+\FF(x_{\beta'}\pm\sigma(x_{\beta'}))$ is contained in another
`homogeneous component' in \eqref{eq:D13}.
\end{itemize}
We conclude that the decomposition in \eqref{eq:D13} is indeed a set grading on
$\cL$.
\end{proof}
Denote the grading given by \eqref{eq:D13} by $\Gamma$. Our aim now is to show that $\Gamma$ cannot be realized as a group-grading. We will do it by computing its diagonal group.
To begin with, note that the automorphism $\sigma$ belongs to
$\Diag(\Gamma)$, and that any element of $\Diag(\Gamma)$ must act as a scalar on
$\cH$. The arguments in the proof of
Proposition \ref{pr:pure} work for $\Gamma$ and hence Equation
\eqref{eq:DiagG} holds: $\Diag(\Gamma)
=\left(\Diag(\Gamma)\cap T_2\right)\times\langle\sigma\rangle$.
For any $\chi\in\Hom(Q/2Q,\{\pm 1\})$, let $\tau_{\chi}$ be the corresponding element in $T_2$ (recall Equation \eqref{eq:T2}). Then, as in the proof of
Proposition \ref{pr:DiagGE}, $\tau_\chi$ is in
$\Diag(\Gamma)$ if and only if $\chi(\alpha+2Q)=\chi(\beta +2Q)$ for all
$\{\alpha,\beta\}\in\frP$, if and only if $\bigl\{(\alpha-\beta)+2Q\mid \{\alpha,\beta\}\in\frP\bigr\}$ is contained in $\ker\chi$. Therefore we get
$\Diag(\Gamma)=T_E\times\langle\sigma\rangle$ for
$E=2Q+\ZZ\bigl\{\alpha-\beta\mid
\{\alpha,\beta\}\in\frP\bigr\}$.
In particular, for $\ell=\{i,j,k,l\}\in\frL$, with $1\leq i<j<k<l\leq 13$, the sum
$\veps_i+\veps_j+\veps_k+\veps_l=(\veps_i+\veps_j)-(\veps_k+\veps_l)+2(\veps_k+\veps_l)$ lies in $E$.
Considering the lines in the first column of \eqref{eq:lines}, and adding the elements
$\veps_i+\veps_j+\veps_k+\veps_l$ for these lines, we check that
$E$ contains the elements $4\veps_1+\sum_{i=2}^{13}\veps_i$. As
$4\veps_1=2\left((\veps_1-\veps_2)+(\veps_1+\veps_2)\right)$ lies in
$2Q\leq E$, it follows that $\sum_{i\neq 1}\veps_i$ lies in $E$.
In the same vein $\sum_{i\neq 2}\veps_i$ lies in $E$, and
subtracting these two elements, we see that $\veps_1-\veps_2$ lies
in $E$. The same argument shows that all the roots of the form
$\veps_i-\veps_j$ lie in $E$. But then we get
$Q=\ZZ\Delta\subseteq E+\ZZ(\veps_{12}+\veps_{13})\subseteq Q$, so $Q=E+\ZZ(\veps_{12}+\veps_{13})$. Also we have
$2(\veps_{12}+\veps_{13})\in 2Q\subseteq E$, and hence the index of
$E$ in $Q$ is at most $2$, and
$\Diag(\Gamma)=T_E\times\langle\sigma\rangle$ is isomorphic to either $C_2$ or $C_2\times C_2$. Actually, the splitting
$V=(\FF u_1+\cdots+\FF u_6)\oplus (\FF v_1+\cdots+\FF v_6)$ gives a
$\ZZ/2$-grading on $\cL$, where the even (respectively odd) part
consists of the elements in $\cL$ that preserve (respectively swap) the
subspaces $\FF u_1+\cdots+\FF u_6$ and $\FF v_1+\cdots+\FF v_6$.
The order two automorphism $\tau$ that fixes the even part and is
$-\id$ on the odd part lies in $T$. It fixes the root spaces
$\cL_{\veps_i-\veps_j}$ and is $-\id$ on the root spaces
$\cL_{\veps_i+\veps_j}$. It
follows that $\tau$ is in $\Diag(\Gamma)\cap T_2$, and we conclude that the diagonal group of $\Gamma$ is exactly
$\Diag(\Gamma)=\langle\tau,\sigma\rangle$.
\smallskip
Alternatively, we can use some software, for instance SageMath
\cite{Sage}, to compute $[Q:E]$. Let
$W$ be the group generated by $\veps_1,\ldots,\veps_{13}$. (Note
$[W:Q]=2$.)
Write in a SageMath cell the following:
\smallskip
{\obeylines
\def\salto{\noindent\phantom{A=matrix([}}
\noindent\texttt{
A=matrix([[2,-2,0,0,0,0,0,0,0,0,0,0,0],[0,2,-2,0,0,0,0,0,0,0,0,0,0],
\salto[0,0,2,-2,0,0,0,0,0,0,0,0,0],[0,0,0,2,-2,0,0,0,0,0,0,0,0],
\salto[0,0,0,0,2,-2,0,0,0,0,0,0,0],[0,0,0,0,0,2,-2,0,0,0,0,0,0],
\salto[0,0,0,0,0,0,2,-2,0,0,0,0,0],[0,0,0,0,0,0,0,2,-2,0,0,0,0],
\salto[0,0,0,0,0,0,0,0,2,-2,0,0,0],[0,0,0,0,0,0,0,0,0,2,-2,0,0],
\salto[0,0,0,0,0,0,0,0,0,0,2,-2,0],[0,0,0,0,0,0,0,0,0,0,0,2,-2],
\salto[0,0,0,0,0,0,0,0,0,0,0,2,2],
\salto[1,1,1,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,1,1,1,0,0,0,0,0,0],
\salto[1,0,0,0,0,0,0,1,1,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,1,1],
\salto[0,1,0,0,1,0,0,1,0,0,1,0,0],[0,1,0,0,0,1,0,0,1,0,0,1,0],
\salto[0,1,0,0,0,0,1,0,0,1,0,0,1],[0,0,1,0,1,0,0,0,1,0,0,0,1],
\salto[0,0,1,0,0,1,0,0,0,1,1,0,0],[0,0,1,0,0,0,1,1,0,0,0,1,0],
\salto[0,0,0,1,1,0,0,0,0,1,0,1,0],[0,0,0,1,0,1,0,1,0,0,0,0,1],
\salto[0,0,0,1,0,0,1,0,1,0,1,0,0]]);
\noindent A.elementary\_divisors()
}}
\smallskip
\noindent where the first six rows correspond to the coordinates of the elements of the basis $2\Delta$ of $2Q$ in the natural basis $\{\veps_1,\ldots,\veps_{13}\}$ of $W$.
The last six rows are the coordinates of the elements
$\veps_i+\veps_j+\veps_k+\veps_l$ for each line $\{i,j,k,l\}\in\frL$.
The outcome of running the SageMath cell above is the following:
\smallskip
\noindent\texttt{
[1,1,1,1,1,1,1,1,1,1,1,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0]}
\smallskip
\noindent giving the elementary divisors of the matrix $A$,
and this shows that the index $[W:E]$ is $4$, and hence the index $[Q:E]$ is $2$.
\smallskip
We conclude that the universal group of $\Gamma$ is also isomorphic to $C_2\times C_2$, because its group of characters is
$\Diag(\Gamma)\simeq C_2\times C_2$, but there are $157$ nonzero
homogeneous components, and hence the natural map $\iota:S\rightarrow U(\Gamma)$ cannot be one-to-one, where $S$ denotes the set of homogeneous components of
$\Gamma$. We have proved the following result:
\begin{theorem}\label{th:D13}
The set grading $\Gamma$ of the simple Lie algebra of type $D_{13}$ in \eqref{eq:D13} cannot be realized as a group grading.
\end{theorem}
Actually, this grading makes sense over an arbitrary field $\FF$ of characteristic not two. Indeed, let $\cL$ be the orthogonal Lie algebra
$\frso(V,\bup)$ of a vector space $V$ over $\FF$ of
dimension $26$, endowed with a nondegenerate symmetric bilinear form $\bup$ of maximal Witt index. That is, $\cL$ is the classical split simple
Lie algebra of type $D_{13}$ over $\FF$.
Pick a basis $\{u_1,\ldots,u_{13},v_1,\ldots,v_{13}\}$ of $V$ with
$\bup(u_i,u_j)=0=\bup(v_i,v_j)$ and $\bup(u_i,v_j)=\delta_{ij}$ for all $i,j$.
As before, the diagonal elements of $\frso(V,\bup)$ relative to this basis form a Cartan subalgebra $\cH$, and the weights (elements of the dual
$\cH^*$) of $V$ relative to $\cH$ are
$\pm\veps_1,\ldots,\pm\veps_{13}$, where
$h.u_i=\veps_i(h)u_i$, $h.v_i=-\veps_i(h)v_i$ for all $h\in\cH$ and $i=1,\ldots,n$.
Up to a scalar, the nondegenerate bilinear form on $\cH^*$ induced
from the trace form on $V$ is given by
$(\veps_i\vert\veps_j)=\delta_{ij}$ (Kronecker delta).
Identify any $x\in \cL$ with its coordinate matrix relative to the basis above, which has the following block form
$
\begin{pmatrix}A&B\\ C&-A^t\end{pmatrix}
$,
where the blocks are $13\times 13$ matrices, with $B$ and $C$ symmetric: $B=B^t$, $C=C^t$.
The root space decomposition of $\cL$ relative to $\cH$ is
\begin{equation}\label{eq:root}
\cL=\cH\oplus\left(\bigoplus_{i\neq j}\cL_{\veps_i-\veps_j}\right)
\oplus\left(\bigoplus_{i<j}\cL_{\veps_i+\veps_j}\right)
\oplus\left(\bigoplus_{i<j}\cL_{-\veps_i-\veps_j}\right)
\end{equation}
where the roots spaces are:
\[
\cL_{\veps_i-\veps_j}
=\FF\begin{pmatrix} E_{ij}&0\\ 0&-E_{ji}\end{pmatrix},\quad
\cL_{\veps_i+\veps_j}
=\FF\begin{pmatrix}0& E_{ij}\\ 0&0\end{pmatrix},\quad
\cL_{-\veps_i-\veps_j}
=\FF\begin{pmatrix} 0&0\\ E_{ij}&0\end{pmatrix},
\]
where $E_{ij}$ is the $13\times 13$-matrix with $1$ in the
$(i,j)$-position and $0$'s elsewhere.
Let $W$ be the free abelian group with generators
$\epsilon_1,\ldots,\epsilon_{13}$
(note the slightly different notation:
$\epsilon_i$ for free generators, and $\veps_i$ for weights), and its
index $2$ subgroup $Q$ freely generated by $\epsilon_1-\epsilon_2,
\epsilon_2-\epsilon_3,\ldots,\epsilon_{12}-\epsilon_{13},
\epsilon_{12}+\epsilon_{13}$.
The root space decomposition \eqref{eq:root} is a $Q$-grading, with
$\cL_{\pm\epsilon_i\pm\epsilon_j}\bydef\cL_{\pm\veps_i\pm\veps_j}$ for all
$i,j$. The automorphism $\sigma: x\mapsto -x^t$ has order $2$ and satisfies $\sigma(\cL_\alpha)=\cL_{-\alpha}$ for all roots $\alpha$.
This automorphism $\sigma$ can be used to define the grading
$\Gamma$ as in \eqref{eq:D13}
The only scalar multiples of a root $\alpha$ are $\pm\alpha$, hence the argument in the proof of Proposition \ref{pr:pure} gives that any automorphism of $\cL$ that acts as a scalar multiple of $\id$ on $\cH$
must act necessarily as $\id$ or $-\id$. Now the arguments in
Lemma \ref{le:phiH-id}
and Proposition \ref{pr:pure} apply to show that $T_2$ is the centralizer of $\sigma$ in $T$ and that \eqref{eq:DiagG} holds here.
On the other hand, any $\varphi\in T_2$ is determined by its
restriction to the root spaces $\cL_{\veps_1-\veps_2},\ldots,
\cL_{\veps_{12}-\veps_{13}}\cL_{\veps_{12}+\veps_{13}}$, and this allows us to identify $T_2$ with the group $\Hom(Q/2Q,\{\pm1\})$.
Proposition \ref{pr:DiagGE} and its proof thus remain valid over an arbitrary field of characteristic not two and, therefore, we conclude that the diagonal
group $\Diag(\Gamma)$ is isomorphic to
$C_2\times C_2$. However, over arbitrary fields the diagonal group is,
up to isomorphism, the group of characters of the universal grading group, but not conversely.
However, for any $\{\alpha,\beta\}\in\frP$ consider the subalgebra
$\cH\oplus\bigl(\FF(x_\alpha+\sigma(x_\alpha))+
\FF(x_\beta+\sigma(x_\beta))\bigr) \oplus
\bigl(\FF(x_\alpha-\sigma(x_\alpha))+
\FF(x_\beta-\sigma(x_\beta))\bigr)$. It consists of three of the homogeneous components of $\Gamma$. The bracket of any two
of these components is nonzero and lies in the other component, because $\alpha\pm\beta$ is not a root.
This shows that all the generators of the universal group have order at most two and, therefore, the universal group is $2$-elementary. But the characteristic of
$\FF$ being not two,
the group of characters of a $2$-elementary group is isomorphic to itself.
Hence the universal group of $\Gamma$ is isomorphic to $C_2\times C_2$, generated by $\sigma$ and by the order two automorphism
$\tau$ that fixes the elements of the form
$\begin{pmatrix}A&0\\ 0&-A^t\end{pmatrix}$ and multiplies by $-1$ the
elements of the form $\begin{pmatrix}0&B\\ C&0\end{pmatrix}$. We thus get the same contradiction leading to Theorem \ref{th:D13}, which is then extended as follows.
\begin{theorem}\label{th:D13_2}
The set grading $\Gamma$ in \eqref{eq:D13} of the classical split simple Lie algebra of type $D_{13}$, over an arbitrary field of characteristic not two, cannot be realized as a group grading.
\end{theorem}
\smallskip
\section{An infinite family of non-group gradings on orthogonal Lie algebras}
\label{se:Steiner}
A careful look at the non-group grading in Section \ref{se:D13} shows
that a key point is the existence of the set of lines in \eqref{eq:lines}.
All the arguments in the previous section can be applied as long as $n>4$ and there is a set $\frL$ of subsets of $4$ elements of
$\{1,\ldots,n\}$, that we will call
lines, such that any
two points lie in a unique line. Any such $\frL$ is called a
$2$-$(n,4,1)$ design, or a Steiner system of type $S(2,4,n)$. (See e.g. the survey paper
\cite{ReidRosa}.)
Actually, all the arguments in the proof of
Proposition \ref{pr:set_grading} work, but a possibility must be added, as there are now three options for the intersection of two lines
$\ell_1$ and $\ell_2$: either $\ell_1\cap\ell_2=\emptyset$, or
$\ell_1\cap\ell_2$ consists of a single point, or $\ell_1=\ell_2$. The extra option:
$\ell_1\cap\ell_2=\emptyset$, is dealt with easily, as in this case
$\ell_1=\{i,j,k,l\}$ and $\ell_2=\{p,q,r,s\}$ for distinct
$i,j,k,l,p,q,r,s$, and hence the subsets of roots $\{\alpha,\beta\}$,
$\{\alpha',\beta'\}$ satisfy that $\alpha,\beta,\alpha',\beta'$ are all orthogonal, and hence the bracket of
$\FF(x_\alpha\pm\sigma(x_\alpha))+
\FF(x_\beta\pm\sigma(x_\beta))$ with
$\FF(x_{\alpha'}\pm\sigma(x_{\alpha'}))
+\FF(x_{\beta'}\pm\sigma(x_{\beta'}))$ is trivial.
Therefore, any Steiner system of type $S(2,4,n)$ allows us to define
a set grading on the simple Lie algebra of type $D_{n}$.
\smallskip
It turns out (see \cite{Hanani}) that a Steiner system of type $S(2,4,n)$ exists if and only if
$n\equiv 1\ \text{or}\ 4\pmod{12}$. One direction is easy: if $\frL$ is
a Steiner system of type $S(2,4,n)$, $n$ must be $\equiv 1\pmod 3$, as $1$ lies in $\frac{n-1}{3}$ lines. On the other hand, if the system has $b$ lines, then necessarily $4b=n\frac{n-1}{3}$, so we have
$n(n-1)\equiv 0\pmod{12}$, and we get $n\equiv 1,4\pmod{12}$.
The lowest such $n$ is $13$, where we get the Steiner system in \eqref{eq:lines}.
If $n$ is congruent to $1$ modulo $12$, and we take the corresponding set grading using \eqref{eq:six} and \eqref{eq:D13}, and compute
$E$ as in the previous section, we check that $E$ contains the
element $\frac{n-1}{3}\veps_1+\sum_{i\neq 1}\veps_i$. But
$\frac{n-1}{3}$ is a multiple of $4$ so, as in the previous section,
we see that $\sum_{i\neq 1}\veps_i$ lies in $E$, and all the subsequent arguments work.
As a consequence, we obtain our last result:
\begin{theorem}
For any $n\equiv 1\pmod{12}$, $n>1$, there is a
non-group grading on the classical split simple Lie algebra of type $D_{n}$ over an arbitrary field $\FF$ of characteristic $\neq 2$. One of the homogeneous components of this grading is a Cartan subalgebra, and all the other homogeneous components have dimension $2$.
\end{theorem}
\bigskip
| 143,463
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TITLE: Primes of the form $n\pm k$
QUESTION [2 upvotes]: Given some arbitrary natural number $n$, can we always find a $k$ such that $n+k$ and $n-k$ are both prime? Has there been any work on finding an upper bound for $k$?
REPLY [3 votes]: Since your question is equivalent to Goldbach, one can try to apply heuristics to estimate what the upper bound on $k$ ought to be (but proving any such upper bound is strictly harder than Goldbach's conjecture). The number of values of $k$ such that $n+k$ and $n-k$ are both prime should be $\gg n/\log^2 n$, and there should be even more solutions when $n$ has many small prime factors.
Therefore, I would expect on $k$ to be $O(\log^2 n)$ on average, so any upper bound should be at least this large. A Cramér-type heuristic based on random subsets of size $n/\log^2 n$ in $[1,n]$ suggests that $k$ should be no larger than $O(\log^3 n)$ with finitely many exceptions (and those could be subsumed into the implied constant).
| 104,804
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TITLE: What is the relationship between method of characteristics and separation of variables?
QUESTION [1 upvotes]: The method of characteristics for a partial differential equation such as
$$u_t + cu_x =0$$
Yields a solution $u(x,t) = f(x-ct)$.
On the other hand, separation of variables $u(x,t) = X(x)T(t)$ tells us that the solution would be
$u(x,t) = e^{\lambda(x-ct)}$
I am a bit confused as to why the first method yields a "more general" solution that the second one. I recognize that I have not specified any initial or boundary conditions and wonder if this may be a reason why I am confused. What is the relationship between both methods and how could the solution of the separation of variables method be extended so that it allows a solution with any function $f(x-ct)$ instead of just the exponential function?
REPLY [2 votes]: The separation gives you one solution (and thus not the most general function). However, as the differential equation is linear, you can add different solutions to get another solution.
In your case, separation gives the solution
$$u_\lambda(x,t) = e^{i \lambda (x-c t)};$$
here, I took the liberty to rename $\lambda \mapsto i \lambda$ with respect to the question.
Now, some very generic class of functions (e.g., $L^2$ which should be additionally sufficiently often differentiable) admit a Fourier representation
$$ f(x) = \int\!d\lambda\, e^{i \lambda x} F(\lambda)\tag{1}$$
with
$$ F(\lambda) = (2\pi)^{-1}\int\!dx \,e^{-i \lambda x} f(x).$$
So given a function $f(x)$, we can build the linear combination (of the solutions provided by the method of separation)
$$ u(x,t) = \int\!d\lambda\, F(\lambda) u_\lambda(x,t)$$
which is again a solution of the differential equation. With property (1), we obtain
$$ u(x,t) = f(x-c t),$$
that is the solution obtained by the method of characteristic.
| 63,903
|
\section{Uniqueness of smooth one-sided ancient rescaled flows} \label{sec:one.sided.flows}
In this section, we characterize smooth ancient flows lying on one side of an asymptotically conical shrinker $\Sigma$, with Gaussian density no larger than twice that of the entropy of $\Sigma$.
\begin{lemma}[One-sided decay] \label{lemma:one.sided.decay}
Let $(S(\tau))_{\tau \leq 0}$ be an ancient rescaled mean curvature flow lying on one side of $\Sigma$ and such that, for $\tau \leq 0$, we can write $S(\tau) := \operatorname{graph}_\Sigma u(\cdot, \tau)$, $u \geq 0$, with
\begin{equation} \label{eq:one.sided.decay.assumption}
\lim_{\tau \to -\infty} \Vert u(\cdot, \tau) \Vert_{3}^{(1)} = 0.
\end{equation}
Then, either $u \equiv 0$, or there exists a nonzero constant $\alpha_1 \in \RR$ such that:
\begin{equation} \label{eq:one.sided.decay.conclusion.1}
\lim_{\tau \to -\infty} e^{\lambda_1 \tau} \Pi_{=\lambda_1} u(\cdot, \tau) = \alpha_1 \varphi_1,
\end{equation}
\begin{equation} \label{eq:one.sided.decay.conclusion.2}
\limsup_{\tau \to -\infty} e^{2\lambda_1 \tau} \Vert \Pi_{=\lambda_1} u(\cdot, \tau) - \alpha_1 e^{-\lambda_1 \tau} \varphi_1 \Vert_{W} < \infty.
\end{equation}
\begin{equation} \label{eq:one.sided.decay.conclusion.3}
\limsup_{\tau \to -\infty} e^{2\lambda_1 \tau} \Vert u(\cdot, \tau) - \Pi_{=\lambda_1} u(\cdot, \tau) \Vert_{W} < \infty,
\end{equation}
\end{lemma}
\begin{proof}
Lemma \ref{lemma:dynamics.condition.check} and \eqref{eq:one.sided.decay.assumption} imply that Lemma \ref{lemma:dynamics}, Corollary \ref{coro:dynamics} are applicable with
\[ \delta(\tau) := \sup_{\sigma \leq \tau} \Vert u(\cdot, \sigma) \Vert_{2,\alpha}^{(1)}. \]
Invoke Corollary \ref{coro:dynamics}. If $u \equiv 0$, there is nothing left to prove. Let us suppose $u \not \equiv 0$.
\begin{claim}
$\mu = \lambda_1$.
\end{claim}
\begin{proof}[Proof of claim]
Note that
\[ 0 \leq u(\cdot, \tau) = \Pi_{=\mu} u(\cdot, \tau) + \Pi_{\neq \mu} u(\cdot, \tau) \implies (\Pi_{=\mu} u(\cdot, \tau))_- \leq |\Pi_{\neq \mu} u(\cdot, \tau)|. \]
By \eqref{eq:dynamics.dominant.mode},
\begin{equation} \label{eq:one.sided.neg.part.bound}
\Vert (\Pi_{=\mu} u(\cdot, \tau))_- \Vert_{W} \leq \Vert \Pi_{\neq \mu} u(\cdot, \tau) \Vert_{W} \leq C \delta(\tau) \Vert \Pi_{=\mu} u(\cdot, \tau) \Vert_{W}, \; \forall \tau \leq \tau_0.
\end{equation}
Denote $h^{(\tau)} := \Vert \Pi_{=\mu} u(\cdot, \tau) \Vert_{W}^{-1} \Pi_{=\mu} u(\cdot, \tau)$. Since $-\lambda_1 \leq \mu \leq 0$, it follows from the Rellich--Kondrachov theorem on $L^2_W(\Sigma)$ that $h^{(\tau)}$ converges after passing to a subsequence to some $\mu$-eigenfunction $h^{(-\infty)}$ with $\Vert h^{(-\infty)} \Vert_{W} = 1$. By \eqref{eq:one.sided.neg.part.bound} and the fact that $\lim_{\tau \to -\infty} \delta(\tau) = 0$, it follows that $h^{(-\infty)} \geq 0$, and the claim follows from elementary elliptic theory.
\end{proof}
In view of $\mu = \lambda_1$, \eqref{eq:dynamics.u.decay.sharp} implies
\begin{equation} \label{eq:one.sided.decay.delta.bound}
\limsup_{\tau \to -\infty} e^{\lambda_1 \tau} \delta(\tau) < \infty.
\end{equation}
Thus,
\[\Vert \tfrac{d}{d\tau} \Pi_{=\lambda_1} u(\cdot, \tau) + \lambda_1 \Pi_{=\lambda_1} u(\cdot, \tau) \Vert_{W} \leq C \delta(\tau) \Vert \Pi_{=\lambda_1} u(\cdot, \tau) \Vert_{W} \]
can be integrated to yield the existence of a limit $\lim_{\tau \to -\infty} e^{\lambda_1 \tau} \Pi_{=\lambda_1} u(\cdot, \tau)$, i.e., \eqref{eq:one.sided.decay.conclusion.1}, and by \eqref{eq:one.sided.decay.delta.bound} also gives \eqref{eq:one.sided.decay.conclusion.2}. Finally, we note that, \eqref{eq:dynamics.dominant.mode} and \eqref{eq:one.sided.decay.delta.bound} imply
\begin{equation} \label{eq:one.sided.decay.diff.lambda1}
\Vert u(\cdot, \tau) - \Pi_{=\lambda_1} u(\cdot, \tau) \Vert_{W} = \Vert \Pi_{>\lambda_1} u(\cdot, \tau) \Vert_{W} \leq C \delta(\tau) \Vert \Pi_{=\lambda_1} u(\cdot, \tau) \Vert_{W} \leq C e^{-2\lambda_1 \tau},
\end{equation}
which implies \eqref{eq:one.sided.decay.conclusion.3}.
\end{proof}
\begin{corollary}[One-sided uniqueness for graphical flows] \label{coro:one.sided.decay.uniqueness}
Up to time translation, there is at most one non-steady ancient rescaled mean curvature flow $(S(\tau))_{\tau \leq 0}$ on one side of $\Sigma$ satisfying \eqref{eq:one.sided.decay.assumption}.
\end{corollary}
\begin{proof}
We assume that $u$, $\bar u \not \equiv 0$ are two such solutions. It follows from Lemma \ref{lemma:one.sided.decay} that we can translate either $u$ or $\bar u$ in time so that
\begin{equation} \label{eq:one.sided.decay.uniqueness.neg.infty}
\lim_{\tau \to -\infty} e^{\lambda_1 \tau} \Vert (\bar u - u)(\cdot, \tau) \Vert_W = 0.
\end{equation}
It will also be convenient to write $\delta(\tau)$, $\bar \delta(\tau)$ for the quantities corresponding to \eqref{eq:dynamics.condition.delta} for $u$, $\bar u$. By Lemmas \ref{lemma:dynamics.condition.check} and \ref{lemma:one.sided.decay},
\begin{equation} \label{eq:one.sided.decay.uniqueness.delta}
\delta(\tau) + \bar \delta(\tau) \leq C_1 e^{-\lambda_1 \tau}, \; \tau \in \RR_-
\end{equation}
for a fixed $C_1$. Finally, we introduce the notation
\[ w := \bar u - u, \; E^w := E(\bar u) - E(u), \]
so that
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.pde}
(\tfrac{\partial}{\partial \tau} - L) w = E^w.
\end{equation}
Using \eqref{eq:linearized.equation.error.decomposition} and the fundamental theorem of calculus,
\begin{align} \label{eq:one.sided.decay.uniqueness.w.error}
E^w
& = \bar u E_1(\cdot, \bar u, \nabla \bar u, \nabla^2 \bar u) + \nabla_\Sigma \bar u \cdot \mathbf{E}_2(\cdot, \bar u, \nabla_\Sigma \bar u, \nabla_\Sigma^2 \bar u) \nonumber \\
& \qquad - u E_1(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) - \nabla_\Sigma u \cdot \mathbf{E}_2(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \nonumber \\
& = w E_1(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \nonumber \\
& \qquad + \nabla_\Sigma w \cdot \mathbf{E}_2(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \nonumber \\
& \qquad + \bar u \big[ E_1(\cdot, \bar u, \nabla \bar u, \nabla^2 \bar u) - E_1(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \big] \nonumber \\
& \qquad + \nabla_\Sigma \bar u \cdot \big[ \mathbf{E}_2(\cdot, \bar u, \nabla_\Sigma \bar u, \nabla_\Sigma^2 \bar u) - \mathbf{E}_2(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \big] \nonumber \\
& = w E_1(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \nonumber \\
& \qquad + \nabla_\Sigma w \cdot \mathbf{E}_2(\cdot, u, \nabla_\Sigma u, \nabla_\Sigma^2 u) \nonumber \\
& \qquad + \Big[ \bar u \int_0^1 D_z E_1(\cdots) \, dt \Big] w \nonumber \\
& \qquad + \Big[ \bar u \int_0^1 D_{\mathbf{q}} E_1(\cdots) \, dt \Big] \cdot \nabla_\Sigma w \nonumber \\
& \qquad + \Big[ \bar u \int_0^1 D_{\mathbf{A}} E_1(\cdots) \, dt \Big] \cdot \nabla^2_\Sigma w \nonumber \\
& \qquad + \Big[ \nabla_\Sigma \bar u \cdot \int_0^1 D_z \mathbf{E}_2(\cdots) \, dt \Big] w \nonumber \\
& \qquad + \Big[ \nabla_\Sigma \bar u \cdot \int_0^1 D_{\mathbf{q}} \mathbf{E}_2(\cdots) \, dt \Big] \cdot \nabla_\Sigma w \nonumber \\
& \qquad + \Big[ \nabla_\Sigma \bar u \cdot \int_0^1 D_{\mathbf{A}} \mathbf{E}_2(\cdots) \, dt \Big] \cdot \nabla^2_\Sigma w,
\end{align}
where, in all six instances, $\cdots$ stands for $(\cdot, u + tw, \nabla_\Sigma u + t \nabla_\Sigma w, \nabla_\Sigma^2 u + t \nabla^2_\Sigma w)$. We take the $L^2_W$ dot product of \eqref{eq:one.sided.decay.uniqueness.w.error} with $w$ and integrate by parts so that, in every term, we have at least two instances of $w$ and $\nabla_\Sigma w$. In particular, we will pick up derivatives of $D_{\mathbf{A}} E_1$ and $D_{\mathbf{A}} \mathbf{E}_2$. Using Lemma \ref{lemma:linearized.equation.error}, \eqref{eq:one.sided.decay.assumption}, and \eqref{eq:one.sided.decay.uniqueness.delta}, we find
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.dot.error}
|\langle w(\cdot, \tau), E^w(\cdot, \tau) \rangle_W| \leq C_2 e^{-\lambda_1 \tau} \Vert w(\cdot, \tau) \Vert^2_{W,1}, \; \tau \in \RR_-,
\end{equation}
for a fixed $C_2$. Here, $\Vert \cdot \Vert_{W,1}$ is the norm induced from \eqref{eq:linearized.equation.weighted.sobolev.dot} with $k=1$.
We use \eqref{eq:one.sided.decay.uniqueness.w.dot.error} to derive two estimates on the evolution of $\Vert w \Vert_W^2$. First, together with \eqref{eq:one.sided.decay.uniqueness.w.pde} and \eqref{eq:linearized.equation.eigenvalues}, it implies
\begin{align*}
\tfrac12 \tfrac{d}{d\tau} \Vert w(\cdot, \tau) \Vert_W^2
& = \langle w(\cdot, \tau), Lw(\cdot, \tau) + E^w(\cdot, \tau) \rangle_W \\
& \leq - \lambda_1 \Vert w(\cdot, \tau) \Vert^2_W + C_2 e^{-\lambda_1 \tau} \Vert w(\cdot, \tau) \Vert^2_{W,1}, \; \tau \in \RR_-,
\end{align*}
which in turn implies
\begin{equation} \label{eq:one.sided.decay.uninqueness.ddt.e2lambda.w.sq}
\tfrac{d}{d\tau} (e^{2\lambda_1 \tau} \Vert w(\cdot, \tau) \Vert^2_W) \leq C_2 e^{\lambda_1 \tau} \Vert w(\cdot, \tau) \Vert^2_{W,1}, \; \tau \in \RR_-.
\end{equation}
Second, recalling the definition of $L$ in \eqref{eq:linearized.equation.linear.operator}, integrating by parts, and using \eqref{eq:one.sided.decay.uniqueness.w.dot.error}, it follows that there exists a sufficiently negative $\tau_0$ such that:
\begin{align}
\tfrac12 \tfrac{d}{d\tau} \Vert w \Vert_W^2
& = - \Vert \nabla_\Sigma w \Vert_W^2 + \langle w, (\tfrac12 + |A_\Sigma|^2)w + E^w \rangle_W \nonumber \\
& \leq - \tfrac12 \Vert \nabla_\Sigma w \Vert_W^2 + C_3 \Vert w \Vert_W^2, \; \tau \leq \tau_0, \label{eq:one.sided.decay.uninqueness.ddt.w.sq}
\end{align}
with a fixed $C_3$.
We next compute the evolution of $\Vert \nabla_\Sigma w \Vert_W^2$. To that end, we need a couple of preliminary computations. By the Gauss equation,
\begin{equation} \label{eq:one.sided.decay.uniqueness.gauss}
\Ric_\Sigma(\nabla_\Sigma w, \nabla_\Sigma w) = H_\Sigma A_\Sigma(\nabla_\Sigma w, \nabla_\Sigma w) - A_\Sigma^2(\nabla_\Sigma w, \nabla_\Sigma w).
\end{equation}
From the definition of the second fundamental form and the shrinker equation, $H_\Sigma = \tfrac12 \mathbf{x} \cdot \nu$,
\begin{equation} \label{eq:one.sided.decay.uniqueness.xgradw}
\nabla_\Sigma (\mathbf{x} \cdot \nabla_\Sigma w) \cdot \nabla_\Sigma w = |\nabla_\Sigma w|^2 - 2H_\Sigma A_\Sigma(\nabla_\Sigma w, \nabla_\Sigma w) + \mathbf{x} \cdot \nabla^2_\Sigma w(\nabla_\Sigma w, \cdot).
\end{equation}
In what follows, we recall the Gaussian density $\rho$, defined in \eqref{eq:linearized.equation.weighted.density}, which satisfies $\nabla \rho = - \tfrac12 \rho \mathbf{x}$. An integration by parts, followed by the Bochner formula $\Delta_\Sigma \nabla_\Sigma w = \nabla_\Sigma \Delta_\Sigma w + \Ric_\Sigma(\nabla_\Sigma w, \cdot)$, \eqref{eq:one.sided.decay.uniqueness.gauss}, \eqref{eq:one.sided.decay.uniqueness.xgradw}, implies:
\begin{align} \label{eq:one.sided.decay.uniqueness.laplacian.trick}
& \int_\Sigma (\Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w)^2 \rho \, d\cH^n \nonumber \\
& \qquad = \int_\Sigma (\Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w) \operatorname{div}_\Sigma(\rho \nabla_\Sigma w) \, d\cH^n \nonumber \\
& \qquad = - \int_\Sigma \nabla_\Sigma (\Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w) \cdot \nabla_\Sigma w \, \rho \, d\cH^n \nonumber \\
& \qquad = - \int_\Sigma (\Delta_\Sigma \nabla_\Sigma w - \Ric_\Sigma(\nabla_\Sigma w, \cdot) - \tfrac12 \nabla_\Sigma (\mathbf{x} \cdot \nabla_\Sigma w)) \cdot \nabla_\Sigma w \, \rho \, d\cH^n \nonumber \\
& \qquad = - \int_\Sigma (\Delta_\Sigma \nabla_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla^2_\Sigma w + A_\Sigma^2(\nabla_\Sigma w, \cdot) - \tfrac12 \nabla_\Sigma w) \cdot \nabla_\Sigma w \, \rho \, d\cH^n \nonumber \\
& \qquad = \int_\Sigma \big[ - \operatorname{div}_\Sigma (\rho \nabla_\Sigma^2 w) + \big( - A_\Sigma^2(\nabla_\Sigma w, \cdot) + \tfrac12 \nabla_\Sigma w \big) \rho \big] \cdot \nabla_\Sigma w \, d\cH^n \nonumber \\
& \qquad = \int_\Sigma (|\nabla_\Sigma^2 w|^2 - A_\Sigma^2(\nabla_\Sigma w, \nabla_\Sigma w) + \tfrac12 |\nabla_\Sigma w|^2) \, \rho \, d\cH^n.
\end{align}
We can now estimate the evolution of $\Vert \nabla_\Sigma w \Vert_\Sigma^2$. Using \eqref{eq:one.sided.decay.uniqueness.w.pde} and the definition of $L$ in \eqref{eq:linearized.equation.linear.operator}:
\begin{align*}
\tfrac12 \tfrac{d}{d\tau} \Vert \nabla_\Sigma w \Vert_W^2
& = \langle \nabla_\Sigma w, \nabla_\Sigma \tfrac{\partial}{\partial \tau} w \rangle_W \\
& = -\langle \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w, \tfrac{\partial}{\partial \tau} w \rangle_W \\
& = -\langle \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w, \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w + |A_\Sigma|^2 w + \tfrac12 w + E^w \rangle_W \\
& = - \Vert \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w \Vert_W^2 + \langle \nabla_\Sigma w, \nabla_\Sigma(|A_\Sigma|^2 w + \tfrac12 w) \rangle_W \\
& \qquad - \langle \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w, E^w \rangle_W \\
& = - \Vert \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w \Vert_W^2 + \Vert (\tfrac12 + |A_\Sigma|^2)^{\frac12} \nabla_\Sigma w \Vert_W^2 + \langle \nabla_\Sigma w, w \nabla_\Sigma |A_\Sigma|^2 \rangle_W \\
& \qquad - \langle \Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w, E^w \rangle_W.
\end{align*}
We claim that this implies:
\begin{equation} \label{eq:one.sided.decay.uninqueness.ddt.gradw.sq}
\tfrac12 \tfrac{d}{d\tau} \Vert \nabla_\Sigma w \Vert_W^2 \leq C_4 \Vert w \Vert_{W,1}^2, \; \tau \leq \tau_0,
\end{equation}
with fixed $C_4$, after possibly choosing a more negative $\tau_0$. Indeed, in the immediately preceding expression, we use Cauchy--Schwarz on the last term to absorb the $\Delta_\Sigma w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma w$ into the first term. The remainder of the first term is used, via \eqref{eq:one.sided.decay.uniqueness.laplacian.trick}, to dominate all $\nabla_\Sigma^2 w$ terms in $E^w$, which we computed in \eqref{eq:one.sided.decay.uniqueness.w.error}; note that these terms have small coefficients for sufficiently negative $\tau$ by virtue of \eqref{eq:one.sided.decay.uniqueness.delta}. This yields \eqref{eq:one.sided.decay.uninqueness.ddt.gradw.sq}.
Together, \eqref{eq:one.sided.decay.uninqueness.ddt.w.sq}, \eqref{eq:one.sided.decay.uninqueness.ddt.gradw.sq} imply that there exist $C_5 \geq 1$, $C_6$ such that
\begin{equation} \label{eq:one.sided.decay.uniqueness.ddt.w.gradw.sq}
\tfrac{d}{d\tau} (\Vert \nabla_\Sigma w \Vert_W^2 + C_5 \Vert w \Vert_W^2) \leq C_6 \Vert w \Vert_W^2, \; \tau \leq \tau_0.
\end{equation}
Integrating \eqref{eq:one.sided.decay.uniqueness.ddt.w.gradw.sq} from $-\infty$ to $\tau$ and using the decay of $w$, we deduce:
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.gradw.sq}
\Vert w(\cdot, \tau) \Vert_{W,1}^2 \leq \Vert \nabla_\Sigma w(\cdot, \tau) \Vert_W^2 + C_5 \Vert w(\cdot, \tau) \Vert_W^2 \leq C_6 \int_{-\infty}^\tau \Vert w(\cdot, s) \Vert_W^2 \, ds, \; \tau \leq \tau_0.
\end{equation}
By \eqref{eq:one.sided.decay.uniqueness.neg.infty}, we may take $\tau_0$ more negative yet so that
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.1}
\Vert w(\cdot, \tau) \Vert_W^2 \leq e^{-2\lambda_1 \tau}, \; \tau \leq \tau_0.
\end{equation}
Thus, by evaluating the integral in \eqref{eq:one.sided.decay.uniqueness.w.gradw.sq} using the crude estimate in \eqref{eq:one.sided.decay.uniqueness.w.small.1}, we find
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.2}
\Vert w(\cdot, \tau) \Vert_{W,1}^2 \leq \frac{C_6}{2|\lambda_1|} e^{-2\lambda_1 \tau}, \; \tau \leq \tau_0,
\end{equation}
with the same $\tau_0$. Integrating \eqref{eq:one.sided.decay.uninqueness.ddt.e2lambda.w.sq} from $-\infty$ to $\tau$, and using \eqref{eq:one.sided.decay.uniqueness.neg.infty} at $-\infty$ and \eqref{eq:one.sided.decay.uniqueness.w.small.2}, we get the following improvement over \eqref{eq:one.sided.decay.uniqueness.w.small.1}:
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.3}
\Vert w(\cdot, \tau) \Vert_W^2 \leq \frac{C_2 C_6}{2 |\lambda_1|^2} e^{-3\lambda_1 \tau}, \; \tau \leq \tau_0,
\end{equation}
with the same $\tau_0$. Now we iterate. Using \eqref{eq:one.sided.decay.uniqueness.w.gradw.sq} again, with \eqref{eq:one.sided.decay.uniqueness.w.small.3} in place of \eqref{eq:one.sided.decay.uniqueness.w.small.1}:
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.4}
\Vert w(\cdot, \tau) \Vert_{W,1}^2 \leq \frac{C_2 C_6^2}{3! |\lambda_1|^3} e^{-3\lambda_1 \tau}, \; \tau \leq \tau_0,
\end{equation}
with the same $\tau_0$. Integrating \eqref{eq:one.sided.decay.uninqueness.ddt.e2lambda.w.sq} from $-\infty$ to $\tau$, and using \eqref{eq:one.sided.decay.uniqueness.w.small.4} rather than \eqref{eq:one.sided.decay.uniqueness.w.small.2}, we get the following improvement over \eqref{eq:one.sided.decay.uniqueness.w.small.3}:
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.5}
\Vert w(\cdot, \tau) \Vert_W^2 \leq \frac{C_2^2 C_6^2}{2 \cdot 3! |\lambda_1|^4} e^{-4\lambda_1 \tau}, \; \tau \leq \tau_0,
\end{equation}
with the same $\tau_0$. Repeating this $k \in \NN$ times altogether (we showed steps $k = 1$, $2$), we find
\begin{equation} \label{eq:one.sided.decay.uniqueness.w.small.6}
\Vert w(\cdot, \tau) \Vert_W^2 \leq \frac{C_2^k C_6^k}{k! (k+1)! |\lambda_1|^{2k}} e^{-(2+k)\lambda_1 \tau}, \; \tau \leq \tau_0,
\end{equation}
with the same $\tau_0$. Fixing $\tau \leq \tau_0$ and sending $k \to \infty$, \eqref{eq:one.sided.decay.uniqueness.w.small.6} gives $w(\cdot, \tau) \equiv 0$.
\end{proof}
| 143,110
|
TITLE: What are the bounds (upper and lower) for $|A+A|$?
QUESTION [3 upvotes]: Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that:
If $A$ is the empty set, then $A+A$ is also empty.
If $A$ is a singleton, then $A+A$ is also a singleton.
If $|A|=2,$ then $|A+A|=3.$
If $|A|=3,$ then $|A+A|$ can be at most $6.$
The set $A=\{1, 2, 3\}$ shows we can have that $|A+A|=5,$ But we can not obtain $4.$
Obvious bounds for $A+A$ are $|A|\le |A+A|\le|A|^2.$
But for large $|A|$ values, it looks like that we can find more sharpe bounds for $|A+A|$ than above ones.
MY QUESTION IS: How can I estimate the size of the set $A+A=\{a+b :a,b\in A\}$ using the size of $A$ ?
REPLY [3 votes]: For general finite subsets of abelian groups we have
$$
|A| \leq |A + A| \leq \frac{|A|(|A| + 1)}{2}
$$
In the case of $\mathbb R$ and $\mathbb C$, we are dealing with torsion-free abelian groups (i.e. they have no finite subgroups) we have a lower bound of $2|A| - 1$, which is attained if and only if $A$ is an arithmetic progression, i.e. $A = \{a,\, a + d,\, a + 2d, \dots,\, a + (N-1)d\}$ for $N = |A|$ and numbers $a, d$.
The upper bound is sharp and occurs if and only if all pairwise sums in $A$ are distinct ($A$ is then called a Sidon set). This happens often in $\mathbb R$ (and $\mathbb C$): for example, if you choose a set of $N$ numbers uniformly random from $[0, 1]$ you will get a Sidon set with probability 1.
| 104,340
|
\begin{document}
\title{Critical branching as a pure death process coming down from infinity
}
\author{ Serik Sagitov \\
Chalmers University of Technology and University of Gothenburg}
\date{}
\maketitle
\begin{abstract}
We consider the critical Galton-Watson process with overlapping generations stemming from a single founder.
Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.
This result brings a new perspective on Vatutin's dichotomy claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.
\end{abstract}
\section{Introduction
}\label{sec:int}
Consider a self-replicating system evolving in the discrete time setting according to the next rules:
\begin{description}
\item[\ \ \,-] the system is founded by a single individual, the founder born at time 0,
\item[\ \ \,-] the founder dies at a random age $L$ and gives a random number $N$ of births at random ages $\tau_j$ satisfying
\begin{equation*}
1\le\tau_1\le \ldots\le \tau_N\le L,
\end{equation*}
\item[\ \ \,-] each new individual lives independently from others according to the same life law as the founder.
\end{description}
An individual which was born at time $t_1$ and dies at time $t_2$ is considered to be alive during the time interval $[t_1,t_2-1]$. Letting $Z(t)$ stand for the number of individuals alive at time $t$, we study the random dynamics of the sequence
$$Z(0)=1, Z(1), Z(2),\ldots,$$
which is a natural extension of the well-known Galton-Watson process, or \textit{GW-process} for short,
see \cite{WG}.
The process $Z(\cdot)$ is the discrete time version of what is usually called the Crump-Mode-Jagers process or the general branching process, see \cite{J}.
To emphasise the discrete time setting, we call it a GW-process with overlapping generations, or \textit{GWO-process} for short.
Denote $a:=\rE(\tau_1+\ldots+\tau_N)$ and observe that $a\ge1$. In this paper, the GWO-process is assumed to satisfy conditions
\begin{equation}
\label{ir}
\rE(N)=1,\quad 0<\rV(N)<\infty,
\end{equation}
$a<\infty$, and
\begin{equation}
\label{d}
t^2\rP(L>t)\to d,\quad 0\le d< \infty,\quad t\to\infty.
\end{equation}
Condition $\rE(N)=1$ says that the reproduction regime is critical, allowing to interpret the parameter $a$ as the \textit{mean generation length}, see \cite{J} and \cite{21}.
In general, the life length $L$ may depend on the offspring number $N$ and the birth times $\tau_j$, so that our standing condition $a<\infty$ tacitly imposes a constraint on the distribution of $L$. Clearly, condition \eqref{d}
implies $\rE(L)<\infty$.
Let
\begin{equation}\label{lov}
b:=\frac{1}{2}
\rV(N),\qquad c:=4bda^{-2},\qquad h:=\frac{a+\sqrt{a^2+4bd}}{2b},
\end{equation}
so that
\begin{equation}\label{stop}
bh^2=ah+d.
\end{equation}
The asymptotic formulas obtained in this paper, are fully described by the triplet $(a,b,d)$, regardless of complicated mutual dependencies between the random variables $\tau_j, N,L$. Moreover, the limiting process of Theorem \ref{thL}, stated in Section \ref{main}, is controlled by a single compound parameter $c$, while Theorem \ref{thQ}, stated in Section \ref{out}, gives a concise asymptotic formula for the survival probability
$$t\rP(Z(t)>0)\to h,\quad t\to\infty,$$
invoking another compound parameter $h$. The proof of Theorem \ref{thL}, given in Section \ref{Lp1}, is based on the proof of Theorem \ref{thQ}, given in Section \ref{out}.
We conclude this section by mentioning the illuminating special case of the \textit{reproduction by splitting} when all offspring of an individual are born simultaneously, so that
$$\rP(\tau_1=\tau_2=\ldots= \tau_N= \tau)=1$$
for some random variable $\tau$.
In the case of critical reproduction by splitting, the mean generation length, $a=\rE(\tau N)$, can be represented as
$$a=\rC(\tau,N)+\rE(\tau).$$
Thus, if $\tau$ and $N$ are positively correlated, the average generation length $a$ may exceed the average life length $\rE(L)$.
There three important examples assuming $\tau\equiv L$: the GW-process, with $L\equiv 1$; the \textit{Bellman-Harris} process \cite{BH}, with $N$ being independent of $L$;
and the \textit{Sevastyanov} process \cite{Sev}, allowing dependence between $N$ and $L$. Notably, condition \eqref{d} was introduced in the pioneering paper \cite{V79} dealing with the Bellman-Harris processes.
A related result to our Theorem \ref{thL} was obtained in \cite{Ya} for the critical Bellman-Harris processes with $d=\infty$.
Turning a specific example of the Sevastyanov process, take
\[\rP(L= t)= p_1 t^{-3}(\ln\ln t)^{-1}, \quad \rP(N=0|L= t)=1-p_2,\quad \rP(N=n_t|L= t)=p_2, \ t\ge2,\]
where $n_t:=\lfloor t(\ln t)^{-1}\rfloor$ and $(p_1,p_2)$ are such that
\[\sum_{t=2}^\infty \rP(L= t)=p_1 \sum_{t=2}^\infty t^{-3}(\ln\ln t)^{-1}=1,\quad \rE(N)=p_1p_2\sum_{t=2}^\infty n_t t^{-3}(\ln\ln t)^{-1}=1.\]
In this case, for some positive constant $c_1$,
\[\rE(N^2)= p_1p_2\sum_{t=1}^\infty n_t^2 t^{-3}(\ln\ln t)^{-1}< c_1\int_2^\infty \frac{d (\ln t)}{(\ln t)^2\ln\ln t}<\infty,\]
implying that condition \eqref{ir} is satisfied. Clearly, condition \eqref{d} holds with $d=0$.
At the same time,
\[a=\rE(NL)= p_1p_2\sum_{t=1}^\infty n_t t^{-2}(\ln\ln t)^{-1}> c_2\int_2^\infty \frac{d (\ln t)}{(\ln t)(\ln\ln t)}=\infty,\]
where $c_2$ is a positive constant. This example demonstrates that for the GWO-process,
conditions \eqref{ir} and \eqref{d} do not automatically imply the condition $a<\infty$.
\section{The main result}\label{main}
\begin{theorem}\label{thL}
For a GWO-process satisfying \eqref{ir}, \eqref{d} and $a<\infty$, there holds a weak convergence of the finite dimensional distributions
\begin{align*}
(Z(ty),0<y<\infty|Z(t)>0)\stackrel{\rm fdd\,}{\longrightarrow} (\eta(y),0<y<\infty),\quad t\to\infty.
\end{align*}
The limiting process is a continuous time pure death process $(\eta(y),0\le y<\infty)$,
whose random evolution is regulated by a single compound parameter $c$ specified in \eqref{lov}.
\end{theorem}
The finite dimensional distributions of $\eta(\cdot)$ are given below in terms of the $k$-dimensional probability generating functions
$\rE(z_1^{\eta(y_1)}\cdots z_k^{\eta(y_k)})$ assuming
\begin{equation}\label{mansur}
0=y_0< y_1< \ldots< y_{j}<1\le y_{j+1}< \ldots< y_k<y_{k+1}=\infty,\quad 0\le j\le k,\quad 0\le z_1,\ldots,z_k<1.
\end{equation}
Here the parameter $j$ controls the location of the pivotal value 1, corresponding to the time of observation $t$ of the underlying GWO-process. If $j\ge1$, then we have
\begin{align*}
\rE(z_1^{\eta(y_1)}\cdots z_k^{\eta(y_k)})=\frac{\sqrt{1+\sum_{i=1}^{j}z_1\cdots z_{i-1}(1-z_{i})\Gamma_i+cz_1\cdots z_{j}y_1^{2} }-\sqrt{1+\sum\nolimits_{i=1}^{k}z_1\cdots z_{i-1}(1-z_{i})\Gamma_i}}{(1+\sqrt{1+c})y_1},
\end{align*}
where $\Gamma_i:=c({y_1}/{y_i} )^2$,
and if $j=0$,
\begin{align*}
\rE(z_1^{\eta(y_1)}\cdots z_k^{\eta(y_k)})=1-\frac{1+\sqrt{1+c\sum\nolimits_{i=1}^{k}z_1\cdots z_{i-1}(1-z_{i})\Gamma_i}}{(1+\sqrt{1+c})y_1}.
\end{align*}
In particular, for $k=1$,
\begin{align*}
\rE(z^{\eta(y)})&= \frac{\sqrt{1+c(1-z)+czy^{2}}-\sqrt{1+c(1-z)}}{(1+\sqrt{1+c})y},\quad 0< y<1,\\
\rE(z^{\eta(y)})&= 1-\frac{1+\sqrt{1+c(1-z)}}{(1+\sqrt{1+c})y},\quad y\ge1.
\end{align*}
Putting here first $z=1$ and then $z=0$, brings
\begin{align*}
\rP(\eta(y)<\infty)&=\frac{\sqrt{1+cy^2}-1}{(1+\sqrt{1+c})y}\cdot1_{\{0< y<1\}}+\Big(1-\frac{2}{(1+\sqrt{1+c})y}\Big)\cdot1_{\{y\ge 1\}},\\
\rP(\eta(y)=0)&=\frac{y-1}{y}\cdot1_{\{y\ge 1\}},
\end{align*}
implying that $\rP(\eta(y)=\infty)>0$ for all $y>0$, and in fact, we may set
$\rP(\eta(0)=\infty)=1.$
To demonstrate that the process $\eta(\cdot)$ is indeed a pure death process, consider the function
\[\rE(z_1^{\eta(y_1)-\eta(y_2)}\cdots z_{k-1}^{\eta(y_{k-1})-\eta(y_{k})}z_k^{\eta(y_k)})\]
determined by
\begin{align*}
\rE(z_1^{\eta(y_1)-\eta(y_2)}\cdots z_{k-1}^{\eta(y_{k-1})-\eta(y_{k})}z_k^{\eta(y_k)})
&=\rE(z_1^{\eta(y_1)}(z_2/z_1)^{\eta(y_2)}\cdots (z_k/z_{k-1})^{\eta(y_k)}).
\end{align*}
This function is given by two expressions
\begin{align*}
\frac{\sqrt{1+\sum\nolimits_{i=1}^{j-1}(1-z_{i})\gamma_i+(1-z_{j})\Gamma_j+cz_j y_1^2}-\sqrt{1+\sum\nolimits_{i=1}^{k} (1-z_{i})\gamma_i}}{(1+\sqrt{1+c})y_1}, \quad &\text{for }j\ge1,\\
\frac{(1+\sqrt{1+c})y_1-1-\sqrt{1+\sum\nolimits_{i=1}^{k} (1-z_{i})\gamma_i}}{(1+\sqrt{1+c})y_1}, \quad &\text{for }j=0,
\end{align*}
where $\gamma_i:=\Gamma_i-\Gamma_{i+1}$ and $\Gamma_{k+1}=0$.
Setting $k=2$, $z_1=z$, and $z_2=1$, we deduce that the function
\[\rE(z^{\eta(y_1)-\eta(y_2)};\eta(y_1)<\infty),\quad 0<y_1<y_2,\quad 0\le z\le1,\]
is given by one of the following three expressions depending on whether $j=2$, $j=1$, and $j=0$,
\begin{align*}
\frac{\sqrt{1+c y_1^2+c(1-z)(1-(y_1/y_2)^2)}
-\sqrt{1+c (1-z)(1-(y_1/y_2)^2)}}{(1+\sqrt{1+c})y_1},\quad &y_2<1, \\
\frac{\sqrt{1+c y_1^2+c(1-z) (1-y_1^2)}
-\sqrt{1+c(1-z)(1-(y_1/y_2)^2)}}{(1+\sqrt{1+c})y_1},\quad &y_1<1\le y_2, \\
1- \frac{1+\sqrt{1+c(1-z)(1-(y_1/y_2)^2)}}{(1+\sqrt{1+c})y_1},\quad &1\le y_1.
\end{align*}
As a function of $z\in[0,1)$, this is a probability generating function, since all its derivatives are non-negative functions. This implies $\rP(\eta(y_1)\ge \eta(y_2))=1$, so that unless the process $\eta(\cdot)$ is sitting at the infinity state, it evolves by negative integer-valued jumps until it gets absorbed at zero.
Of crucial interest is the conditional probability generating function
\[\rE(z^{\eta(y_1)-\eta(y_2)}| \eta(y_1)<\infty),\quad 0<y_1<y_2,\quad 0\le z\le1,\]
given by one of the following three expressions
\begin{align*}
\frac{\sqrt{1+c y_1^2+c(1-z)(1-(y_1/y_2)^2)}
-\sqrt{1+c (1-z)(1-(y_1/y_2)^2)}}{\sqrt{1+c y_1^2}-1},\quad &y_2<1, \\
\frac{\sqrt{1+c y_1^2+c(1-z) (1-y_1^2)}
-\sqrt{1+c(1-z)(1-(y_1/y_2)^2)}}{\sqrt{1+c y_1^2}-1},\quad &y_1<1\le y_2, \\
1- \frac{\sqrt{1+c(1-z)(1-(y_1/y_2)^2)}-1}{(1+\sqrt{1+c})y_1-2},\quad &1\le y_1.
\end{align*}
In particular, setting here $z=0$, we obtain
\[\rP(\eta(y_1)-\eta(y_2)=0| \eta(y_1)<\infty)= \left\{
\begin{array}{llr}
\frac{\sqrt{1+c(1+y_1^2-(y_1/y_2)^2)}-\sqrt{1+c(1-(y_1/y_2)^2)}}{\sqrt{1+c y_1^2}-1} & \text{for} & 0<y_1< y_2<1, \\
\frac{\sqrt{1+c}-\sqrt{1+c(1-(y_1/y_2)^2)}}{\sqrt{1+c y_1^2}-1} & \text{for} & 0<y_1<1\le y_2, \\
1- \frac{\sqrt{1+c(1-(y_1/y_2)^2)}-1}{(1+\sqrt{1+c})y_1-2} & \text{for} & 1\le y_1<y_2.
\end{array}
\right.
\]
Notice that given $0<y_1\le1$,
\[\rP(\eta(y_1)-\eta(y_2)=0| \eta(y_1)<\infty)\to 0,\quad y_2\to\infty,\]
which is expected because of $\eta(y_1)\ge\eta(1)\ge1$ and $\eta(y_2)\to0$ as $y_2\to\infty$.
\begin{figure}[h]
\begin{center}
\includegraphics[width=6cm]{density1.png}\qquad \includegraphics[width=6cm]{density.png}
\end{center}
\caption{The dash-dot line is the probability density function of $T$, solid line is the probability density function of $T_0$. The left panel illustrates the case $c=5$, and the right panel illustrates the case $c=15$.}
\label{trump}
\end{figure}
According to the above mentioned formulas, the random times
\[T=\sup\{u: \eta(u)=\infty\},\quad T_0=\inf\{u:\eta(u)=0\}\]
have the marginal distributions
\begin{align*}
\rP(T\le y)&=\frac{\sqrt{1+cy^2}-1}{(1+\sqrt{1+c})y}\cdot1_{\{0\le y<1\}}+\Big(1-\frac{2}{(1+\sqrt{1+c})y}\Big)\cdot1_{\{y\ge 1\}},\\
\rP(T_0\le y)&=\frac{y-1}{y}\cdot1_{\{y\ge 1\}}.
\end{align*}
The distribution of $T_0$ is free from the parameter $c$ and has the Pareto probability density function
\[f_0(y)=y^{-2}1_{\{y>1\}}.\]
If $d=0$, then $c=0$ and
$\rP(T=T_0)=1$.
If $d>0$, then $T\le T_0$, and the distribution of $T$ has the following probability density function
\[
f(y)=\left\{
\begin{array}{llr}
\frac{1}{(1+\sqrt{1+c})y^2} (1-\frac{1}{\sqrt{1+cy^2}})& \text{for} & 0\le y<1, \\
\frac{2}{(1+\sqrt{1+c})y^2} & \text{for} & y\ge1,
\end{array}
\right.
\]
having a positive jump at $y=1$ of size $f(1)-f(1-)=(1+c)^{-1/2}$. Observe that, see Figure \ref{trump},
\[\frac{f(1)-f(1-)}{f(1)}\to\frac{1}{2},\quad c\to\infty.\]
Intuitively, the limiting pure death process counts the long-living individuals in the GWO-process, whose life length is of order $t$. These long-living individual may have descendants, however none of them would live long enough to be detected by the finite dimensional distributions at the relevant time scale, see Lemma 2 below.
Theorem \ref{thL} suggests a new perspective on Vatutin's dichotomy, see \cite{V79}, claiming that the long term survival of a critical age-dependent branching process is due to either a large number of short-living individuals or a small number of long-living individuals.
In terms of the random times $T\le T_0$, Vatutin's dichotomy discriminates between two possibilities: if $T>1$, then $\eta(1)=\infty$, meaning that the GWO-process has survived due to a large number of individuals, while if $T\le 1<T_0$, then $1\le \eta(1)<\infty$ meaning that the GWO-process has survived due to a small number of individuals.
\section{Asymptotic formula for the survival probability}\label{out}
This section deals with the survival probability of the GWO-process $Q(t):=\rP(Z(t)>0)$.
Denote
$P(t):=\rP(Z(t)=0),$
so that
$$P(0)=0,\quad P(t+1)\ge P(t),\quad t\ge0.$$
The GWO-process can be represented as the sum
\begin{equation}\label{CD}
Z(t)=1_{\{L>t\}}+\sum\nolimits_{j=1}^{N} Z_j(t-\tau_j),\quad t\ge0,
\end{equation}
involving $N$ independent daughter processes $Z_j(\cdot)$ generated by the founder individual at the birth times $\tau_j$, $j=1,\ldots,N$.
Here it is assumed that $Z_j(t)=0$ for $t<0$ and $j=1,\ldots,N$.
The branching property \eqref{CD} implies the relation
\[ 1_{\{Z(t)=0\}}=1_{\{L\le t\}}\prod\nolimits_{j=1}^{N} 1_{\{Z_j (t-\tau_j)=0\}},\]
saying that the GWO-process goes extinct by the time $t$ if, on one hand, the founder is dead at time $t$ and, on the other hand, all daughter processes are extinct by the time $t$.
After taking expectations of both sides, we can write
\begin{equation}\label{ejp}
P(t)=\rE\Big(\prod\nolimits_{j=1}^{N}P(t-\tau_j);L\le t\Big).
\end{equation}
As shown later on, this non-linear equation for $P(\cdot)$ entails the following simple asymptotic formula for $Q(t)=1-P(t)$.
\begin{theorem}\label{thQ}
Consider a GWO-process satisfying \eqref{ir}, \eqref{d} and $a<\infty$.
Then $Q(t)\sim ht^{-1}$, where $h$ is the compound parameter defined in \eqref{lov}.
\end{theorem}
\begin{corollary}\label{co1}
Consider a Bellan-Harris process satisfying \eqref{ir} and \eqref{d}. Then $a=\rE(L)$ is finite and $Q(t)\sim ht^{-1}$, with $h$ given by \eqref{lov}.
\end{corollary}
\begin{corollary}\label{co2}
Consider a GW-process satisfying \eqref{ir}. Then
$Q(t)\sim (bt)^{-1}$.
\end{corollary}
Corollary \ref{co1} is a straightforward version of Theorem \ref{thQ} adapted to the Bellman-Harris process. This result was first stated as a corollary of \cite[Theorem 3]{V79}.
In its turn, Corollary \ref{co2} is a straightforward version of Corollary \ref{co1}, recovering the classical formula for the critical GW-process, see \cite{AN}.
Paper \cite{T} states more general results in a setting with $\rV(N)\le\infty$.
In particular, the statement of Theorem \ref{thQ} can be extracted from \cite[Corollary B]{T}. Nonetheless, here we give a direct autonomous proof of Theorem \ref{thQ}, which paves the way to the proof of the main result of this paper, Theorem 1.
\subsection{Outline of the proof of Theorem \ref{thQ}}\label{ou}
The proof of Theorem \ref{thQ} is quite intricate. We start by stating four lemmas and two propositions. Let
\begin{align}
\Phi(z)&:=\rE((1-z)^ N-1+Nz),\quad 0\le z\le 1, \label{AL}\\
W(t)&:=(1-ht^{-1})^{N}+Nht^{-1}-\sum\nolimits_{j=1}^{N}Q(t-\tau_j)-\prod\nolimits_{j=1}^{N} P(t-\tau_j), \label{Wt}\\
D(u,t)&:=\rE\Big(1-\prod\nolimits_{j=1}^{N}P(t-\tau_j);\,u<L\le t\Big)+\rE\Big((1-ht^{-1})^{N} -1+Nht^{-1};L> u\Big), \label{Dut}\\
\rE_t(X)&:=\rE(X;L\le t ),\quad t\ge 1.\label{Et}
\end{align}
\begin{lemma}\label{fQd}
Given $0<u<t$, \eqref{AL}, \eqref{Wt}, \eqref{Dut}, and \eqref{Et}, we have
\begin{align*}
\Phi(ht^{-1})= \rP(L> t)+\rE_u\Big(\sum\nolimits_{j=1}^{N}Q(t-\tau_j)\Big)-Q(t)+\rE_u(W(t))+D(u,t).
\end{align*}
\end{lemma}
\begin{lemma}\label{L3}
If \eqref{ir} and \eqref{d} hold, then $\rE(N;L>ty)=o(t^{-1})$ as $t\to\infty$.
\end{lemma}
\begin{lemma}\label{L2}
If \eqref{ir}, \eqref{d}, and $a<\infty$ hold, then for any fixed $0<y<1$,
\begin{align*}
\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{1}{t-\tau_j}-\frac{1}{t}\Big)\Big)\sim at^{-2},\quad t\to\infty.
\end{align*}
\end{lemma}
\begin{lemma}\label{L4}
If $0\le f_j,g_j\le 1$ for $ j=1,\ldots,k$, then
\[ \prod\nolimits_{j=1}^k(1-g_j)-\prod\nolimits_{j=1}^k(1-f_j)=\sum\nolimits_{j=1}^k (f_j-g_j)r_j, \]
where $0\le r_j\le1$ and
\begin{align*}
1-r_j=\sum\nolimits_{i=1}^{j-1}g_i+\sum\nolimits_{i=j+1}^{k}f_i-R_j,
\end{align*}
for some $R_j\ge0$.
If moreover, $f_j\le q$ and $g_j\le q$, then
$$1- r_j\le(k-1)q,\qquad R_j\le kq,\qquad R_j\le k^2q^2.$$
\end{lemma}
\begin{proposition}\label{Lx}
If \eqref{ir}, \eqref{d}, and $a<\infty$ hold, then $\limsup_{t\to\infty} tQ(t)<\infty$.
\end{proposition}
\begin{proposition}\label{Ly}
If \eqref{ir}, \eqref{d}, and $a<\infty$ hold, then $\liminf_{t\to\infty} tQ(t)>0$ as $t\to\infty$.
\end{proposition}
\vspace{0.25cm}
According to these two propositions,
\begin{equation}\label{ca}
0<q_1\le tQ(t)\le q_2<\infty,\quad t\ge t_0.
\end{equation}
The claim $tQ(t)\to h$ is derived using \eqref{ca} by accurately removing asymptotically negligible terms from the relation for $Q(\cdot)$ stated in Lemma 1, after setting $u=ty$ with a fixed $0<y<1$, and then choosing a sufficiently small $y$. In particular, as an intermediate step, we will show that
\begin{align}
Q(t)= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}Q(t-\tau_j)\Big)+\rE_{ty}(W(t))-aht^{-2}+o(t^{-2}),\quad t\to\infty. \label{rys}
\end{align}
Then, restating our goal as $\phi(t)\to 0$ in terms of the function $\phi(t)$, defined by
\begin{equation}\label{cal}
Q(t)=\frac{h +\phi(t)}{t},\quad t\ge1,
\end{equation}
we rewrite \eqref{rys} as
\begin{align}
\frac{h +\phi(t)}{t}&= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\frac{h +\phi(t-\tau_j)}{t-\tau_j}\Big)+\rE_{ty}(W(t))-aht^{-2}+o(t^{-2}),\quad t\to\infty. \label{eye}
\end{align}
It turns out that the three terms involving $h$, outside $W(t)$, effectively cancel each other, yielding
\begin{align}
\frac{\phi(t)}{t}&= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\frac{\phi(t-\tau_j)}{t-\tau_j}+W(t)\Big)+o(t^{-2}),\quad t\to\infty.\label{luh}
\end{align}
Treating $W(t)$ in terms of Lemma \ref{L4}, brings
\begin{align}
\phi(t)&= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\phi(t-\tau_j) r_j(t)\frac{t}{t-\tau_j}\Big)+o(t^{-1}), \label{afo}
\end{align}
where $r_j(t)$ is a counterpart of $r_j$ in Lemma \ref{L4}.
To derive from here the desired convergence $\phi(t)\to0$, we will adapt a clever trick from Chapter 9.1 of \cite{Seva}, which was further developed in \cite{V79} for the Bellman-Harris process, with possibly infinite $\rV(N)$. Define a non-negative function $m(t)$ by
$$m(t):=|\phi(t)|\, \ln t,\quad t\ge 2. $$
Multiplying \eqref{afo} by $\ln t$ and using the triangle inequality, we obtain
\begin{align}
m(t)\le \rE_{ty}\Big(\sum\nolimits_{j=1}^{N} m(t-\tau_j)r_j(t) \frac{t\ln t}{(t-\tau_j)\ln(t-\tau_j)}\Big)+v(t),\label{elin}
\end{align}
where $v(t)\ge 0$ and $v(t)=o(t^{-1}\ln t)$ as $t\to\infty$. It will be shown that this leads to $m(t)=o(\ln t)$, thereby concluding the proof of Theorem 2.
\subsection{Proof of lemmas and propositions}\label{lemmas}
\begin{proof} {\sc of Lemma \ref{fQd}}.
For $0<u<t$, relation \eqref{ejp} gives
\begin{align}\label{Qln}
P(t)=\rE_u\Big(\prod\nolimits_{j=1}^{N}P(t-\tau_j) \Big)+\rE\Big(\prod\nolimits_{j=1}^{N}P(t-\tau_j);u<L\le t\Big).
\end{align}
and similarly, by \eqref{AL},
\begin{align*}
\Phi(ht^{-1})
&=\rE_u\Big((1-ht^{-1})^{N}-1+Nht^{-1}\Big)+\rE\Big((1-ht^{-1})^{N}-1 +Nht^{-1};L> u\Big).
\end{align*}
Adding the latter relation to
\begin{align*}
1
&=\rP(L\le u)+\rP(L> t)+\rP(u<L\le t),
\end{align*}
and subtracting \eqref{Qln} from the sum, we get
\begin{align*}
\Phi(ht^{-1})+Q(t)=\rE_u\Big((1-ht^{-1})^{N} +Nht^{-1}-\prod\nolimits_{j=1}^{N}P(t-\tau_j)\Big)+\rP(L> t)+D(u,t),
\end{align*}
with $D(u,t)$ defined by \eqref{Dut}.
After a rearrangement, we obtain the statement of the lemma.
\end{proof}
\begin{proof} {\sc of Lemma \ref{L3}}.
For any fixed $\epsilon>0$,
\begin{align*}
\rE(N;L>t)=\rE(N;N\le t\epsilon,L>t)+\rE(N;1<N(t\epsilon)^{-1},L>t)\le t\epsilon\rP(L>t)+(t\epsilon)^{-1}\rE(N^2;L>t).
\end{align*}
Thus, by \eqref{ir} and \eqref{d},
\begin{align*}
\limsup_{t\to\infty} (t\rE(N;L>t))\le d\epsilon,
\end{align*}
and the assertion follows as $\epsilon\to0$.
\end{proof}
\begin{proof} {\sc of Lemma \ref{L2}}.
Put
\begin{align*}
B_t(y)&:= t^2\,\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{1}{t-\tau_j}-\frac{1}{t}\Big)\Big)-a.
\end{align*}
For any $0<u<ty$, using
\[a=\rE_u(\tau_1+\ldots+\tau_N)+A_u,\quad A_u:=\rE(\tau_1+\ldots+\tau_N;L> u),\]
we get
\begin{align*}
B_t(y)&= \rE_u\Big(\sum\nolimits_{j=1}^{N} \frac{t}{t-\tau_j}\tau_j\Big)+\rE\Big(\sum\nolimits_{j=1}^{N} \frac{t}{t-\tau_j}\tau_j\,;u<L\le ty\Big)-\rE_u(\tau_1+\ldots+\tau_N)-A_u
\\
&=\rE\Big(\sum\nolimits_{j=1}^{N}\frac{\tau_j}{1-\tau_j/t};u<L\le ty\Big)+\rE_u\Big(\sum\nolimits_{j=1}^{N}\frac{\tau_j^2}{t-\tau_j}\Big)-A_u.
\end{align*}
For the first term on the right hand side, we have $\tau_j\le L\le ty$, so that
\begin{align*}
\rE\Big(\sum\nolimits_{j=1}^{N}\frac{\tau_j}{1-\tau_j/t};u<L\le ty\Big)\le(1-y)^{-1}A_u.
\end{align*}
For the second term, $\tau_j\le L\le u$ and therefore
\begin{align*}
\rE_u\Big(\sum\nolimits_{j=1}^{N}\frac{\tau_j^2}{t-\tau_j}\Big)\le\frac{u^2}{t-u}\rE_u(N)\le\frac{u^2}{t-u}.
\end{align*}
This yields
\[-A_u\le B_t(y)\le (1-y)^{-1}A_u+\frac{u^2}{t-u},\quad 0<u<ty<t,\]
implying
\[-A_u\le \liminf_{t\to\infty} B_t(y)\le\limsup_{t\to\infty} B_t(y)\le (1-y)^{-1}A_u.\]
Since $A_u\to0$ as $u\to\infty$, we conclude that $B_t\to0$ as $t\to\infty$.
\end{proof}
\begin{proof} {\sc of Lemma \ref{L4}}.
Let
\begin{equation*}
r_j:=(1-g_1)\ldots (1-g_{j-1})(1-f_{j+1})\ldots (1-f_k),
\end{equation*}
then $0\le r_j\le1$ and the first stated equality is obtained by telescopic summation of
\begin{align*}
(1-g_1)\prod\nolimits_{j=2}^{k}(1-f_j)-\prod\nolimits_{j=1}^k(1-f_j)&=(f_1-g_1)r_1,\\
(1-g_1)(1-g_2)\prod\nolimits_{j=3}^{k}(1-f_j)- (1-g_1)\prod\nolimits_{j=2}^{k}(1-f_j)&=(f_2-g_2)r_2,\ldots,\\
\prod\nolimits_{j=2}^{k}(1-g_j)-\prod\nolimits_{j=1}^{k-1}(1-g_j)(1-f_k)&=(f_k-g_k)r_k.
\end{align*}
The second stated equality is obtained with
\begin{align*}
R_j&:=\sum_{i=j+1}^{k}f_i(1-(1-f_{j+1})\ldots (1-f_{i-1}))+\sum_{i=1}^{j-1}g_i(1-(1-g_1)\ldots (1-g_{i-1})(1-f_{j+1})\ldots (1-f_k)),
\end{align*}
by performing telescopic summation of
\begin{align*}
1-(1-f_{j+1})&=f_{j+1},\\
(1-f_{j+1})-(1-f_{j+1})(1-f_{j+2})&=f_{j+2}(1-f_{j+1}),\ldots,\\
\prod\nolimits_{i=j+1}^{k-1}(1-f_j)- \prod\nolimits_{i=j+1}^{k}(1-f_j)&=f_k\prod\nolimits_{i=j+1}^{k-1}(1-f_j),\\
\prod\nolimits_{i=j+1}^{k}(1-f_j)-(1-g_1)\prod\nolimits_{i=j+1}^{k}(1-f_j)&=g_1\prod\nolimits_{i=j+1}^{k}(1-f_j),\ldots,\\
\prod\nolimits_{i=1}^{j-2}(1-g_i)\prod\nolimits_{i=j+1}^{k}(1-f_j)- \prod\nolimits_{i=1}^{j-1}(1-g_i)\prod\nolimits_{i=j+1}^{k}(1-f_j)&=g_{j-1} \prod\nolimits_{i=1}^{j-2}(1-g_i)\prod\nolimits_{i=j+1}^{k}(1-f_j).
\end{align*}
By the above definition of $R_j$, we have $R_j\ge0$. Furthermore, given $f_j\le q$ and $g_j\le q$, we get
\[R_j\le \sum\nolimits_{i=1}^{j-1}g_i+\sum\nolimits_{i=j+1}^{k}f_i\le (k-1)q.
\]
It remains to observe that
\begin{align*}
1-r_j\le 1-(1-q)^{k-1}\le (k-1)q,
\end{align*}
and from the definition of $R_j$,
\[R_j\le q\sum\nolimits_{i=1}^{k-j-1}(1-(1-q)^{i})+q\sum\nolimits_{i=1}^{j-1}(1-(1-q)^{k-j+i-1})\le q^2\sum\nolimits_{i=1}^{k-2}i\le k^2q^2.\]
\end{proof}
\begin{proof} {\sc of Proposition \ref{Lx}}.
By the definition of $\Phi(\cdot)$, we have
$$\Phi(Q(t))+P(t)=\rE_u\Big(P(t)^{N} \Big)+\rP(L> u)-\rE\Big(1-P(t)^ N;\,L> u\Big),$$
for any $0<u<t$. This and \eqref{Qln} yield
\begin{align}
\Phi(Q(t))&=\rE_u\Big(P(t)^{N}-\prod\nolimits_{j=1}^{N}P(t-\tau_j)\Big)+\rP(L> u) \nonumber\\
&-\rE\Big(1-P(t)^ N;\,L> u\Big)-\rE\Big(\prod\nolimits_{j=1}^{N}P(t-\tau_j);u<L\le t\Big). \label{Nad}
\end{align}
An upper bound follows
\begin{align*}
\Phi(Q(t))&\le \rE_u\Big(P(t)^{N}-\prod\nolimits_{j=1}^{N}P(t-\tau_j)\Big)+\rP(L> u),
\end{align*}
which together with Lemma \ref{L4} and monotonicity of $Q(\cdot)$ entail
\begin{align}\label{13}
\Phi(Q(t))\le \rE_u\Big(\sum\nolimits_{j=1}^{N}(Q(t-\tau_j)-Q(t))\Big)+\rP(L>u).
\end{align}
Borrowing an idea from \cite{T}, suppose, on the contrary, that
$$t_n:=\min\{t: tQ(t)\ge n\}$$
is finite for any natural $n$. It follows that
$$Q(t_n)\ge \frac{n}{t_n},\qquad Q(t_n-u)<\frac{n}{t_n-u},\quad 1\le u\le t_n-1.$$
Putting $t=t_n$ into \eqref{13} and using monotonicity of $\Phi(\cdot)$, we find
\begin{eqnarray*}
\Phi(nt_n^{-1})\le \Phi(Q(t_n))\le \rE_u\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{n}{t_n-\tau_j}-\frac{n}{t_n}\Big)\Big)+\rP(L> u).
\end{eqnarray*}
Setting here $u=t_n/2$ and applying Lemma \ref{L2}, we arrive at the relation
$$\Phi(nt_n^{-1})=O(nt_n^{-2}),\quad n\to\infty.$$
Observe that under condition \eqref{ir}, the L'Hospital rule gives
\begin{equation}\label{L1}
\Phi(z)\sim bz^2,\quad z\to0.
\end{equation}
The resulting contradiction, $n^{2}t_n^{-2}=O(nt_n^{-2})$ as $n\to\infty$, finishes the proof of the proposition.
\end{proof}
\begin{proof} {\sc of Proposition \ref{Ly}}.
Relation \eqref{Nad} implies
\begin{align*}
\Phi(Q(t))\ge \rE_u\Big(P(t)^{N}-\prod\nolimits_{j=1}^{N}P(t-\tau_j)\Big)-\rE\Big(1-P(t)^ N;\,L> u\Big).
\end{align*}
By Lemma \ref{L4},
\begin{align*}
P(t)^{N}-\prod\nolimits_{j=1}^{N}P(t-\tau_j)= \sum_{j=1}^{N}(Q(t-\tau_j)-Q(t))r_j^*(t),
\end{align*}
where $0\le r_j^*(t)\le 1$ is a counterpart of term $r_j$ in Lemma \ref{L4}. Due to monotonicity of $P(\cdot)$, we have, again referring to Lemma \ref{L4},
$$1-r_j^*(t)\le (N-1)Q(t-L).$$
Thus, for $0<y<1$,
\begin{align}\label{cont}
\Phi(Q(t))&\ge \rE_{ty}\Big(\sum_{j=1}^{N}(Q(t-\tau_j)-Q(t))r_j^*(t) \Big)-\rE\Big(1-P(t)^ N;\,L> ty\Big).
\end{align}
The assertion $\liminf_{t\to\infty} tQ(t)>0$ is proven by contradiction. Assume that $\liminf_{t\to\infty} tQ(t)=0$, so that
$$t_n:=\min\{t: tQ(t)\le n^{-1}\}$$
is finite for any natural $n$. Plugging $t=t_n$ in \eqref{cont} and using
$$Q(t_n)\le \frac{1}{nt_n},\quad Q(t_n-u)-Q(t_n)\ge \frac{1}{n(t_n-u)}-\frac{1}{nt_n},\quad 1\le u\le t_n-1,$$
we get
$$\Phi\Big(\frac{1}{nt_n}\Big)\ge n^{-1}\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{1}{t_n-\tau_j}-\frac{1}{t_n}\Big)r_j^*(t_n)\Big)-\frac{1}{nt_n}\rE(N;\,L> t_ny).$$
Given $L\le ty$, we have
\begin{align*}
1-r_j^*(t)\le NQ(t(1-y))\le N\frac{q_2}{t(1-y)},
\end{align*}
where the second inequality is based on the already proven part of \eqref{ca}. Therefore,
$$\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{1}{t_n-\tau_j}-\frac{1}{t_n}\Big)(1-r_j(t_n))\Big)\le \frac{q_2y}{t_n^2(1-y)^2}\rE(N^2),$$
and we derive
\begin{align*}
nt_n^2\Phi(\tfrac{1}{nt_n})&\ge t_n^2\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{1}{t_n-\tau_j}-\frac{1}{t_n}\Big)\Big)
-\frac{\rE(N^2)q_2y}{(1-y)^2}-t_n\rE(N;\,L> t_ny).
\end{align*}
Sending $n\to\infty$ and applying \eqref{L1}, Lemma \ref{L3}, and Lemma \ref{L2},
we arrive at the inequality
$$0\ge a-yq_2\rE(N^2)(1-y)^{-2},\quad 0<y<1,$$
which is false for sufficiently small $y$.
\end{proof}
\subsection{Proof of \ $\boldsymbol{ tQ(t)\to h}$}\label{end}
\begin{proof} {\sc of \eqref{luh}}.
Fix an arbitrary $0<y<1$.
Lemma 1 with $u=ty$, gives
\begin{align}
\Phi(h t^{-1})= \rP(L> t)+\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}Q(t-\tau_j)\Big)-Q(t)+\rE_{ty}(W(t))+D(ty,t). \label{mam}
\end{align}
Let us show that
\begin{align}
D(ty,t)=o(t^{-2}),\quad t\to\infty. \label{cov}
\end{align}
Using Lemma \ref{L3} and \eqref{ca}, we find that for an arbitrarily small $\epsilon>0$,
\[\rE\Big(1-\prod\nolimits_{j=1}^{N}P(t-\tau_j);\,ty<L\le t(1-\epsilon)\Big)=o(t^{-2}),\quad t\to\infty.\]
On the other hand,
\begin{align*}
\rE\Big(1-\prod\nolimits_{j=1}^{N}P(t-\tau_j);\,t(1-\epsilon)<L\le t\Big)\le \rP(t(1-\epsilon)<L\le t),
\end{align*}
so that in view of \eqref{d},
\[\rE\Big(1-\prod\nolimits_{j=1}^{N}P(t-\tau_j);\,ty<L\le t\Big)=o(t^{-2}),\quad t\to\infty.\]
This, \eqref{Dut} and Lemma \ref{L3} entail \eqref{cov}.
Combining \eqref{mam}, \eqref{cov},
and
$$\rP(L> t)-\Phi(h t^{-1})=dt^{-2}-bh^2t^{-2}+o(t^{-2})\stackrel{\eqref{stop}}{=}-aht^{-2}+o(t^{-2}),\quad t\to\infty,$$
we derive \eqref{rys}, which in turn gives \eqref{eye}. The latter implies \eqref{luh} since by Lemmas \ref{L3} and \ref{L4},
\[ \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\frac{h }{t-\tau_j}\Big)-\frac{h}{t}=\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t-\tau_j}-\frac{h}{t}\Big)\Big)
-ht^{-1}\rE(N;L\ge ty)=aht^{-2}+o(t^{-2}).
\]
\end{proof}
\begin{proof} {\sc of \eqref{afo}}.
Turning to \eqref{luh}, observe that the random variable
$$
W(t)=(1-h t^{-1})^{N}-\prod\nolimits_{j=1}^{N}\Big(1-\frac{h +\phi(t-\tau_j)}{t-\tau_j}\Big)+\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t}-\frac{h +\phi(t-\tau_j)}{t-\tau_j}\Big),
$$
can be represented in terms of Lemma \ref{L4} as
$$
W(t)=\prod\nolimits_{j=1}^{N}(1-f_j(t))-\prod\nolimits_{j=1}^{N}(1-g_j(t))+\sum\nolimits_{j=1}^{N}(f_j(t)-g_j(t))=\sum\nolimits_{j=1}^{N}(1-r_j(t))(f_j(t)-g_j(t)),
$$
by assigning
\begin{align}\label{sal}
f_j(t):=h t^{-1},\quad g_j(t):=\frac{h +\phi(t-\tau_j)}{t-\tau_j}.
\end{align}
Here $0\le r_j(t)\le 1$ and
\begin{align}\label{stal}
1-r_j(t)\stackrel{ \eqref{ca}}{\le} Nq_2t^{-1},\quad t\ge1.
\end{align}
After plugging into \eqref{luh} the expression
$$
W(t)=\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t}-\frac{h }{t-\tau_j}\Big)(1-r_j(t))-\sum\nolimits_{j=1}^{N}\frac{\phi(t-\tau_j)}{t-\tau_j}(1-r_j(t)),
$$
we get
\begin{align*}
\frac{\phi(t)}{t}&= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\frac{\phi(t-\tau_j)}{t-\tau_j}r_j(t)\Big)+\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t-\tau_j}-\frac{h}{t}\Big)(1-r_j(t) )\Big)+o(t^{-2}),\quad t\to\infty.
\end{align*}
The latter expectation is non-negative, and for an arbitrary $\epsilon>0$, it has the following upper bound
\begin{align*}
\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t-\tau_j}-\frac{h}{t}\Big)(1-r_j(t) )\Big)
\stackrel{ \eqref{stal}}{\le} q_2\epsilon\rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\Big(\frac{h }{t-\tau_j}-\frac{h}{t} \Big)\Big)+\frac{q_2h}{(1-y)t^2}\rE(N^2;N> t\epsilon).
\end{align*}
Thus, in view of Lemma \ref{L2},
\begin{align*}
\frac{\phi(t)}{t}&= \rE_{ty}\Big(\sum\nolimits_{j=1}^{N}\frac{\phi(t-\tau_j)}{t-\tau_j}r_j(t)\Big)+o(t^{-2}),\quad t\to\infty.
\end{align*}
Multiplying this relation by $t$, we arrive at \eqref{afo}.
\end{proof}
\begin{proof} {\sc of $\phi(t)\to 0$}.
If the non-decreasing function
$$M(t):=\max_{1\le j\le t} m(j)$$
is bounded from above, then $\phi(t)=O(\frac{1}{\ln t})$ proving that $\phi(t)\to 0$ as $t\to\infty$. If $M(t)\to\infty$ as $t\to\infty$, then there is an integer-valued sequence $0<t_1<t_2<\ldots,$ such that the sequence $M_n:=M(t_n)$ is strictly increasing and converges to infinity.
In this case,
\begin{equation}\label{liv}
m(t)\le M_{n-1}<M_n,\quad 1\le t< t_n,\quad m(t_n)=M_n,\quad n\ge1.
\end{equation}
Since $|\phi(t)|\le \frac{M_{n}}{\ln t_{n}}$ for $t_n\le t<t_{n+1}$, to finish the proof of $\phi(t)\to 0$, it remains to verify that
\begin{equation}\label{dog}
M_{n}=o(\ln t_{n}),\quad n\to\infty.
\end{equation}
Putting $t=t_n$ in \eqref{elin} and using \eqref{liv}, we find
\begin{align*}
M_n\le M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}r_j(t_n)\frac{t_n\ln t_n}{(t_n-\tau_j)\ln(t_n-\tau_j)}\Big)+(t_n^{-1}\ln t_n)o_n.
\end{align*}
Here and elsewhere, $o_n$ stands for a non-negative sequence such that $o_n\to0$ as $n\to\infty$. In different formulas, the sign $o_n$ represents different such sequences.
Since
$$
0\le \frac{t\ln t}{(t-u)\ln (t-u)}-1\le \frac{u(1+\ln t)}{(t-u)\ln (t-u)},\quad 1\le u\le t-2,
$$
it follows that
\begin{align*}
M_n-M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}r_j(t_n)\Big)&\le M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\frac{\tau_j(1+\ln t_n)}{t_n(1-y)\ln (t_n(1-y))}\Big)+(t_n^{-1}\ln t_n)o_n,
\end{align*}
which together with Lemma 2 give
\begin{align}\label{alt}
M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}(1-r_j(t_n))\Big)&\le a(1-y)^{-1}t_n^{-1}M_n +t_n^{-1}(M_n+\ln t_n)o_n.
\end{align}
Now it is time to unpack the term $r_j(t)$. By Lemma \ref{L4} with \eqref{sal},
$$
1-r_j(t)=\sum_{i=1}^{j-1}\frac{h +\phi(t-\tau_i)}{t-\tau_i}+(N-j)\frac{h }{t}-R_j(t),
$$
where provided $\tau_j\le ty$,
$$
0\le R_j(t)\le Nq_2t^{-1}(1-y)^{-1},\quad R_j(t)\le N^2q_2^2t^{-2}(1-y)^{-2}.
$$
This allows us to unpack \eqref{alt} in the form
\begin{align*}
M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}&\Big(\sum_{i=1}^{j-1}\frac{h +\phi(t_n-\tau_i)}{t_n-\tau_i}+(N-j)\frac{h }{t_n}\Big)\Big)\\
&\le M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N} R_j(t_n)\Big)+a(1-y)^{-1}t_n^{-1}M_n +t_n^{-1}(M_n+\ln t_n)o_n.
\end{align*}
To estimate the last expectation, observe that if $\tau_j\le ty$, then for any $\epsilon>0$,
$$
R_j(t)\le Nq_2t^{-1}(1-y)^{-1} 1_{\{N>t\epsilon\}}+ N^2q_2^2t^{-2} (1-y)^{-2}1_{\{N\le t\epsilon\}},
$$
implying
$$\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}R_j(t_n)\Big)
\le q_2t_n^{-1}(1-y)^{-1}\rE(N^{2} ; N> t_n\epsilon)+ q_2^2\epsilon t_n^{-1}(1-y)^{-2}\rE(N^2),$$
so that
\begin{align*}
M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\Big(\sum\nolimits_{i=1}^{j-1}\frac{h +\phi(t_n-\tau_i)}{t_n-\tau_i}+(N-j)\frac{h }{t_n}\Big)\Big)
\le a(1-y)^{-1}t_n^{-1}M_n +t_n^{-1}(M_n+\ln t_n)o_n.
\end{align*}
Since
\[\sum\nolimits_{j=1}^{N}\sum\nolimits_{i=1}^{j-1}\Big(\frac{h}{t_n-\tau_i}- \frac{h }{t_n}\Big)\ge0,\]
we obtain
\begin{align*}
M_n\rE_{t_ny}\Big(\sum\nolimits_{j=1}^{N}\Big(\sum_{i=1}^{j-1}\frac{\phi(t_n-\tau_i)}{t_n-\tau_i}+(N-1)\frac{h }{t_n})\Big)\Big)
\le a(1-y)^{-1}t_n^{-1}M_n +t_n^{-1}(M_n+\ln t_n)o_n.
\end{align*}
Given $\tau_j\le L\le t_ny$, the inequality
$$\frac{\phi(t_n-\tau_i)}{t_n-\tau_i}\stackrel{}{\ge} \frac{q_1-h}{t_n(1-y)}$$
holds for sufficiently large $n$, due to \eqref{ca} and \eqref{cal}. This gives
\[\sum\nolimits_{j=1}^{N}\Big(\sum_{i=1}^{j-1}\frac{\phi(t_n-\tau_i)}{t_n-\tau_i}+(N-1)\frac{h }{t_n}\Big)\ge \Big(h+\frac{q_1-h }{2(1-y)}\Big)t_n^{-1}N(N-1),\]
which after multiplying by $t_nM_n$ and taking expectations, yields
\begin{align*}
\Big(h+\frac{q_1-h }{2(1-y)}\Big)M_n\rE_{t_ny}(N(N-1))
\le a(1-y)^{-1}M_n +(M_n+\ln t_n)o_n.
\end{align*}
Finally, since
$$ \rE_{t_ny}(N(N-1))\to2b,\quad n\to\infty,$$
we derive that for any $0<\epsilon<y<1$, there is a finite $n_\epsilon$ such that for all $n>n_\epsilon$,
$$M_n\Big(2bh(1-y)+bq_1-bh-a-\epsilon\Big) \le \epsilon\ln t_n.$$
By \eqref{stop}, we have $bh\ge a$, and therefore,
$$2bh(1-y)+bq_1-bh-a-\epsilon\ge bq_1-2bhy-y.$$
Thus, choosing $y=y_0$ such that $bq_1-2bhy_0-y_0=\frac{bq_1}{2}$, we see that
$$\limsup_{n\to\infty}\frac{M_n}{\ln t_n} \le \frac{2\epsilon}{bq_1},$$
which entails \eqref{dog} as $\epsilon\to0$, concluding the proof of $\phi(t)\to 0$.
\end{proof}
\section{Proof of Theorem \ref{thL}}\label{Lp1}
For the $k$-dimensional probability generating function
\[\rE(z_1^{Z(t_1)}\cdots z_k^{Z(t_k)})=\sum_{i_1=0}^\infty\ldots\sum_{i_k=0}^\infty\rP(Z(t_1)=i_1,\ldots, Z(t_k)=i_k)z_1^{i_1}\cdots z_k^{i_k},\]
with $0< t_1\le \ldots\le t_k$ and $z_1,\ldots,z_k\in[0,1]$, we will use the following notational agreements. We denote
\[P_k(\bar t,\bar z):=P_k(t_1,\ldots,t_{n};z_1,\ldots,z_{k}):=\rE(z_1^{Z(t_1)}\cdots z_k^{Z(t_k)}),\]
and write for $t\ge0$,
\[P_k(t+\bar t,\bar z):=P_k(t+t_1,\ldots,t+t_{k};z_1,\ldots,z_{k}).\]
Moreover,
for $0< y_1<\ldots<y_k$, we write
\[P_k(t\bar y,\bar z):=P_k(ty_1,\ldots,ty_{k};z_1,\ldots,z_{k}),\]
and assuming $0< y_1<\ldots<y_k<1$,
\[P_k^*(t,\bar y,\bar z):=\rE(z_1^{Z(ty_1)}\cdots z_{k}^{Z(ty_{k})};Z(t)=0)=P_{k+1}(ty_1,\ldots,ty_k,t;z_1,\ldots,z_k,0).
\]
These notational agreements will be similarly applied to the functions
\begin{equation}\label{Q*}
Q_k(\bar t,\bar z):=1-P_k(\bar t,\bar z),\quad Q_k^*(t,\bar y,\bar z):=1-P_k^*(t,\bar y,\bar z).
\end{equation}
Our special interest is in the function
\begin{equation}\label{krik}
Q_k(t):=Q_k(t+\bar t,\bar z),\quad 0= t_1< \ldots< t_k, \quad z_1,\ldots,z_k\in[0,1),
\end{equation}
to be viewed as a counterpart of the function $Q(t)$ treated by Theorem 2. Put
\begin{equation}\label{hk}
h_k:=h\frac{1+\sqrt{1+cg_k}}{1+\sqrt{1+c}},\quad g_k:= g_k(\bar y,\bar z):=\sum_{i=1}^{k}z_1\cdots z_{i-1}(1-z_{i})y_{i}^{-2},
\end{equation}
where $h$ and $c$ are defined in \eqref{lov}.
The key step of the proof of Theorem 1 is to show that for any given $1=y_1<y_2<\ldots<y_k$,
\begin{equation}\label{dm}
tQ_k(t)\to h_k,\quad t_i:=t(y_i-1), \quad i=1,\ldots,k,\quad t\to\infty.
\end{equation}
This is done following the steps of our proof of $tQ(t)\to h$.
Unlike $Q(t)$, the function $Q_k(t)$ is not monotone over $t$. However, monotonicity of $Q(t)$ was used in the proof of Theorem 2 only in the proof of \eqref{ca}. The corresponding statement
$$ 0<q_1\le tQ_k(t)\le q_2<\infty,\quad t\ge t_0,
$$
follows from the bounds $(1-z_1)Q(t)\le Q_k(t)\le Q(t)$, which hold due to monotonicity of the underlying generating functions over
$z_1,\ldots,z_{n}$. Indeed,
\[Q_k(t)\le Q_k(t, t+t_2,\ldots,t+t_{k};0,\ldots,0)= Q(t),\]
and on the other hand,
\[Q_k(t)= Q_k(t,t+t_2,\ldots,t+t_{k};z_1,\ldots,z_k)= \rE(1-z_1^{Z(t)}z_2^{Z(t+t_2)}\cdots z_k^{Z(t+t_k)})\ge \rE(1-z_1^{Z(t)}),\]
where
\[ \rE(1-z_1^{Z(t)})\ge \rE(1-z_1^{Z(t)};Z(t)\ge1)\ge (1-z_1)Q(t).\]
\subsection{Proof of \ $\boldsymbol{tQ_k(t)\to h_k}$}\label{Lup}
The branching property \eqref{CD} of the GWO-process gives
\[ \prod_{i=1}^{k} z_i^{Z(t_i)}=\prod_{i=1}^{k} z_i^{1_{\{L>t_i\}}}\prod\nolimits_{j=1}^{N} z_i^{Z_j(t_i-\tau_j)}.\]
Given $0< t_1<\ldots<t_k< t_{k+1}=\infty$, we use
\begin{align*}
\prod_{i=1}^{k} z_i^{1_{\{L>t_i\}}}&=1_{\{L\le t_1\}}+\sum_{i=1}^{k}z_1\cdots z_{i}1_{\{t_{i}<L\le t_{i+1}\}},
\end{align*}
to deduce the following counterpart of \eqref{ejp}
\begin{align*}
P_k(\bar t,\bar z)&=\rE_{t_1}\Big(\prod_{j=1}^{N}P_k(\bar t-\tau_j,\bar z)\Big)+\sum_{i=1}^{k}z_1\cdots z_{i}\rE\Big(\prod_{j=1}^{N}P_k(\bar t-\tau_j,\bar z);
t_{i}<L\le t_{i+1}\Big),
\end{align*}
which entails
\begin{align}\label{apes}
P_k(\bar t,\bar z)&=\rE_{t_1}\Big(\prod_{j=1}^{N}P_k(\bar t-\tau_j,\bar z)\Big)+\sum_{i=1}^{k}z_1\cdots z_{i} \rP(t_{i}<L\le t_{i+1})) \nonumber\\
&-\sum_{i=1}^{k}z_1\cdots z_{i} \rE\Big(1-\prod_{j=1}^{N}P_k(\bar t-\tau_j,\bar z); t_{i}<L\le t_{i+1})\Big).
\end{align}
Using this relation we establish the following counterpart of Lemma \ref{fQd}.
\begin{lemma}\label{fad}
Consider function \eqref{krik} and put
$P_k(t):=1-Q_k(t)=P_k(t+\bar t,\bar z)$.
For $0<u<t$, the relation
\begin{align}
\Phi(h_k t^{-1})&= \rP(L> t)-\sum_{i=1}^{k}z_1\cdots z_{i}\rP(t+t_i<L\le t+t_{i+1}) \nonumber \\
&+\rE_u\Big(\sum\nolimits_{j=1}^{N}Q_k(t-\tau_j)\Big)-Q_k(t)+\rE_u(W_k(t))+D_k(u,t), \label{arr}
\end{align}
holds with $t_{k+1}=\infty$,
\begin{align} \label{tWt}
W_k(t):=(1-h_k t^{-1})^{N}+Nh_k t^{-1}-\sum\nolimits_{j=1}^{N}Q_k(t-\tau_j)-\prod\nolimits_{j=1}^{N}P_k(t-\tau_j)
\end{align}
and
\begin{align} \label{tDut}
D_k(u,t):=\ &\rE\Big(1-\prod\nolimits_{j=1}^{N}P_k(t-\tau_j);u<L\le t\Big)+\rE\Big((1-h_k t^{-1})^{N} -1+Nh_k t^{-1};L> u\Big) \nonumber\\
&+\sum_{i=1}^{k}z_1\cdots z_{i} \rE\Big(1-\prod_{j=1}^{N}P_k(t-\tau_j); t+t_{i}<L\le t+t_{i+1}\Big).
\end{align}
\end{lemma}
\begin{proof}
According to \eqref{apes},
\begin{align*}
P_k(t)&=\rE_u\Big(\prod_{j=1}^{N}P_k(t-\tau_j)\Big)+\rE\Big(\prod\nolimits_{j=1}^{N}P_k(t-\tau_j);u<L\le t\Big)
\\
&+\sum_{i=1}^{k}z_1\cdots z_{i} \rP(t+t_{i}<L\le t+t_{i+1})-\sum_{i=1}^{k}z_1\cdots z_{i} \rE\Big(1-\prod_{j=1}^{N}P_k(t-\tau_j); t+t_{i}<L\le t+t_{i+1}\Big).
\end{align*}
By the definition of $\Phi(\cdot)$,
\begin{align*}
\Phi(h_k t^{-1})+1
&=\rE_u\Big((1-h_k t^{-1})^{N}+Nh_k t^{-1}\Big)+\rP(L> t)\\
&+\rE\Big((1-h_k t^{-1})^{N} -1+Nh_k t^{-1};L> u\Big)+\rP(u<L\le t),
\end{align*}
and after subtracting the two last equations, we get
\begin{align*}
\Phi(h_k t^{-1})+Q_k(t)&=\rE_u\Big((1-h_k t^{-1})^{N} +Nh_k t^{-1}-\prod\nolimits_{j=1}^{N}P_k(t-\tau_j)\Big)+\rP(L> t)\\
&+\sum_{i=1}^{k}z_1\cdots z_{i} \rP(t+t_{i}<L\le t+t_{i+1})+D_k(u,t)
\end{align*}
with $D_k(u,t)$ satisfying \eqref{tDut}.
After a rearrangement, relation \eqref{arr} follows together with \eqref{tWt}.
\end{proof}
With Lemma \ref{fad} in hand, convergence \eqref{dm} is proven applying almost exactly the same argument used in the proof of $tQ(t)\to h$. An important new feature emerges due to the additional term in the asymptotic relation defining the limit $h_k$. Let $1=y_1<y_2<\ldots<y_k<y_{k+1}=\infty$. Since
\begin{align*}
\sum\nolimits_{i=1}^{k}z_1\cdots z_{i}\rP(ty_{i}<L\le ty_{i+1})\sim d t^{-2}\sum_{i=1}^{k}z_1\cdots z_{i}(y_{i}^{-2}-y_{i+1}^{-2}),
\end{align*}
we see that
\begin{align*}
\rP(L> t)-\sum\nolimits_{i=1}^{k}z_1\cdots z_{i}\rP(ty_{i}<L\le ty_{i+1})\sim dg_k t^{-2},
\end{align*}
where $g_k$ is defined by \eqref{hk}.
Assuming $0\le z_1,\ldots,z_k<1$, we ensure that $g_k>0$, and as a result, we arrive at a counterpart of the quadratic equation \eqref{stop},
\[
bh_k^2=ah_k+dg_k,
\]
which gives
\[
h_k=\frac{a+\sqrt{a^2+4bdg_k}}{2b}=h\frac{1+\sqrt{1+cg_k}}{1+\sqrt{1+c}},\]
justifying our definition \eqref{hk}.
We conclude that for $k\ge1$,
\begin{equation}\label{love}
\frac{Q_k(t\bar y,\bar z)}{Q(t)}\to \frac{1+\sqrt{1+c\sum\nolimits_{i=1}^{k}z_1\cdots z_{i-1}(1-z_{i})y_{i}^{-2}}}{1+\sqrt{1+c}},\quad 1=y_1<\ldots< y_k,\quad
0\le z_1,\ldots,z_k<1.
\end{equation}
\subsection{Conditioned generating functions}\label{Lp2}
To finish the proof of Theorem 1, consider the generating functions conditioned on the survival of the GWO-process.
Given \eqref{mansur} with $j\ge1$, we have
\begin{align*}
Q(t)\rE&(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)}|Z(t)>0)=\rE(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)};Z(t)>0)\\
&=P_k(t\bar y,\bar z)-\rE(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)};Z(t)=0)\stackrel{\eqref{Q*}}{=}Q_j^*(t,\bar y,\bar z)-Q_k(t\bar y,\bar z),
\end{align*}
and therefore,
\[\rE(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)}|Z(t)>0)=\frac{Q_j^*(t,\bar y,\bar z)}{Q(t)}-\frac{Q_k(t\bar y,\bar z)}{Q(t)}.\]
Similarly, if \eqref{mansur} holds with $j=0$, then
\[\rE(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)}|Z(t)>0)=1-\frac{Q_k(t\bar y,\bar z)}{Q(t)}.\]
Letting $t'=ty_1$, we get
\[\frac{Q_k(t\bar y,\bar z)}{Q(t)}=\frac{Q_k(t',t'y_2/y_1,\ldots,t'y_k/y_1)}{Q(t')}\frac{Q(ty_1)}{Q(t)},\]
and applying relation \eqref{love},
\begin{equation*}
\frac{Q_k(t\bar y,\bar z)}{Q(t)}\to \frac{1+\sqrt{1+\sum\nolimits_{i=1}^{k}z_1\cdots z_{i-1}(1-z_{i})\Gamma_i}}{(1+\sqrt{1+c})y_1},\quad \Gamma_i:=c({y_1}/{y_i} )^2.
\end{equation*}
On the other hand, since
\[Q_j^*(t,\bar y,\bar z)=Q_{j+1}(ty_1,\ldots,ty_j,t;z_1,\ldots,z_j,0), \quad j\ge1,\]
we also get
\begin{equation*}
\frac{Q_j^*(t,\bar y,\bar z)}{Q(t)}\to \frac{1+\sqrt{1+\sum\nolimits_{i=1}^{j}z_1\cdots z_{i-1}(1-z_{i})\Gamma_i+cz_1\cdots z_{j}y_1^2}}{(1+\sqrt{1+c})y_1}.
\end{equation*}
We conclude that as stated in Section \ref{main},
\begin{align*}
\rE(z_1^{Z(ty_1)}\cdots z_k^{Z(ty_k)}|Z(t)>0)\to \rE(z_1^{\eta(y_1)}\cdots z_k^{\eta(y_k)}).
\end{align*}
| 179,594
|
TITLE: Double integral $ \iint \limits_D \frac{y}{x^2+(y+1)^2}dxdy$, $D$=$\{(x,y): x^2+y^2 \le1 , y\ge0\}$
QUESTION [16 upvotes]: Solve $$ \iint \limits_D \frac{y}{x^2+(y+1)^2}dxdy \ \ \ \ . . . \ (*)$$
where $D$=$\{$$(x,y): x^2+y^2 \le1 , y\ge0 $$\}$
$$
$$
Here is my attempt.
$$\begin{align}
&(1).\ \ \ (*)=\int_{-1}^1 \int_{0}^{\sqrt{1-x^2}}\frac{y}{x^2+(y+1)^2}dydx \\
&(2).\ \ \ (*)= \int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{y}{x^2+(y+1)^2}dxdy \\
&(3). \ \ \int\frac{y+1}{x^2+(y+1)^2}dx = \arctan\left(\frac{x}{y+1}\right) + C \\
&(4). \ \ \ (*)=\int_{0}^{\pi} \int_{0}^{1}\frac{r^2sin\theta}{r^2+2rsin\theta+1}drd\theta \\\\
\end{align}$$
I used $(1)$, $(4)$ and $(2)$ with $(3)$,
but didn't solve yet.
$$$$
Did I make a mistake?
Could you give me some advice, please?
How can I solve this integral...
Thank you for your attention to this matter.
$$$$
P.S.
Here is result of wolframalpha
$$$$
$$ $$
Additionally... I did like this.. maybe useless :-(
$$\begin{align}
(*)
& = \int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{y}{x^2+(y+1)^2}dxdy \\\\
&=\int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{y+1}{x^2+(y+1)^2}dxdy + \int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{1}{x^2+(y+1)^2}dxdy \\\\
&=\int_{0}^1 \left(\arctan\left(\frac{\sqrt{1-y^2}}{y+1}\right) - \arctan\left(\frac{-\sqrt{1-y^2}}{y+1}\right)\right)dy \\
& \ \ \ \ + \int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{1}{x^2+(y+1)^2}dxdy \\\\
&=\int_{0}^1 \left(\arctan\left(\sqrt\frac{1-y}{1+y} \ \right) - \arctan\left(-\sqrt\frac{1-y}{1+y} \ \right)\right)dy \\
& \ \ \ \ +\int_{0}^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\frac{1}{x^2+(y+1)^2}dxdy \\\\
&= terrible?! \\
\end{align}$$
$$
$$
$$ $$
---------------------------------------------------------------------------
This picture is for asking to Christian Blatter
(I am really sorry, if I bother you guys for this picture.)
REPLY [4 votes]: The integrand function $\frac{y}{x^2+(y+1)^2}$ suggest to put
$$\left\{
\begin{align}
x&=r\cos\theta\\
y+1&=r\sin\theta
\end{align}\right.
$$
so that $x^2+(y+1)^2=r^2$ and the Jacobian is $r$.
From $y\ge 0$ we have $y=r\sin\theta-1\ge 0$ that is $r\ge\frac{1}{\sin\theta}$ and from $x^2+y^2\le 1$ we have $$x^2+y^2=r^2\cos^2\theta+(r\sin\theta-1)^2=r^2-2r\sin\theta+1\le 1$$ and then $r(r-2\sin\theta)\le0$ so that $r\le 2\sin\theta$.
So we have
$$\boxed{
r_{\min}=\frac{1}{\sin\theta}\le r\le 2\sin\theta=r_{\max}}
$$
For $y=0$ (i.e. $r\sin\theta =1$) we have $-1\le x\le 1$, that is $-1\le r\cos\theta\le 1$ and then $-1\le\tan\theta\le 1$; thus
$$\boxed{
\theta_{\min}=\frac{\pi}{4}\le \theta\le \frac{3\pi}{4}=\theta_{\max}}
$$
or $\frac{-\pi}{4}\le \theta\le \frac{+\pi}{4}$ if you prefer.
The figure help to show all we have done.
So the integrand in polar coordinates becomes $f(r,\theta)=\frac{r\sin\theta-1}{r^2}$ and the integral becomes
$$
\mathcal{I}=\int_{\theta_{\min}}^{\theta_{\max}}\int_{r_{\min}}^{r_{\max}}
f(r,\theta)r\,\mathrm d r\,\mathrm d\theta=
\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\int_{\frac{1}{\sin\theta}}^{2\sin\theta}\left(\sin\theta-\frac{1}{r}\right) \mathrm d r\,\mathrm d\theta
$$
The integral in $r$ is easy to evaluate
$$\begin{align}
\int_{\frac{1}{\sin\theta}}^{2\sin\theta}\left(\sin\theta-\frac{1}{r}\right) \mathrm d r
&=
\left[\sin\theta\, r-\log r\right]_{\frac{1}{\sin\theta}}^{2\sin\theta}\\
&=\sin\theta\left[2\sin\theta-\tfrac{1}{\sin\theta}\right]-\left[\log(2\sin\theta)-\log\left(\tfrac{1}{\sin\theta}\right)\right]\\
&=-\cos(2\theta)-\log\left(2\sin^2\theta\right)
\end{align}
$$
Then the integral in $\theta$ is
$$\begin{align}
\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \left[-\cos(2\theta)-\log\left(2\sin^2\theta\right)\right]\mathrm d \theta
&=
\left[-\frac{1}{2}\sin(2\theta)\, \right]_{\frac{\pi}{4}}^{\frac{3\pi}{4}}-\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \log\left(2\sin^2\theta\right)\mathrm d \theta=1+J
\end{align}
$$
where
$$
J=-\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \log\left(2\right)\mathrm d \theta-\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \log\left(\sin^2\theta\right)\mathrm d \theta=
-\frac{\pi}{2}\log 2-2C+\pi\log 2=\frac{\pi}{2}\log 2-2C
$$
observig that
$$\begin{align}
-\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \log\left(\sin^2\theta\right)\mathrm d \theta &=
-\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \log\left(\cos^2\theta\right)\mathrm d \theta=-2\int_{0}^{\frac{\pi}{4}} \log\left(\cos^2\theta\right)\mathrm d \theta\\
&=-4\int_{0}^{\frac{\pi}{4}} \log\left(\cos\theta\right)\mathrm d \theta=-4\left(\frac{C}{2}-\frac{\pi}{4}\log 2\right)\\
&=-2C+\pi\log 2
\end{align}
$$
where $C$ is the Catalan's constant (see for exaple here).
Finally we have
$$\large\color{blue}{
\mathcal I=1+\frac{\pi}{2}\log 2-2C}
$$
| 200,146
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Ferry Routes
Mini Cruises
Experience
Customer Services
DFDS IS "WORLD'S LEADING FERRY OPERATOR" FOR THE EIGHTH TIME
Hamburg, 06 December 2018 . The Danish shipping company DFDS was honored for the eighth time in a row with the World Travel Award for Best Shipping Company. At the 25th World Travel Awards on December 1, 2018 in Lisbon, Portugal, the ferry company once again received the coveted award.
The prestigious World Travel Awards are considered the Oscars of the travel and tourism industry, and since 1993 have excelled in excellence in the sector. The awarding of the title among the nominated travel companies will be decided by a jury of leading industry experts and consumers.
"We are very pleased to have been recognized as one of the world's leading ferry operators for another year. We would like to thank our loyal customers and partners who re-selected DFDS, "says Pete Akermann, Marketing Director.
"Our goal is to provide our customers with outstanding customer service and an exceptional ferry experience, whether on our cruise ferries, eg from Amsterdam to Newcastle, or on our flexible, hourly crossings from Calais and Dunkirk to Dover," says Akermann.
"We attach great importance to the feedback from our customers in order to further develop and improve our service and our offers. This year's investments in the new catering concept on the Amsterdam-Newcastle route, as well as the upcoming renovations of the dining rooms on board our ships on the Dunkirk-Dover route are proof of that. We want to continue our growth course and continue to invest in our ships and our product in the future to ensure the satisfaction of our customers, "explains Akermann.!
| 375,900
|
With a job as a copywriter for a PR firm under his belt, his career as an author could begin - his first publication was in The Arkham Collector in 1968. He is probably best known for his science fiction (the Humanx Commonwealth set of books) and his novelizations of many motion pictures (Star Wars trilogy, Alien, et al).
His love of travel - especially to exotic locations - often comes through in his novels. On this planet, he has traveled to: Tahiti, French Polynesia, Europe, Asia, Tanzania, Kenya, the "Green Hell" of the Peruvian jungle, Australia, the Lechugilla Cave, the Batoka Gorge in Zambezi, Namibia, the Andes, Peru, Brazil, Papua New Guinea.
As with many authors, he has also lectured (in this case, at UCLA and Los Angeles City College), although he and his wife Joann now live in Prescott, AZ.
Online Information:
Web:
Anthology
Log in or register to write something here or to contact authors.
Need help? accounthelp@everything2.com
| 324,911
|
RS700 Nationals’ Day Three Report, Results and Vid
Congratulations to Stephen Carr from Brightlingsea SC for winning the spinnaker raffle. The spinnaker was generously donated by Hyde Sails
Read about day three of the RS700 Volvo Noble Marine National Championships supported by Hyde Sails in Pete Purkiss’ report here
Third Eastbourne vid here
Second Eastbourne vid here
First Eastbourne vid here:
Eastbourne photos up on their facebook page here
Photo gallery incoming...
Thank you to Harken for sponsoring today’s fab daily prizes
Day two’s report up on Y&Y here
#RS700sailing! #SailitLiveitLoveit #RoosterKit #InYourElement #RSsailing #HydeSails @RoosterKit
Happy RS700 Sailing!
Cheers
Clare
| 261,792
|
Your Search Results
13 fonts match your search/filter criteria. You are only showing fonts with the classifications 'Blackletter', 'Monospaced', 'Pixel', OR 'Typewriter', and with the tag 'Historical'... To clear this filter, click here or adjust the options in the sidebar.
| 145
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TITLE: Averaging for nonlinear systems
QUESTION [2 upvotes]: I am trying to figure out how the following result has been obtained.
Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system:
$$ \dot{ \hat{x} }(t) = k a \sin ( \omega t ) J( \hat{x}(t) + a \sin ( \omega t) ) $$
with constant parameters $k, a, \omega$.
Consider Taylor series expansion:
$$ J( \hat{x}(t) + a \sin ( \omega t) ) = J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) } a \sin ( \omega t ) + o(a^2) $$
Now since
$$ \dot{ \hat{x} }(t) = k a \sin ( \omega t ) \left( J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) } a \sin ( \omega t )+ o(a^2) \right) $$
is a periodic ( ??? ) time-varying system, averaging may be applied:
$$ \dot{ \hat{x } }(t) = \frac{1}{T} \int_{0}^{T} k a \sin ( \omega t ) \left( J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) }a \sin ( \omega t ) + o(a^2) \right) dt $$
where $T = \frac{2 \pi}{ \omega }$.
And then:
$$ \frac{1}{T} \int_{0}^{T} k a \sin ( \omega t ) \left( J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) }a \sin ( \omega t ) + o(a^2) \right) dt = \frac{ka^2}{2} \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) } + o(a^4) $$
How was this integral computed?
I can only see a reason if $J$ is periodic as well or even constant in $[0, T]$. Otherwise, I can't see how the integrand is periodic and how $o(a^4)$ appears.
I found this result here on page 7.
Remark: I didn't denote the $ \hat{x} $ in the averaged system with another symbol, but I feel that it would have been more correct to use something like $ \hat{x}_{ \text{av} } $ to avoid ambiguity.
REPLY [2 votes]: I looked through the paper you referenced and even the book they reference within $[17]$. I too cannot see why the above is true unless $J$ is periodic or constant. So I think there is a heavy abuse of notation here ("By fixing $\hat{x}(t)$ to a dummy variable $z$") and what they are really saying is as follows:
Instead of using this expression:
$$ \dot{ \hat{x} }(t) = k a \sin ( \omega t ) \left( J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) }a \sin ( \omega t ) + O(a^2) \right) $$
Define $y(s)$ to be the solution of the following differential equation
$$ \frac{d}{ds}(y(s)) = k a \sin ( \omega (s+t) ) \left( J( \hat{x}(t) ) + \frac{ \partial J }{ \partial \hat{x} } \Bigg|_{ \hat{x} = \hat{x}(t) }a \sin ( \omega (s+t) ) + O(a^2) \right) $$
Here we have taken the Taylor series in $x$ of $J(x)$, expanded about $\hat{x}(t)$, and evaluated at $\hat{x}(t) + a \sin ( \omega ( s + t ) )$. From that point of view, $y(0) = \hat{x}(t)$ and $y$ as a function of $s$ is periodic with period $T$.
The integration that comes next then makes sense. Note the integral is $O(a^4)$ because
$$\int_0^T a \sin (\omega (s + t)) a^k \sin^k(\omega(s + t)) ds$$ is zero iff $k$ is even.
| 74,306
|
TITLE: Orthonormal zero Function
QUESTION [1 upvotes]: I have this exercise
Let H be a Hilbert space with orthonormal basis $\{e_n | n\in N\}$ and let
$f_n = e_n + e_{n+1}$
If $\langle f,f_n \rangle = 0$ for all $n$ how do I prove that $f=0$
I think i can do it for a even n for an odd one i cant... Can anyone help?
REPLY [0 votes]: You can also directly consider the Fourier coefficients of the basis. The assumption says $ ⟨f,e_{n+1}⟩=- ⟨f,e_n⟩$, so $$⟨f,e_n⟩=(-1)^{n-1}⟨f,e_1⟩$$ and per the answer of T. Bongers, the asymptotic behavior following from Parseval
$$
\|f\|^2=\sum_{n=1}^\infty |⟨f,e_n⟩|^2
$$
demands that all coefficients are zero.
| 37,658
|
2020-05-26
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Graves disease and goiter
Diffuse toxic goiter (Bazed disease, Graves' disease) is a disease characterized by a widespread enlargement of the thyroid gland and… — Frequencies: 4
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Posts Tagged ‘intensive blemish balm’
Intensive Blemish Balm: What it Does to Help You
So you know that you want to get something that is going to help you cover up blemishes on your face, but without having to use full-on foundation. Well in this case there are a few different types of beauty product that you could choose from, but the blemish balm cream is probably the best that you could get.
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Where to Find it
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This is one great beauty product that you can always take along with you whether you are going grocery shopping or going out on a date with that special someone. In fact, you will probably want to keep a couple tubes of this stuff at home so that you always have a replacement.
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Where to Buy Blemish Balm
There are lots of different beauty products that a woman should have in her arsenal, but when it really comes down to it, there are a few that are really important and which you are just not going to want to go without. The intensive blemish balm is one of these, a beauty product that is worth its weight in gold and which you are never going to want to be without.
Now it is just a matter of you learning about where you can go to buy blemish balm, and fortunately because it is so popular there are quite a few different places that you can head to if you want to get some of this blemish balm for yourself.
The Skin Specialist
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Their blemish balm works to moisturize the skin but also to act as a sort of foundation for blemishes on the skin. This way you do not have to wear full coverage foundation but are still able to cover up those blemishes and redder areas on the skin.
Health Chemist
Another store that you can go through if you want to buy blemish balm is Health Chemist. You can definitely trust to buy blemish balm here because they are such a well known and reputable beauty supply store. They know that you want to stay looking beautiful and feel great about yourself, and this is why they offer all the latest and greatest beauty products.
They actually have a few different types of the blemish balm product that you can check out, and their prices are actually affordable as well.
Keep in mind that there are actually a few at home recipes that you can use as well if you like to make this blemish balm. The benefit of this would be that you would know exactly what ingredients are going into the product so for instance if you are really into the natural or organically made beauty products, this may be a great option for you.
Beauty products are usually not that easy to make, but it will be worth it when you have a beauty balm that you know is completely natural.
| 96,046
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20 Luxury Mid Century Flush Mount Light Pics By sanditirta92 On March 12, 2018 ★★★★★20 Luxury Mid Century Flush Mount Light Pics,5 / 5 ( 1votes )You need to enable JavaScript to vote Ceiling Light Fixture Covers Fresh Wood Beaded Chandelier Exciting from mid century flush mount light , image source: searchengineoptimization4.info Gallery of 20 Luxury Mid Century Flush Mount Light Pics «» Related Post "20 Luxury Mid Century Flush Mount
| 41,992
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TITLE: Is a sign-preserving operator on $L^2$ a multiplication?
QUESTION [7 upvotes]: Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$.
$T$ is assumed to be sign-preserving in the sense that
$$
v(x) \cdot (Tv)(x) \ge0
$$
for $\mu$-almost all $x\in \Omega$ and all $v\in L^2(\mu)$.
Does this imply that $T$ is a multiplication? That is, does there exist $\phi\in L^\infty(\mu)$ such that $Tv = u\cdot v$?
I could show the following property:
$$
\chi_{A^c} \cdot (T\chi_A) = 0
$$
$\mu$-almost everywhere for all characteristic functions $\chi_A$ of $A\in\Sigma$. This would prove the question for $\mathbb R^n$ or $l^2(\mathbb N)$. I was not able to prove the question in the general case.
REPLY [5 votes]: First, if $uv=0$ then $uTv=0$ a.e.: indeed, applying the hypothesis with the function $\epsilon u+\epsilon^{-1}v$ and evaluating on $\{u\neq 0\}$, we get
$$0\le \epsilon^2uTu+\epsilon^{-2}vTv+uTv+vTu=\epsilon^2 uTu+uTv$$
a.e. and deduce the claim (sending $\epsilon\to 0$). This gives the property $(*)$ that you showed.
If the space is $\sigma$-finite, using $(*)$ we can easily reduce to the case that the measure is finite. Now take $v:=T1$. We claim that $Tu=uv$ (a.e.) when $u$ is simple: since $T$ is linear, it suffices to show this when $u=1_A$ is a characteristic function. But
$$v=T1=T1_A+T1_{A^c}$$
and $T1_{A^c}$ vanishes (a.e.) on $A$, hence $T1_A=v$ (a.e.) on $A$. Since $T1_A$ vanishes (a.e.) on $A^c$, the claim follows. Taking $A:=\{|v|>\|T\|\}$, if $\mu(A)>0$ we see that $\|T1_A\|_{L^2}>\|T\|\|1_A\|_{L^2}$, contradiction. So $v\in L^\infty$. The statement now follows since simple functions are dense.
It seems false to me without assuming the space to be $\sigma$-finite: take $\Omega:=[0,1]^2$, with $\mu:=\mathcal{H}^1$ (1-dimensional Hausdorff measure) and the $\sigma$-algebra $\mathcal A$ generated by horizontal and vertical slices ($\{s\}\times[0,1]$ and $[0,1]\times\{t\}$).
Now with little work you can show that all elements of $\mathcal A$, up to adding and removing negligible sets, are of the form
$$\bigcup\Big(\{s_i\}\times[0,1]\Big)\cup\bigcup\Big([0,1]\times\{t_j\}\Big),$$
where both unions are (at most) countable. Hence, $L^2(\mu)$ splits as a direct sum $V\oplus W$, where $V$ consists of functions of the form $f=\sum_i a_i 1_{\{s_i\}\times[0,1]}$ and $W$ of similar "vertical" functions.
Now declare $T$ to act by multiplication by $0$ on $V$ and multiplication by $1$ on $W$. It's easy to see that there is no consistent choice of $v$.
If you don't want atoms in the counterexample, take instead the $\sigma$-algebra generated by sets of the form $\{s\}\times E'$ and $E\times\{t\}$, where $s,t$ range in $[0,1]$ and $E,E'$ vary among Borel subsets of $[0,1]$. In this case, measurable sets have the form
$$\bigcup(\{s_i\}\times E_i')\cup\bigcup(E_j\times\{t_j\})\cup (E\times E')\cup N,$$
where $E_j,E$ are Borel subsets of $[0,1]\setminus\bigcup\{s_i\}$, $E_i',E'$ are Borel subsets of $[0,1]\setminus\bigcup\{t_j\}$, and finally $N$ is any subset of $\Big(\bigcup\{s_i\}\Big)\times\Big(\bigcup\{t_j\}\Big)$.
Once you have this, you can argue as before.
| 131,842
|
\begin{document}
\title{\textsc{Weighted Projective Lines and Rational Surface Singularities}}
\author{Osamu Iyama}
\address{Osamu Iyama\\ Graduate School of Mathematics\\ Nagoya University\\ Chikusa-ku, Nagoya, 464-8602, Japan}
\email{iyama@math.nagoya-u.ac.jp}
\author{Michael Wemyss}
\address{Michael Wemyss, School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow, G12 8QW,
UK.}
\email{michael.wemyss@glasgow.ac.uk}
\thanks{Osamu Iyama was supported by JSPS Grant-in-Aid for Scientific Research (B)24340004, (B)16H03923, (C)23540045 and (S)15H05738, and Michael Wemyss was supported by EPSRC grant~EP/K021400/1.}
\begin{abstract}
In this paper we study rational surface singularities $R$ with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen--Macaulay modules. We do this by realising $R$ as a certain $\mathbb{Z}$-graded Veronese subring $S^{\ox}$ of the homogeneous coordinate ring $S$ of the Geigle--Lenzing weighted projective line $\mathbb{X}$, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on $\mathbb{X}$. We then give a second proof that these are special CM modules by comparing $\qgr S^{\ox}$ and $\coh\bX$, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that $\qgr S^{\ox}$ is equivalent to $\qgr\Gamma$ where $\Gamma$ is the corresponding reconstruction algebra, and that the degree zero piece of $\Gamma$ coincides with Ringel's canonical algebra. This implies that $\Gamma$ contains the canonical algebra and furthermore $\qgr\Gamma$ is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.
\end{abstract}
\maketitle
\parindent 20pt
\parskip 0pt
\section{Introduction}
\subsection{Motivation and Overview}\label{motivation section}
It is well known that any rational surface singularity has only finitely many indecomposable special CM modules, but it is in general a difficult task to classify and describe them explicitly.
In this paper we use the combinatorial structure encoded in the homogeneous coordinate ring $S$ of the Geigle--Lenzing weighted projective line $\bX$ to solve this problem for a large class of examples arising from star shaped dual graphs, extending our previous work \cite{IW} to cover a much larger class of varieties. In the process, we link $S$, its Veronese subrings, the reconstruction algebra and the canonical algebra, through a range of categorical equivalences.
A hint of a connection between rational surface singularities and the canonical algebra can be found in the lecture notes \cite{Ringel}. In his study of the representation theory of the canonical algebra $\Lambda_{\bp,\bl}$, Ringel drew pictures \cite[p196]{Ringel} including the following one for type $\tilde{E}_7$:
\begin{equation}
\begin{array}{c}
\begin{tikzpicture}[xscale=0.8,yscale=0.8]
\draw[gray] (0,-2) -- (0.5,-2.5);
\draw[gray] (0,-1) -- (1.5,-2.5);
\draw[gray] (0,0) -- (2.5,-2.5);
\draw[gray] (0.5,0.5) -- (3.5,-2.5);
\draw[gray] (1.5,0.5) -- (4.5,-2.5);
\draw[gray] (2.5,0.5) -- (5.5,-2.5);
\draw[gray] (3.5,0.5) -- (6.5,-2.5);
\draw[gray] (4.5,0.5) -- (7.5,-2.5);
\draw[gray] (5.5,0.5) -- (8.5,-2.5);
\draw[gray] (6.5,0.5) -- (9.5,-2.5);
\draw[gray] (7.5,0.5) -- (10.5,-2.5);
\draw[gray] (8.5,0.5) -- (11.5,-2.5);
\draw[gray] (9.5,0.5) -- (12.5,-2.5);
\draw[gray] (10.5,0.5) -- (12.5,-1.5);
\draw[gray] (11.5,0.5) -- (12.5,-0.5);
\draw[gray] (0,0) -- (0.5,0.5);
\draw[gray] (0,-1) -- (1.5,0.5);
\draw[gray] (0,-2) -- (2.5,0.5);
\draw[gray] (0.5,-2.5) -- (3.5,0.5);
\draw[gray] (1.5,-2.5) -- (4.5,0.5);
\draw[gray] (2.5,-2.5) -- (5.5,0.5);
\draw[gray] (3.5,-2.5) -- (6.5,0.5);
\draw[gray] (4.5,-2.5) -- (7.5,0.5);
\draw[gray] (5.5,-2.5) -- (8.5,0.5);
\draw[gray] (6.5,-2.5) -- (9.5,0.5);
\draw[gray] (7.5,-2.5) -- (10.5,0.5);
\draw[gray] (8.5,-2.5) -- (11.5,0.5);
\draw[gray] (9.5,-2.5) -- (12.5,0.5);
\draw[gray] (10.5,-2.5) -- (12.5,-0.5);
\draw[gray] (11.5,-2.5) -- (12.5,-1.5);
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\node (F7) at (7.5,-2.5) [vertex] {};
\node (F8) at (8.5,-2.5) [vertex] {};
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\node (F10) at (10.5,-2.5) [vertex] {};
\node (F11) at (11.5,-2.5) [vertex] {};
\node (F12) at (12.5,-2.5) [vertex] {};
\draw (0.5,0.5) circle (3.5pt);
\node at (0.5,0.85) {$\scriptstyle 0$};
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\node at (4.5,0.85) {$\scriptstyle a_1$};
\draw (6.5,0.5) circle (3.5pt);
\node at (6.5,0.85) {$\scriptstyle c_2$};
\draw (8.5,0.5) circle (3.5pt);
\node at (8.5,0.85) {$\scriptstyle a_2$};
\draw (12.5,0.5) circle (3.5pt);
\node at (12.5,0.85) {$\scriptstyle \omega$};
\draw (3.5,-2.5) circle (3.5pt);
\node at (3.5,-2.85) {$\scriptstyle c_1$};
\draw (6.5,-2.5) circle (3.5pt);
\node at (6.5,-2.85) {$\scriptstyle b$};
\draw (9.5,-2.5) circle (3.5pt);
\node at (9.5,-2.85) {$\scriptstyle c_3$};
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\draw[->,line width=1pt] (F3) -- (R6);
\draw[->,line width=1pt] (R6) -- (F9);
\draw[->,line width=1pt] ($(F9) + (25:4.5pt)$) -- ($(R12) + (-115:4.5pt)$);
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\draw[->,line width=1pt,rounded corners] ($(R0) + (-25:4.5pt)$) -- ($(C4) + (25:2pt)$) -- ($(C5)+ (90:2pt)$) -- ($(C6)+ (90:2pt)$) -- ($(R4)+ (-155:4.5pt)$);
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\end{tikzpicture}
\end{array}\label{RingelPicture}
\end{equation}
What is remarkable is that this picture is identical to one the authors drew in \cite[8.2]{IW} when classifying special CM modules for a certain family of quotient singularities $\mathbb{C}[[x,y]]^G$ with $G\leq \GL(2,\KK)$, namely
\[
\begin{array}{c}
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\draw[gray] (2.5,0.5) -- (5.5,-2.5);
\draw[gray] (3.5,0.5) -- (6.5,-2.5);
\draw[gray] (4.5,0.5) -- (7.5,-2.5);
\draw[gray] (5.5,0.5) -- (8.5,-2.5);
\draw[gray] (6.5,0.5) -- (9.5,-2.5);
\draw[gray] (7.5,0.5) -- (10.5,-2.5);
\draw[gray] (8.5,0.5) -- (11.5,-2.5);
\draw[gray] (9.5,0.5) -- (12.5,-2.5);
\draw[gray] (10.5,0.5) -- (12.5,-1.5);
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\draw[gray] (1.5,-2.5) -- (4.5,0.5);
\draw[gray] (2.5,-2.5) -- (5.5,0.5);
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\draw[gray] (6.5,-2.5) -- (9.5,0.5);
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\draw[gray] (8.5,-2.5) -- (11.5,0.5);
\draw[gray] (9.5,-2.5) -- (12.5,0.5);
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\node (F0) at (0.5,-2.5) [vertex] {};
\node (F1) at (1.5,-2.5) [vertex] {};
\node (F2) at (2.5,-2.5) [vertex] {};
\node (F3) at (3.5,-2.5) [vertex] {};
\node (F4) at (4.5,-2.5) [vertex] {};
\node (F5) at (5.5,-2.5) [vertex] {};
\node (F6) at (6.5,-2.5) [vertex] {};
\node (F7) at (7.5,-2.5) [vertex] {};
\node (F8) at (8.5,-2.5) [vertex] {};
\node (F9) at (9.5,-2.5) [vertex] {};
\node (F10) at (10.5,-2.5) [vertex] {};
\node (F11) at (11.5,-2.5) [vertex] {};
\node (F12) at (12.5,-2.5) [vertex] {};
\draw (0.5,0.5) circle (4.5pt);
\draw (4.5,0.5) circle (4.5pt);
\draw (6.5,0.5) circle (4.5pt);
\draw (8.5,0.5) circle (4.5pt);
\draw (12.5,0.5) circle (4.5pt);
\draw (3.5,-2.5) circle (4.5pt);
\draw (6.5,-2.5) circle (4.5pt);
\draw (9.5,-2.5) circle (4.5pt);
\end{tikzpicture}
\end{array}
\]
Further, although they were not drawn in \cite{IW}, the arrows in \eqref{RingelPicture} are implicit in the calculation of the quiver of the corresponding reconstruction algebra \cite[\S4]{WemGL2}. This paper grew out of trying to give a conceptual explanation for this coincidence, since a connection between the mathematics underpinning the two pictures did not seem to be known.
In fact, the connection turns out to be explained by a very general phenomenon. Recall first that one of the basic properties of the canonical algebra $\Lambda_{\bp,\bl}$ is that there is always a derived equivalence \cite{GL1}
\[
\Db(\coh\bX_{\bp,\bl})\simeq \Db(\mod \Lambda_{\bp,\bl})
\]
where $\coh\bX_{\bp,\bl}$ is the weighted projective line of Geigle--Lenzing (for details, see \S\ref{geom intro}). Thus, to explain the above coincidence, we are led to consider the possibility of linking the weighted projective line, viewed as a Deligne--Mumford stack, to the study of rational surface singularities. However, the weighted projective line $\bX_{\bp,\bl}$ cannot itself be the stack that we are after, since it only has dimension one, and rational surface singularities have, by definition, dimension two.
We need to increase the dimension, and the most naive way to do this is to consider the total space of a line bundle over $\bX_{\bp,\bl}$. We thus choose any member of the grading group $\ox\in\bL$ and consider the total space stack $\bTot(\cO_{\bX}(-\ox))$ (for definition, see \S\ref{geom intro}). From tilting on this and its coarse moduli, under mild assumptions we prove that the Veronese subring $S^{\ox}:=\bigoplus_{i\in\bZ}S_{i\ox}$ is a weighted homogeneous rational surface singularity, giving the first concrete connection between the above two settings. Furthermore, from the stack $\bTot(\cO_{\bX}(-\ox))$ we then describe the special CM $S^{\ox}$-modules, and give precise information regarding the minimal resolution of $\Spec S^{\ox}$ and its derived category. Using this, through a range of categorical equivalences we are then able to relate $\CM^{\bZ}\!S^{\ox}$ and $\vect\bX$, which finally allows us to explain categorically why the above two pictures must be the same.
We now describe our results in detail.
\subsection{Veronese Subrings and Special CM modules}\label{geom intro}
Throughout, let $\KK$ denote an algebraically closed field of characteristic zero. For any $n\geq 0$, choose positive integers $p_1,\hdots,p_n$ with all $p_i\geq 2$ and set $\bp:=(p_1,\hdots,p_n)$.
Furthermore, choose pairwise distinct points $\lambda_1,\hdots,\lambda_n$ in $\bP^1$, and denote $\bl:=(\lambda_1,\hdots,\lambda_n)$. Let $\ell_i(t_0,t_1)\in\KK[t_0,t_1]$ be the linear form defining $\lambda_i$, and write
\[
S_{\bp,\bl}=S:=\frac{\KK[t_0,t_1,x_1,\hdots,x_n]}{(x_i^{p_i}-\ell_i(t_0,t_1)\mid 1\leq i\leq n)}.
\]
Moreover, let $\bL=\bL(p_1,\hdots,p_n)$ denote the abelian group generated by the elements $\ox_1,\hdots,\ox_t$ subject to the relations
$p_1\ox_1=p_2\ox_2=\cdots =p_n\ox_n=:\oc$. With this input $S$ is an $\bL$-graded algebra with $\deg x_i:=\ox_i$ and $\deg t_j:=\oc$, and $\bL$ is a rank one abelian group, possibly containing torsion.
Often we normalize $\bl$ so that $\lambda_1=1$, $\lambda_2=\infty$ and $\lambda_3=1$, however it is important for changing parameters later that we allow ourselves flexibility.
From this, consider the stack
\[
\bX_{\bp,\bl}=\bX:=[(\Spec S_{\bp,\bl}\backslash 0)/\Spec \KK\bL],
\]
with coarse moduli space denoted $X_{\bp,\bl}=X$. It is well known that $X\cong \bP^1$, regardless of $\bp$ and $\bl$ (see \ref{notation throughout}\eqref{notation throughout 2b}).
To increase dimension we then choose $0\neq \ox\in\bL_+$ (for definition see \ref{notation throughout}\eqref{notation throughout 2}) and consider both the Veronese subring $S^{\ox}:=\bigoplus_{i\in\bZ}S_{i\ox}$ and the total space stack
\[
\bTot(\cO_{\bX_{\bp,\bl}}(-\ox)):=[(\Spec S_{\bp,\bl}\backslash 0\times\Spec\KK[t])/\Spec k\bL],
\]
where $\bL$ acts on $t$ with weight $-\ox$. Writing $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form (see \ref{notation throughout}\eqref{notation throughout 2}), we show in \ref{sings on T} that the coarse moduli space $T^{\ox}$ is a surface containing a $\bP^1$, and on that $\bP^1$ complete locally the singularities of $T^{\ox}$ are of the form
\begin{equation}\label{T picture}
\begin{array}{c}
\begin{tikzpicture}[scale=1.5]
\draw (-0.2,0) to [bend left=5] node[pos=0.75,above] {$\scriptstyle \bP^1$} (4.2,0);
\filldraw [black] (0.0,0.01) circle (1pt);
\filldraw [black] (1,0.08) circle (1pt);
\filldraw [black] (2,0.11) circle (1pt);
\filldraw [black] (4,0.01) circle (1pt);
\node at (0,-0.15) {$\scriptstyle \frac{1}{p_1}(1,-a_1)$};
\node at (0,0.2) {$\scriptstyle \lambda_1$};
\node at (1,-0.1) {$\scriptstyle \frac{1}{p_2}(1,-a_2)$};
\node at (1,0.275) {$\scriptstyle \lambda_2$};
\node at (2,-0.075) {$\scriptstyle \frac{1}{p_3}(1,-a_3)$};
\node at (2,0.3) {$\scriptstyle \lambda_3$};
\node at (3,-0.1) {$\scriptstyle \hdots$};
\node at (4,-0.15) {$\scriptstyle \frac{1}{p_n}(1,-a_n)$};
\node at (4,0.2) {$\scriptstyle \lambda_n$};
\end{tikzpicture}
\end{array}
\end{equation}
where for notation see \ref{cyclic quot def}. There is a natural map $\psi\colon T^{\ox}\to\Spec S^{\ox}$ (\S\ref{stack}), and we let $\varphi\colon Y^{\ox}\to T^{\ox}$ denote the minimal resolution of $T^{\ox}$. This datum can be summarized by the following commutative diagram
\begin{equation}
\begin{array}{c}
\begin{tikzpicture}
\node (top 1) at (0,0) {$\bTot(\cO_{\bX}(-\ox))$};
\node (top 2) at (2.5,0) {$\bX$};
\node (bottom 1) at (0,-1.5) {$T^{\ox}$};
\node (bottom 2) at (2.5,-1.5) {$X\cong\mathbb{P}^1$};
\node (min) at (-1.5,-1) {$Y^{\ox}$};
\node (V) at (0,-2.5) {$\Spec S^{\ox}$};
\draw[->] (top 1) -- node[left] {$\scriptstyle g$} (bottom 1);
\draw[->] (top 2) -- node[right] {$\scriptstyle f$} (bottom 2);
\draw[->] (top 1) -- node[above] {$\scriptstyle q$} (top 2);
\draw[->] (bottom 1) -- node[above] {$\scriptstyle p$} (bottom 2);
\draw[->] (min) -- node[above] {$\scriptstyle \varphi$} (bottom 1);
\draw[->] (bottom 1) -- node[right] {$\scriptstyle \psi$} (V);
\draw[->,densely dotted] (min) -- node[below left] {$\scriptstyle \pi$} (V);
\end{tikzpicture}
\end{array}\label{stack diagram 1}
\end{equation}
We remark that the coarse moduli space $T^{\ox}$ is a singular line bundle in the sense of Dolgachev \cite[\S4]{Dolgachev} and Pinkham \cite[\S3]{Pinkham}, which also appears in the work of Orlik--Wagreich \cite{OW} and many others. However, the key difference in our approach is that the grading group giving the quotient is $\bL$ not $\bZ$, and indeed it is the extra combinatorial structure of $\bL$ that allows us to extract the geometry much more easily.
Write $\cE:=\bigoplus_{i\in [0,\oc\,]}\cO_{\bX}(i)$ for the Geigle--Lenzing tilting bundle on $\bX$ \cite{GL1}. Our first main theorem is the following:
\begin{thm}\label{tilting intro}
If $0\neq\ox=\sum_{i=1}^na_i\ox_i+a\oc\in\bL_+$, then with notation as in \eqref{stack diagram 1},
\begin{enumerate}
\item\label{tilting intro 1}\textnormal{(=\ref{tilting on coarse T})} $p^*(\cO_{\bP^1}\oplus\cO_{\bP^1}(1))$ is a tilting bundle on $T^{\ox}$. In particular $H^i(\cO_{T^{\ox}})=0$ for all $i\geq 1$, and $S^{\ox}$ is a rational surface singularity.
\item\label{tilting intro 2}\textnormal{(=\ref{tilting on stack T})} $q^*\cE$ is a tilting bundle on $\bT^{\ox}:=\bTot(\cO_{\bX}(-\ox))$ such that
\[
\begin{tikzpicture}[xscale=1]
\node (a1) at (0,0) {$\Db(\Qcoh\bT^{\ox})$};
\node (a2) at (5,0) {$\Db(\Mod\End_{\bT^{\ox}}(q^*\cE))$};
\node (b1) at (0,-1.5) {$\Db(\Qcoh\bX)$};
\node (b2) at (5,-1.5) {$\Db(\Mod\Lambda_{\bp,\bl})$};
\draw[->] (a1) -- node[above] {$\scriptstyle \RHom_{\bT^{\ox}}(q^*\cE,-)$} node[below] {$\scriptstyle \sim$} (a2);
\draw[->] (b1) -- node[above] {$\scriptstyle \RHom_{\bX}(\cE,-)$} node[below] {$\scriptstyle \sim$} (b2);
\draw[->] (a1) -- node[left] {$\scriptstyle \Rq_*$} (b1);
\draw[->] (a2) -- node[right] {$\mbox{\scriptsize res}$} (b2);
\end{tikzpicture}
\]
commutes, where $\Lambda_{\bp,\bl}$ is the canonical algebra of Ringel.
\item\label{tilting intro 3}\textnormal{(=\ref{stackminres})}
There is a fully faithful embedding $
\Db(\coh Y^{\ox})\hookrightarrow \Db(\coh\bT^{\ox})$.
\item\label{tilting intro 4}\textnormal{(=\ref{stackminres 2A})} If further $(p_i,a_i)=1$ for all $1\leq i\leq n$, then the embedding in \eqref{tilting intro 3} is an equivalence if and only if every $a_i$ is $1$, that is $\ox=\sum_{i=1}^{n}\ox_i+a\oc$.
\end{enumerate}
\end{thm}
The coprime assumption in \ref{tilting intro}\eqref{tilting intro 4} is not restrictive, since we show in \S\ref{changing parameters} that we can always replace $\bX_{\bp,\bl}$ by some equivalent $\bX_{\bp'\!,\bl'}$ for which the coprime assumption holds. See \ref{change parameters so coprime} for full details.
Combining the above tilting result with \eqref{T picture} and a combinatorial argument, we are in fact able to determine the precise dual graph (for definition see \ref{dual graph defin}) of the morphism $\pi$ in \eqref{stack diagram 1}. Recall that for each $\frac{1}{p_i}(1,-a_i)$ in \eqref{T picture} with $a_i\neq 0$, we can consider the Hirzebruch--Jung continued fraction expansion
\begin{equation}
\frac{p_i}{p_i-a_i}=
\alpha_{i1}-\frac{1}{\alpha_{i2} - \frac{1}{\alpha_{i3} -
\frac{1}{(...)}}} :=[\alpha_{i1},\hdots,\alpha_{im_i}],\label{HJ intro}
\end{equation}
with each $\alpha_{ij}\geq 2$; see \S\ref{iseries section} for full details.
\begin{thm}[{=\ref{degen lemma}, \ref{middle SI number}}]\label{dual graph general intro}
Let $0\neq\ox\in\bL_+$ and as above write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. Then the dual graph of the morphism $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ is
\begin{equation}\label{key dual graph}
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=0.8]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-3,1) [vertex]{};
\node (A2) at (-3,2) [vertex] {};
\node (A3) at (-3,3) [vertex] {};
\node (A4) at (-3,4) [vertex] {};
\node (B1) at (-1.5,1) [vertex] {};
\node (B2) at (-1.5,2) [vertex] {};
\node (B3) at (-1.5,3) [vertex] {};
\node (B4) at (-1.5,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,4.25) {};
\node at (0,-0.2) {$\scriptstyle -\beta$};
\node at (-2.6,1) {$\scriptstyle -\alpha_{11}$};
\node at (-2.6,2) {$\scriptstyle -\alpha_{12}$};
\node at (-2.35,3) {$\scriptstyle -\alpha_{1m_1-1}$};
\node at (-2.45,4) {$\scriptstyle -\alpha_{1m_1}$};
\node at (-1.1,1) {$\scriptstyle -\alpha_{21}$};
\node at (-1.1,2) {$\scriptstyle -\alpha_{22}$};
\node at (-0.85,3) {$\scriptstyle -\alpha_{2m_2-1}$};
\node at (-0.95,4) {$\scriptstyle -\alpha_{2m_2}$};
\node at (0.4,1) {$\scriptstyle -\alpha_{31}$};
\node at (0.4,2) {$\scriptstyle -\alpha_{32}$};
\node at (0.65,3) {$\scriptstyle -\alpha_{3m_3-1}$};
\node at (0.55,4) {$\scriptstyle -\alpha_{3m_3}$};
\node at (2.45,1) {$\scriptstyle -\alpha_{n1}$};
\node at (2.45,2) {$\scriptstyle -\alpha_{n2}$};
\node at (2.7,3) {$\scriptstyle -\alpha_{nm_n-1}$};
\node at (2.6,4) {$\scriptstyle -\alpha_{nm_n}$};
\draw (A1) -- (0);
\draw (B1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (B2) -- (B1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (B4) -- (B3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\end{tikzpicture}
\end{array}
\end{equation}
where the arm $[\alpha_{i1},\hdots,\alpha_{im_i}]$ corresponds to $i\in\{1,\ldots,n\}$ with $a_i\neq0$, and the $\alpha_{ij}$ are given by the Hirzebruch--Jung continued fractions in \eqref{HJ intro}.
Furthermore, denoting $v=\#\{i\mid a_i\neq 0\}$ for the number of arms, we have $\beta=a+v$.
\end{thm}
We first establish in \ref{when Y min res} that $\pi$ is the minimal resolution if and only if $\ox\notin[0,\oc\,]$. Theorem~\ref{dual graph general intro} is then proved by splitting into the two cases $\ox\notin[0,\oc\,]$ and $\ox\in[0,\oc\,]$, with the verification in both cases being rather different. Note that the case $\ox\in [0,\oc\,]$ is degenerate as $[0,\oc\,]$ is a finite interval, containing only those $\ox$ of the form $a_i\ox_i$ for some $i$ and some $0\leq a_i<p_i$. In this paper we are mostly interested in special CM modules and these are defined using the minimal resolution: this is why below the condition $\ox\notin[0,\oc\,]$ often appears.
We remark that for $0\neq \ox\in\bL_+$, $S^{\ox}$ is rarely a quotient singularity, and it is even more rare for it to be ADE. Nevertheless, the dual graphs of all quotient singularities $\KK^{2}/G$ (where $G$ is a small subgroup of $\GL(2,\KK)$) are known \cite{Brieskorn}, and so whether $S^{\ox}$ is a quotient singularity can, if needed, be immediately determined by \ref{dual graph general intro}, after contracting ($-1$)-curves if necessary.
One key observation in this paper is that the construction of a minimal resolution $Y^{\ox}\to\Spec S^{\ox}$ allows us not only to construct many rational surface singularities with prescribed dual graphs (by taking suitable Veronese subrings of $S$), but furthermore we can use the stack structure to determine the special CM $S^{\ox}$-modules. Throughout, we denote by $\SCM S^{\ox}$ the category of special CM $S^{\ox}$-modules; for definitions see \S\ref{Prelim ReconAlg}.
As notation, recall that the $i$-series associated to the Hirzebruch--Jung continued fraction expansion $\frac{r}{a}=[\alpha_1,\hdots,\alpha_m]$ is defined as $i_0=r$, $i_1=a$ and $i_{t}=\alpha_{t-1}i_{t-1}-i_{t-2}$ for all $t$ with $2\leq t\leq m+1$, and we denote
\[
I(r,a):=\{i_0,i_1,\hdots,i_{m+1}\}.
\]
As convention $I(r,r)=\emptyset$. Also, for $\oy\in\bL$, we write $S(\oy)^{\ox}:=\bigoplus_{i\in\bZ}S_{\oy+i\ox}$.
\begin{thm}[{=\ref{specials determined thm}, \ref{specials lag approach thm}}]\label{specials general intro}
If $\ox\in\bL_+$ with $\ox\notin [0,\oc\,]$, write $\ox$ in normal form $\ox=\sum_{i=1}^na_i\ox_i+a\oc$.
Then
\[
\SCM S^{\ox}=\add\{S(u\ox_j)^{\ox}\mid j\in [1,n], u\in I(p_j,p_j-a_j) \}.
\]
\end{thm}
This allows us to construct both $R=S^{\ox}$, and its special CM modules, for (almost) every star shaped dual graph. We remark that this is the first time that special CM modules have been classified in any example with infinite CM representation type, and indeed, due to the non--tautness of the dual graph, in an uncountable family of examples. For simplicity in this paper, we restrict the explicitness to certain families of examples, and refer the reader to \S\ref{Sect 5.2} for more details.
By construction, all the special CM $S^{\ox}$-modules have a natural $\bZ$-grading, and we let $N$ denote their sum. By definition the \emph{reconstruction algebra} is defined to be $\Gamma_{\!\ox}:=\End_{S^{\ox}}(N)$, and in this setting it inherits a $\bN$-grading from the grading of the special CM modules in \ref{specials general intro}. In general, it is not generated in degree one over its degree zero piece, but nevertheless the degree zero piece is always some canonical algebra of Ringel. We state the first half of the following result vaguely, giving a much more precise description of the parameters in \ref{deg 0 general prop}.
\begin{prop}\label{deg 0 intro}
Suppose that $x\in\bL_+$ with $\ox\notin [0,\oc\,]$.
\begin{enumerate}
\item\label{deg 0 intro 1} \textnormal{(=\ref{deg 0 general prop})} The degree zero part of $\Gamma_{\!\ox}$ is isomorphic to the canonical algebra $\Lambda_{\bq,\bm}$, for some suitable parameters $(\bq,\bm)$.
\item\label{deg 0 intro 2} \textnormal{(=\ref{grading inherited})} For $\os:=\sum_{i=1}^n\ox_i$, then $\Gamma_{\!\os}$ is generated in degree one over its degree zero piece. Moreover the degree zero piece is the canonical algebra $\Lambda_{\bp,\bl}$.
\end{enumerate}
\end{prop}
\subsection{Geigle--Lenzing Weighted Projective Lines via Rational Surface Singularities}
Motivated by the above, and also the fact that when studying curves it should not matter how we embed them into surfaces (and thus be independent of any self-intersection numbers that appear), we then investigate when $\qgr^\bZ\! S^{\ox}\simeq \coh\bX$.
In very special cases, $\coh\mathbb{X}_{\bp,\bl}$ is already known to be equivalent to $\qgr^\bZ\! R$ for some connected graded commutative ring $R$ \cite[8.4]{GL91}. The nicest situation is when the star-shaped dual graph is of Dynkin type, and further $R$ is the ADE quotient singularity associated to the Dynkin diagram via the McKay correspondence (with a slightly non-standard grading). However, all the previous attempts to link the weighted projective line to rational singularities have taken all self-intersection numbers to be $-2$, which is well-known to restrict the possible configurations to ADE Dynkin type.
One of our main results is the following, which does not even require that $\ox\in\bL_+$.
\begin{thm}[{=\ref{WPL as qgrZ}}]
Suppose that $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ is not torsion, and denote $R:=S^{\ox}$. Then the following conditions are equivalent.
\begin{enumerate}
\item The natural functor $(-)^{\ox}\colon\CM^\bL\!S\to\CM^\bZ\!R$ is an equivalence.
\item The natural functor $(-)^{\ox}\colon\qgr^\bL\!S\to\qgr^\bZ\!R$ is an equivalence.
\item For any $\oz\in\bL$, the ideal $I^{\oz}:=S(\oz)^{\ox}\cdot S(-\oz)^{\ox}$ of $R$ satisfies $\dim_k(R/I^{\oz})<\infty$.
\item $(p_i,a_i)=1$ for all $1\le i\le n$.
\end{enumerate}
\end{thm}
Thus for a non-torsion element $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ of $\bL$, there is an equivalence $\qgr^\bZ\!S_{\bp,\bl}^{\ox}\simeq \coh\bX_{\bp,\bl}$ if and only if $(p_i,a_i)=1$ for all $i$ with $1\leq i\leq n$. Thus, by choosing a suitable $\ox$, the weighted projective line can be defined using only connected $\bN$-graded rational surface singularities. Also, we remark that in the case $(p_i,a_i)\neq 1$ we still have that $\qgr S^{\ox}$ is equivalent to some weighted projective line, but the parameters are no longer $(\bp,\bl)$. We leave the details to \S\ref{qgr Veronese section}.
Combining the above gives our next main result.
\begin{cor}[{=\ref{WPL as qgrZ}, \ref{qgrR via qgrLambda}}]\label{S=R=Gamma intro}
Let $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. If $(p_i,a_i)=1$ for all $1\leq i\leq n$, then
\[
\coh\bX_{\bp,\bl}\simeq \qgr^\bZ\! S^{\ox}\simeq\qgr^\bZ\!\Gamma_{\!\ox},
\]
and further $\Gamma_{\!\ox}$ is an $\bN$-graded ring, with zeroth piece a canonical algebra.
\end{cor}
In the case when $(p_i,a_i)\neq 1$ we have a similar result but again there is a change of parameters, so we refer the reader to \ref{qgrR via qgrLambda} for details. Combining \ref{S=R=Gamma intro} with \ref{deg 0 intro}\eqref{deg 0 intro 2}, we can view the weighted projective line $\bX_{\bp,\bl}$ as an Artin--Zhang noncommutative projective scheme over the canonical algebra $\Lambda_{\bp,\bl}$ \cite{Minamoto}.
\subsection{Some Particular Veronese Subrings}
We then investigate in detail the particular Veronese subrings $S^{\os_a}$ for $\os_a:=\os+a\oc$ for some $a\geq 0$, and the special case $\os:=\sum_{i=1}^n\ox_i$. We call $S^{\os_a}$ the \emph{$a$-Wahl Veronese subring}, and in this case, the singularities in \eqref{T picture} are all of the form $\frac{1}{p_i}(1,-1)$, which are cyclic Gorenstein and so have a resolution consisting of only ($-2$)-curves. Thus resolving the singularities in \eqref{T picture}, by \ref{dual graph general intro} we see that the dual graph of the minimal resolution of $\Spec S^{\os_a}$ is
\begin{equation}\label{s Veron dual graph}
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=1]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-3,1) [vertex] {};
\node (A2) at (-3,2) [vertex] {};
\node (A3) at (-3,3) [vertex] {};
\node (A4) at (-3,4) [vertex] {};
\node (B1) at (-1.5,1) [vertex] {};
\node (B2) at (-1.5,2) [vertex] {};
\node (B3) at (-1.5,3) [vertex] {};
\node (B4) at (-1.5,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,4.25) {};
\node at (0,-0.2) {$\scriptstyle -n-a$};
\node at (-2.7,1) {$\scriptstyle -2$};
\node at (-2.7,2) {$\scriptstyle -2$};
\node at (-2.7,3) {$\scriptstyle -2$};
\node at (-2.7,4) {$\scriptstyle -2$};
\node at (-1.2,1) {$\scriptstyle -2$};
\node at (-1.2,2) {$\scriptstyle -2$};
\node at (-1.2,3) {$\scriptstyle -2$};
\node at (-1.2,4) {$\scriptstyle -2$};
\node at (0.3,1) {$\scriptstyle -2$};
\node at (0.3,2) {$\scriptstyle -2$};
\node at (0.3,3) {$\scriptstyle -2$};
\node at (0.3,4) {$\scriptstyle -2$};
\node at (2.3,1) {$\scriptstyle -2$};
\node at (2.3,2) {$\scriptstyle -2$};
\node at (2.3,3) {$\scriptstyle -2$};
\node at (2.3,4) {$\scriptstyle -2$};
\draw (A1) -- (0);
\draw (B1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (B2) -- (B1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (B4) -- (B3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(2,1) -- (2,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_n-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(0,1) -- (0,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_3-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(-1.5,1) -- (-1.5,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_2-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(-3,1) -- (-3,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_1-1$};
\end{tikzpicture}
\end{array}
\end{equation}
where there are $n$ arms, and the number of vertices on arm $i$ is $p_i-1$. It turns out that these particular Veronese subrings have many nice properties; not least by \ref{tilting intro}\eqref{tilting intro 4} they are precisely the Veronese subrings for which
\[
\Db(\coh Y^{\ox})\hookrightarrow \Db(\coh\bT^{\ox})
\]
is an equivalence. In \S\ref{domestic section} we investigate $S^{\os_a}$ in the case when $(p_1,p_2,p_3)$ forms a Dynkin triple, in which case $S^{\os_a}$ is isomorphic to a quotient singularity by some finite subgroup of $\GL(2,\KK)$ of type $\mathbb{D}$, $\mathbb{T}$, $\mathbb{O}$ or $\mathbb{I}$ (see \ref{Veronese=quotient} for details). In this situation $S^{\os_a}$ and its reconstruction algebra have a very nice relationship to the preprojective algebra of the canonical algebra, and this is what turns out to explain the motivating coincidence from \S\ref{motivation section} in \ref{equiv last intro} below.
For arbitrary parameters $(\bp,\bl)$, the Veronese subring $S^{\os}$ has a particularly nice form.
\begin{thm}[{=\ref{S is generated}}]
For any $\mathbb{X}_{\bp,\bl}$, $S^{\os}$ is generated by the homogeneous elements
\begin{eqnarray*}
\mathsf{u}_i&:=&\left\{\begin{array}{ll}x_1^{p_1+p_2}x_3^{p_2}\hdots x_n^{p_2}&i=1,\\
x_2^{p_1+p_2}x_3^{p_1}\hdots x_n^{p_1}&i=2,\\
-x_1^{p_i}x_2^{p_2+p_i}x_3^{p_i}\hdots \widehat{x_i}\hdots x_n^{p_i}&3\leq i\leq n,\end{array}\right.\\
\mathsf{v}&:=&x_1x_2\hdots x_n.
\end{eqnarray*}
\end{thm}
\begin{prop}[{=\ref{dual graph assignment}}]
With notation as above, the modules $S(u\ox_j)^{\os}$ appearing in \ref{specials general intro} are precisely the following ideals of $S^{\os}$, and furthermore they correspond to the dual graph of the minimal resolution of $\Spec S^{\os}$ \eqref{s Veron dual graph} in the following way:
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=1,bend angle=30, looseness=1]
\node (0) at (0,0) {$\scriptstyle (\mathsf{v}^{p_2},\mathsf{u}_1)$};
\node (A1) at (-3.25,1) {$\scriptstyle (\mathsf{v}^{p_2+1},\mathsf{u}_1)$};
\node (A2) at (-3.25,2) {$\scriptstyle (\mathsf{v}^{p_2+2},\mathsf{u}_1)$};
\node (A3) at (-3.25,3) {$\scriptstyle (\mathsf{v}^{p_2+p_1-2},\mathsf{u}_1)$};
\node (A4) at (-3.25,4) {$\scriptstyle (\mathsf{v}^{p_2+p_1-1},\mathsf{u}_1)$};
\node (B1) at (-1.5,1) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-1})$};
\node (B2) at (-1.5,2) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-2})$};
\node (B3) at (-1.5,3) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^2)$};
\node (B4) at (-1.5,4) {$\scriptstyle (\mathsf{u}_1,\mathsf{v})$};
\node (C1) at (0,1) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-1})$};
\node (C2) at (0,2) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-2})$};
\node (C3) at (0,3) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{2})$};
\node (C4) at (0,4) {$\scriptstyle (\mathsf{u}_3,\mathsf{v})$};
\node (n1) at (2,1) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-1})$};
\node (n2) at (2,2) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-2})$};
\node (n3) at (2,3) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{2})$};
\node (n4) at (2,4) {$\scriptstyle (\mathsf{u}_n,\mathsf{v})$};
\node at (-3.25,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,2.5) {$\hdots$};
\draw [-] (A1)+(-30:8.5pt) -- ($(0) + (160:11.5pt)$);
\draw [-] (B1) --(0);
\draw [-] (C1) --(0);
\draw [-] (n1) --(0);
\draw [-] (A2) -- (A1);
\draw [-] (B2) --(B1);
\draw [-] (C2) --(C1);
\draw [-] (n2) --(n1);
\draw [-] (A4) -- (A3);
\draw [-] (B4) --(B3);
\draw [-] (C4) --(C3);
\draw [-] (n4) -- (n3);
\end{tikzpicture}
\end{array}
\]
\end{prop}
The relations between $\mathsf{u}_1,\hdots,\mathsf{u}_n,\mathsf{v}$ turn out to be easy to describe, and remarkably have already appeared in the literature. It is well-known \cite[3.6]{Wahl} that there is a family of rational surface singularities $R_{\bp,\bl}$ where the dual graph of the minimal resolution of $\Spec R_{\bp,\bl}$ is precisely \eqref{dual graph} with $a=0$. Indeed, in \cite{Wahl} $R_{\bp,\bl}$ is defined as follows: given the same data $(\bp,\bl)$ as above normalised so that $\lambda_1=(1:0)$, $\lambda_2=(0:1)$ and $\lambda_3,\hdots,\lambda_n\in\KK^*$ are pairwise distinct, we can consider the commutative $\KK$-algebra $R_{\bp,\bl}$, generated by $u_1,\hdots,u_n,v$ subject to the relations given by the $2\times 2$ minors of the matrix
\[
\left(
\begin{array}{ccccc}
u_2&u_3&\hdots&u_{n}&v^{p_2}\\
v^{p_1}&\lambda_3u_3+v^{p_3}&\hdots&\lambda_nu_n+v^{p_n}&u_1
\end{array}
\right)
\]
This is a connected $\mathbb{N}$-graded ring graded by $\deg v:=1$, $\deg u_1:=p_2$, $\deg u_2:=p_1$ and $\deg u_i:=p_i$ for all $3\leq i\leq n$.
We show that $S^{\os}$ recovers precisely the above $R_{\bp,\bl}$.
\begin{thm}[{=\ref{Veronese result}}]\label{R and S intro}
There is an isomorphism $R_{\bp,\bl}\cong S^{\os}$ of $\bZ$-graded algebras given by $u_i\mapsto \mathsf{u}_i$ for $1\leq i\leq n$ and $v\mapsto\mathsf{v}$.
\end{thm}
Thus the Veronese method we develop in this paper for constructing rational surface singularities recovers as a special case the example of \cite{Wahl}, but in a way suitable for arbitrary labelled star-shaped graphs, and also in a way suitable for obtaining the special CM modules.
We then present the reconstruction algebra of $R_{\bp,\bl}\cong S^{\os}$, since again in this situation it has a particularly nice form. In principle, using \ref{specials general intro}, we can do this for any Veronese $S^{\ox}$ with $0\neq\ox\in\bL_+$, but for notational ease we restrict ourselves to the case $\ox=\os$.
\begin{thm}[=\ref{recon relations}]
The reconstruction algebra $\Gamma_{\bp,\bl}$ of $R_{\bp,\bl}$ can be written explicitly as a quiver with relations. It is the path algebra of the double of the quiver $Q_{\bp}$ of the canonical algebra, subject to the relations induced by the canonical relations, and furthermore at every vertex, all 2-cycles that exist at that vertex are equal.
\end{thm}
We refer the reader to \ref{recon relations} for more details, but remark that the reconstruction algebra was originally invented in order to extend the notion of a preprojective algebra to a more general geometric setting. In our situation here, the reconstruction algebra is not quite the preprojective algebra of the canonical algebra $\Lambda_{\bp,\bl}$, but the relations in \ref{recon relations} are mainly of the same form as the preprojective relations; the reconstruction algebra should perhaps be thought of as a better substitute.
In the last section of the paper, finally we can explain the coincidence of the two motivating pictures, as a consequence of the following result.
\begin{thm}[=\ref{equiv last}]\label{equiv last intro}
Let $R$ be the $(m-3)$-Wahl Veronese subring associated with $(p_1,p_2,p_3)=(2,3,3)$, $(2,3,4)$ or $(2,3,5)$ and $m\ge3$, and $\mathfrak{R}$ its completion. Let $G\leq \bL$ be the cyclic group generated by $(h(m-2)+1)\ow$, where $h=6$, $12$ or $30$ respectively. Then
\begin{enumerate}
\item There are equivalences $\vect\bX\simeq\CM^\bZ\!R$ and
\[
F\colon (\vect\bX)/G\xrightarrow{\sim}\CM\mathfrak{R}.
\]
\item For the canonical tilting bundle $\mathcal{E}$ on $\bX$, we have $\SCM\mathfrak{R}=\add F\mathcal{E}$.
\end{enumerate}
\end{thm}
\medskip
\noindent
\textbf{Acknowledgements.} The authors thank Kazushi Ueda and Atsushi Takahashi for many helpful comments and remarks.
They also thank Mitsuyasu Hashimoto, Ryo Takahashi and Yuji Yoshino for valuable discussions on reflexive modules.
\medskip
\noindent
\textbf{Conventions.} Throughout, $\KK$ denotes an algebraically closed field of characteristic zero. All modules will be right modules, and for a ring $A$ write $\mod A$ for the category of finitely generated right $A$-modules. If $G$ is an abelian group and $A$ is a noetherian $G$-graded ring, $\gr^G\! A$ will denote the category of finitely generated $G$-graded right $A$-modules. Throughout when composing maps $fg$ will mean $f$ then $g$, similarly for arrows $ab$ will mean $a$ then $b$. Note that with this convention $\Hom_R(M,N)$ is an $\End_R(M)^{\op}$-module and an $\End_R(N)$-module. For $M\in\mod A$ we denote $\add M$ to be the full subcategory consisting of summands of finite direct sums of copies of $M$.
\section{Preliminaries}\label{prelim}
\subsection{Notation} We first fix notation. For $n\geq 0$, choose positive integers $p_1,\hdots,p_n$ with all $p_i\geq 2$, and set $\bp:=(p_1,\hdots,p_n)$. Likewise, for pairwise distinct points $\lambda_1,\hdots,\lambda_n\in\mathbb{P}^1$, and set $\bl:=(\lambda_1,\hdots,\lambda_n)$. Let $\ell_i(t_0,t_1)\in\KK[t_0,t_1]$ be the linear form defining $\lambda_i$.
\begin{notation}\label{notation throughout}
To this data we associate
\begin{enumerate}
\item\label{notation throughout 2} The abelian group $\bL=\bL(p_1,\hdots,p_n)$ generated by the elements $\ox_1,\hdots,\ox_n$ subject to the relations
$p_1\ox_1=p_2\ox_2=\cdots =p_n\ox_n=:\oc$. Note that $\bL$ is an ordered group with positive elements $\bL_+=\sum_{i=1}^n\mathbb{Z}_{\geq 0}\ox_i$. Since $\bL/\bZ\oc\cong\prod_{i=1}^n\bZ/p_i\bZ$ canonically, each $\ox\in\bL$ can be written uniquely in \emph{normal form} as $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ with $0\leq a_i<p_i$ and $a\in\mathbb{Z}$. Then $\ox$ belongs to $\bL_+$ if and only if $a\geq0$.
\item\label{notation throughout 2b} The commutative $\KK$-algebra $S_{\bp,\bl}$ defined as
\[
S_{\bp,\bl}:=\frac{\KK[t_0,t_1,x_1,\hdots,x_n]}{(x_i^{p_i}-\ell_i(t_0,t_1)\mid 1\leq i\leq n)}.
\]
As in the introduction, this is $\bL$-graded by $\deg x_i:=\ox_i$, and defines the weighted projective line $\mathbb{X}_{\bp,\bl}:=[(\Spec S\backslash 0)/\Spec \KK\bL]$ and its coarse moduli space $X_{\bp,\bl}:=(\Spec S\backslash 0)/\Spec \KK\bL$.
The open cover $\Spec S\backslash0=U_0\cup U_1$ with $U_i:=\Spec S_{t_i}$ induces an open cover $X_{\bp,\bl}=X_0\cup X_1$ with $X_i:=\Spec (S_{t_i})_0$, where $(S_{t_i})_0$ is the degree zero part of $S_{t_i}$.
Since by inspection $(S_{t_i})_0=\KK[t_{1-i}/t_i]$, it follows that $X_{\bp,\bl}\cong\bP^1$.
\end{enumerate}
When $n\geq2$, often we choose $\lambda_1=(1:0)$ and $\lambda_2=(0:1)$, in which case $\ell_1=t_1$, $\ell_2=t_0$ and $\ell_i=\lambda_it_0-t_1$ for $3\le i\le n$, and there is a presentation
\[
S_{\bp,\bl}=\frac{\KK[x_1,\hdots,x_n]}{(x_i^{p_i}+x_1^{p_1}-\lambda_ix_2^{p_2}\mid 3\leq i\leq n)}.
\]
Moreover, when $n\geq 3$, we can further associate
\begin{enumerate}[resume]
\item The quiver
\[
\begin{array}{cc}
Q_{\bp}:= &
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=1]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-3,1) [vertex] {};
\node (A2) at (-3,2) [vertex] {};
\node (A3) at (-3,3) [vertex] {};
\node (A4) at (-3,4) [vertex] {};
\node (B1) at (-1.5,1) [vertex] {};
\node (B2) at (-1.5,2) [vertex] {};
\node (B3) at (-1.5,3) [vertex] {};
\node (B4) at (-1.5,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,5) [cvertex] {};
\draw [->] (A1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(0);
\draw [->] (B1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(0);
\draw [->] (C1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(0);
\draw [->] (n1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_n$}(0);
\draw [->] (A2) -- node[right] {$\scriptstyle x_1$} (A1);
\draw [->] (B2) -- node[right] {$\scriptstyle x_2$}(B1);
\draw [->] (C2) -- node[right] {$\scriptstyle x_3$}(C1);
\draw [->] (n2) -- node[right] {$\scriptstyle x_n$}(n1);
\draw [->] (A4) -- node[right] {$\scriptstyle x_1$} (A3);
\draw [->] (B4) -- node[right] {$\scriptstyle x_2$}(B3);
\draw [->] (C4) -- node[right] {$\scriptstyle x_3$}(C3);
\draw [->] (n4) -- node[right] {$\scriptstyle x_n$}(n3);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(A4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(B4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(C4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_n$}(n4);
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(2,1) -- (2,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_n-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(0,1) -- (0,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_3-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(-1.5,1) -- (-1.5,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_2-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(-3,1) -- (-3,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_1-1$};
\end{tikzpicture}
\end{array}
\end{array}
\]
(where there are $n$ arms, and the number of vertices on arm $i$ is $p_i-1$).
\item The \emph{canonical algebra} $\Lambda_{\bp,\bl}$, namely the path algebra of the quiver $Q_{\bp}$ subject to the relations
\[
I:=\langle x_1^{p_1}-\lambda_ix_2^{p_2}+x_i^{p_i}\mid 3\leq i\leq n\rangle.
\]
There is a degenerate definition of the canonical algebra if $1\leq n\leq 2$; see \cite{GL1}.
\item\label{notation throughout 5} The commutative $\KK$-algebra $R_{\bp,\bl}$, generated by $u_1,\hdots,u_n,v$ subject to the relations given by the $2\times 2$ minors of the matrix
\[
\left(
\begin{array}{ccccc}
u_2&u_3&\hdots&u_{n}&v^{p_2}\\
v^{p_1}&\lambda_3u_3+v^{p_3}&\hdots&\lambda_nu_n+v^{p_n}&u_1
\end{array}
\right)
\]
This is a connected $\mathbb{N}$-graded ring graded by $\deg u_1:=p_2$, $\deg u_2:=p_1$, $\deg v:=1$, and $\deg u_i:=p_i$ for all $3\leq i\leq n$.
\end{enumerate}
We will also consider
\begin{enumerate}[resume]
\item Star-shaped graphs of the form
\begin{equation}\label{dual graph}
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=1]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-3,1) [vertex]{};
\node (A2) at (-3,2) [vertex] {};
\node (A3) at (-3,3) [vertex] {};
\node (A4) at (-3,4) [vertex] {};
\node (B1) at (-1.5,1) [vertex] {};
\node (B2) at (-1.5,2) [vertex] {};
\node (B3) at (-1.5,3) [vertex] {};
\node (B4) at (-1.5,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,4.25) {};
\node at (0,-0.2) {$\scriptstyle -\beta$};
\node at (-2.6,1) {$\scriptstyle -\alpha_{11}$};
\node at (-2.6,2) {$\scriptstyle -\alpha_{12}$};
\node at (-2.35,3) {$\scriptstyle -\alpha_{1n_1-1}$};
\node at (-2.45,4) {$\scriptstyle -\alpha_{1n_1}$};
\node at (-1.1,1) {$\scriptstyle -\alpha_{21}$};
\node at (-1.1,2) {$\scriptstyle -\alpha_{22}$};
\node at (-0.85,3) {$\scriptstyle -\alpha_{2n_2-1}$};
\node at (-0.95,4) {$\scriptstyle -\alpha_{2n_2}$};
\node at (0.4,1) {$\scriptstyle -\alpha_{31}$};
\node at (0.4,2) {$\scriptstyle -\alpha_{32}$};
\node at (0.65,3) {$\scriptstyle -\alpha_{3n_3-1}$};
\node at (0.55,4) {$\scriptstyle -\alpha_{3n_3}$};
\node at (2.45,1) {$\scriptstyle -\alpha_{v1}$};
\node at (2.45,2) {$\scriptstyle -\alpha_{v2}$};
\node at (2.7,3) {$\scriptstyle -\alpha_{vn_v-1}$};
\node at (2.6,4) {$\scriptstyle -\alpha_{vn_v}$};
\draw (A1) -- (0);
\draw (B1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (B2) -- (B1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (B4) -- (B3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\end{tikzpicture}
\end{array}
\end{equation}
where there are $v\geq 2$ arms, each $n_i\geq 1$, each $\alpha_{ij}\geq 2$ and $\beta\geq 1$. Later, we will assume $\beta\geq v$.
\end{enumerate}
\end{notation}
Let us briefly recall here some basic properties of $S=S_{\bp,\bl}$ from \cite{GL1} (see also \cite[3.15]{HIMO}) that we will require later.
It is elementary that $S$ is an \emph{$\bL$-domain}, i.e.\ a product of non-zero homogeneous elements is again non-zero. Recall that an element $x\in S$ is called an \emph{$\bL$-prime} if $S/(x)$ is an $\bL$-domain.
\begin{prop}\label{L-factorial}
$S$ is \emph{$\bL$-factorial}, i.e.\ every non-zero homogeneous element in $S$ is a product of $\bL$-prime elements in $S$.
\end{prop}
\subsection{Preliminaries on Rational Surface Singularities}
We briefly review some combinatorics associated to rational surface singularities. Recall that a commutative $\KK$-algebra $R$ is called a \emph{rational surface singularity} if $\dim R=2$ and there exists a resolution $f\colon X\to\Spec R$ such that $\mathbf{R}f_*\mathcal{O}_X=\mathcal{O}_R$. If this property holds for one resolution, it holds for all resolutions \cite[5.10]{KM}.
In our setting later $R$ will be a rational surface singularity with a unique singular point, at the origin. Completing at this maximal ideal to give $\mathfrak{R}$, in the minimal resolution $Y\to\Spec \mathfrak{R}$ the fibre above the origin is well-known to be a tree (i.e.\ a finite connected graph with no cycles) of $\mathbb{P}^1$s denoted $\{ E_i\}_{i\in I}$. Their self-intersection numbers satisfy $E_i\cdot E_i\leq -2$, and moreover the intersection matrix $(E_i\cdot E_j)_{i,j\in I}$ is negative definite. We encode the intersection matrix in the form of the labelled dual graph:
\begin{defin}\label{dual graph defin}
Suppose that $\{ E_i \}_{i\in I}$ is a collection of $\mathbb{P}^1$s forming the exceptional locus in a resolution of some affine rational surface singularity. The dual graph is defined as follows: for each curve draw a vertex, and join two vertices if and only if the corresponding curves intersect. Furthermore label every vertex with the self-intersection number of the corresponding curve.
\end{defin}
Thus, given a complete local rational surface singularity, we obtain a labelled tree. Conversely, suppose that $T$ is a tree, with vertices denoted $E_1,\hdots,E_n$, labelled with integers $w_1,\hdots,w_n$. To this data we associate the symmetric matrix $M_T=(b_{ij})_{1\leq i,j\leq n}$ with $b_{ii}$ defined by $b_{ii}:=w_i$, and $b_{ij}$ (with $i\neq j$) defined to be the number of edges linking the vertices $E_i$ and $E_j$. We denote the free abelian group generated by the vertices $E_i$ by $\cZ$, and call its elements \emph{cycles}. The matrix $M_T$ defines a symmetric bilinear form $(-,-)$ on $\cZ$ and in analogy with the geometry, we will often write $Y\cdot Z$ instead of $(Y,Z)$, and consider
\[
\cZ_{\top}:=\{ Z=\sum_{i=1}^na_iE_i\in\cZ\mid Z\neq 0, \mbox{ all }a_i\geq 0, \mbox{ and }Z\cdot E_i\leq 0 \mbox{ for all } i \}.
\]
If there exists $Z\in\cZ_{\top}$ such that $Z\cdot Z<0$, then automatically $M_T$ is negative definite \cite[Prop 2(ii)]{Artin}. In this case, $\cZ_{\top}$ admits a unique smallest element $Z_f$, called the \emph{fundamental cycle}. Whenever all the coefficients in $Z_f$ are one, the fundamental cycle is said to be \emph{reduced}.
We now consider the case of the labelled graph \eqref{dual graph} and calculate some combinatorics that will be needed later. Denoting the set of vertices of \eqref{dual graph} by $I$, considering $Z:=\sum_{i\in I}E_i$ it is easy to see that
\begin{equation}\label{Zf dot Ei}
\begin{array}{cc}
(-Z\cdot E_i)_{i\in I}=&
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=0.75]
\node (0) at (0,0) {$\scriptstyle \beta-v$};
\node (A1) at (-3,1) {$\scriptstyle \alpha_{11}-2$};
\node (A2) at (-3,2) {$\scriptstyle \alpha_{12}-2$};
\node (A3) at (-3,3) {$\scriptstyle \alpha_{1n_1-1}-2$};
\node (A4) at (-3,4) {$\scriptstyle \alpha_{1n_1}-1$};
\node (B1) at (-1.5,1) {$\scriptstyle \alpha_{21}-2$};
\node (B2) at (-1.5,2) {$\scriptstyle \alpha_{22}-2$};
\node (B3) at (-1.5,3) {$\scriptstyle \alpha_{2n_2-1}-2$};
\node (B4) at (-1.5,4) {$\scriptstyle \alpha_{2n_2}-1$};
\node (C1) at (0,1) {$\scriptstyle \alpha_{31}-2$};
\node (C2) at (0,2) {$\scriptstyle \alpha_{32}-2$};
\node (C3) at (0,3) {$\scriptstyle \alpha_{3n_3-1}-2$};
\node (C4) at (0,4) {$\scriptstyle \alpha_{3n_3}-1$};
\node (n1) at (2,1) {$\scriptstyle \alpha_{v1}-2$};
\node (n2) at (2,2) {$\scriptstyle \alpha_{v2}-2$};
\node (n3) at (2,3) {$\scriptstyle \alpha_{vn_v-1}-2$};
\node (n4) at (2,4) {$\scriptstyle \alpha_{vn_v}-1$};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,4.25) {};
\draw (A1) -- (0);
\draw (B1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (B2) -- (B1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (B4) -- (B3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\end{tikzpicture}
\end{array}
\end{array}
\end{equation}
and so $Z$ satisfies $Z\cdot E_i\leq 0$ for all $i\in I$ if and only if $\beta\geq v$. Since $\cZ_{\top}$ does not contain elements smaller than $Z$, the fundamental cycle $Z_f$ is given by $Z=\sum_{i\in I}E_i$ if and only if $\beta\geq v$. In this case $Z_f$ is reduced.
We remark that in general there will be many singularities with dual graph \eqref{dual graph}, and indeed a labelled graph $T$ is called \emph{taut} if there exists a unique (up to isomorphism in the category of complete local $\KK$-algebras) rational surface singularity which has $T$ for its dual graph of its minimal resolution. It is well known that the labelled graph \eqref{dual graph} is taut if and only if $n=3$ \cite{Laufer}.
\subsection{Preliminaries on Reconstruction Algebras}\label{Prelim ReconAlg}
Let $R$ be a rational surface singularity. A CM $R$-module $M$ is called \emph{special} if $\Ext^1_R(M,R)=0$ \cite{IW}, and we write $\SCM R$ for the category of special CM $R$-modules.
The following local-to-global lemma is useful. In particular, if $R$ has a unique singular point $\m$, to conclude that $\add M=\SCM R$ it suffices to check this complete locally at $\m$.
\begin{lemma}\label{Check locally gen}
Let $R$ be a rational surface singularity, and $M\in \CM R$. If $\add \widehat{M}_\m=\SCM \widehat{R}_\m$ for all $\m\in\Max R$, then $\add M=\SCM R$.
\end{lemma}
\begin{proof}
Since Ext groups localise and complete, certainly $M\in\SCM R$ and thus $\add M\subseteq \SCM R$. Next, let $X\in\SCM R$. Then $\add\widehat{X}_\m\subseteq\SCM \widehat{R}_{\m}$ for all $\m\in\Max R$, so by assumption $\add\widehat{X}_\m\subseteq\add \widehat{M}_\m$ for all $\m\in\Max R$. By \cite[2.26]{IW4} we conclude that $\add X\subseteq\add M$, so $X\in\add M$ and thus $\add M\supseteq \SCM R$ also holds.
\end{proof}
The following asserts that a global additive generator of $\SCM R$ exists, regardless of the number of points in the singular locus.
\begin{thm}[{=\cite{VdB1d}}]\label{recon Db min res}
Let $R$ be a rational surface singularity, and $\pi\colon X\to \Spec R$ the minimal resolution. Then
\begin{enumerate}
\item There exists $M\in\SCM R$ such that $\SCM R=\add M$.
\item There is a triangle equivalence $\Db(\mod\End_R(M))\cong\Db(\coh X)$.
\end{enumerate}
\end{thm}
\begin{proof}
This is known but usually only stated when $R$ is complete, so for the convenience of the reader we provide a proof. By \cite[3.2.5]{VdB1d} there is a progenerator $\cO_X\oplus\cM$ for the category of perverse sheaves (with perversity $-1$), which induces an equivalence
\[
\Db(\mod\End_X(\cO_X\oplus\cM))\cong\Db(\coh X).
\]
There is an isomorphism $\End_X(\cO_X\oplus\cM)\cong\End_R(R\oplus \pi_*\cM)$ by \cite[4.1]{DW2}. Furthermore, $\cO_X\oplus\cM$ remains a progenerator under flat base change \cite[3.1.6]{VdB1d}, so $\add \widehat{M}_\m=\SCM \widehat{R}_\m$ by \cite{Wunram,IW}. The result then follows using \ref{Check locally gen}.
\end{proof}
\begin{defin}
For any $M\in \SCM R$ such that $\SCM R=\add M$, we call $\End_R(M)$ the reconstruction algebra.
\end{defin}
In this global setting, the reconstruction algebra is only defined up to Morita equivalence. Only after completing $R$, or in certain other settings (see \ref{Gamma convention remark}), will there be a canonical choice.
When $\mathfrak{R}$ is a complete local rational surface singularity with minimal resolution $X\to\Spec\mathfrak{R}$, there is a much more explicit description of the additive generator of $\SCM \mathfrak{R}$. Let $\{E_i\mid i\in I\}$ denote the exceptional curves, then for each $i\in I$, by \cite{Wunram} there exists a CM $\mathfrak{R}$-module $M_i$ such that $H^1(\mathcal{M}_i^\vee)=0$ and $c_1(\mathcal{M}_i)\cdot E_j=\delta_{ij}$ hold, where $\mathcal{M}_i:=\pi^*(M_i)^{\vee\vee}$ for $(-)^\vee=\sHom_X(-,\cO_X)$. It is shown in \cite[1.2]{Wunram} that $\{M_i\mid i\in I\}$ are precisely the indecomposable non-free objects in $\SCM \mathfrak{R}$, hence $\mathfrak{R}\oplus\bigoplus_{i\in I}M_i$ is the natural additive generator for $\SCM\mathfrak{R}$.
\begin{defin}
Let $\mathfrak{R}$ be a complete local rational surface singularity. We call $\Gamma:=\End_{\mathfrak{R}}(\mathfrak{R}\oplus(\bigoplus_{i\in I}M_i))$
the \emph{reconstruction algebra} of $\mathfrak{R}$.
\end{defin}
\begin{remark}\label{Gamma convention remark}
If $R$ is a rational surface singularity with a unique singular point, and there exist $L_i\in \CM R$ such that $\widehat{L}_i\cong M_i$ for all $i$, then we also use the letter $\Gamma$ to denote the particular reconstruction algebra
\[
\Gamma:=\End_R(R\oplus\bigoplus_{i\in I}L_i)
\]
of $R$. Such $L_i$ are not guaranteed to exist, in general.
\end{remark}
In the complete local setting, the quiver of $\Gamma$, and the number of its relations, is completely determined by the intersection theory.
\begin{thm}[{=\cite[3.3]{WemGL2}}]\label{GL2 for R}
Let $\mathfrak{R}$ be a complete local rational surface singularity. The quiver and the numbers of relations of $\Gamma$ is given as follows: for every $i\in I$ associate a vertex labelled $i$ corresponding to $M_i$, and also associate a vertex labelled $\begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}$ corresponding to $\mathfrak{R}$. Then the number of arrows and relations between the vertices is
\[
\begin{tabular}{*4c}
\toprule
&&Number of arrows&Number of relations\\
\midrule
$i\rightarrow j$&&$(E_i\cdot E_j)_+$ & $(-1-E_i\cdot E_j)_+$\\
$\begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}\rightarrow\begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}$&&$0$&$-Z_K\cdot Z_f+1=-1-Z_f\cdot Z_f$\\
$i\rightarrow \begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}$&&$-E_i\cdot Z_f$&$0$\\
$\begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}\rightarrow i$&& $((Z_K-Z_f)\cdot E_i)_+$& $((Z_K-Z_f)\cdot E_i)_-$\\
\bottomrule\\
\end{tabular}
\]
where for $a\in\bZ$
\[
a_+:=\left\{\begin{array}{ccc}a& \t{if}& a\geq 0\\ 0 & \t{if}&a<0\end{array}\right. \quad \t{and}\quad a_-=\left\{\begin{array}{rcc}0& \t{if} &a\geq 0\\ -a & \t{if}&a<0\end{array}\right. ,
\]
and the canonical cycle $Z_K$ is by definition the rational cycle defined by the condition $Z_K\cdot E_i=E_i^2+2$ for all $i\in I$.
\end{thm}
\subsection{Hirzebruch--Jung Continued Fraction Combinatorics}\label{iseries section} We review briefly the notation and combinatorics surrounding dimension two cyclic quotient singularities.
\begin{defin}\label{cyclic quot def}
For $r,a\in\bN$ with $r>a$ the group
$G=\frac{1}{r}(1,a)$ is defined by
\[
G= \left\langle \zeta:=\left(\begin{array}{cc} \ve & 0\\ 0& \ve^a
\end{array}\right)\right\rangle \leq \GL(2,\KK),
\]
where $\ve$ is a primitive $r^{\rm th}$ root of unity.
By abuse of notation, we also denote the corresponding quotient singularity $S^G$ for $S=\KK[x,y]$ by $\frac{1}{r}(1,a)$.
\end{defin}
\begin{remark}
In the literature it is often assumed that the greatest common divisor $(r,a)$ is $1$, which is equivalent to the group having no pseudoreflections. However we do not make this assumption, since in our construction later singularities with pseudoreflections naturally appear.
\end{remark}
Provided that $a\neq 0$, we consider the Hirzebruch--Jung continued fraction expansion of $\frac{r}{a}$, namely
\[
\frac{r}{a}=\alpha_1-\frac{1}{\alpha_2 - \frac{1}{\alpha_3 -
\frac{1}{(...)}}} :=[\alpha_1,\hdots,\alpha_n]
\]
with each $\alpha_i\geq 2$. The labelled Dynkin diagram
\[
\begin{tikzpicture}[xscale=1.2]
\node (1) at (1,0) [vertex] {};
\node (2) at (2,0) [vertex] {};
\node (3) at (4,0) [vertex] {};
\node (4) at (5,0)[vertex] {};
\node at (3,0) {$\cdots$};
\node (1a) at (0.9,-0.25) {$\scriptstyle -\alpha_{1}$};
\node (2a) at (1.9,-0.25) {$\scriptstyle - \alpha_{2}$};
\node (3a) at (3.9,-0.25) {$\scriptstyle -\alpha_{n-1}$};
\node (4a) at (4.9,-0.25) {$\scriptstyle - \alpha_{n}$};
\draw [-] (1) -- (2);
\draw [-] (2) -- (2.6,0);
\draw [-] (3.4,0) -- (3);
\draw [-] (3) -- (4);
\end{tikzpicture}
\]
is precisely the dual graph of the
minimal resolution of $\KK^{2}/\frac{1}{r}(1,a)$ \cite[Satz8]{Riemen}. Note that \cite{Riemen} assumed the condition $(r,a)=1$, but the result holds generally: if we write $h:=(r,a)$, then the quotient singularities $\frac{1}{r}(1,a)$ and $\frac{1}{r/h}(1,a/h)$ are isomorphic, and furthermore both have the same Hirzebruch--Jung continued fraction expansion.
\begin{defin}\label{combdata}
For integers $1\leq a<r$ as above, denote the continued fraction expansion $\frac{r}{a}=[\alpha_1,\hdots,\alpha_n]$. We associate combinatorial series defined as follows:
\begin{enumerate}
\item The $i$-series is defined as $i_0=r$, $i_1=a$ and
\[
i_{t}=\alpha_{t-1}i_{t-1}-i_{t-2}
\]
for all $t$ with $2\leq t\leq n+1$.
\item Denote the $i$-series of $\frac{r}{r-a}$ by $\mathsf{i}_t$, then the $j$-series is defined $j_t=\mathsf{i}_{n+1-t}$ for all $t$ with $0\leq t\leq n+1$.
\end{enumerate}
\end{defin}
As in the introduction, we denote $I(r,a):=\{i_0,i_1,\hdots,i_{m+1}\}$,
where by convention $I(r,r)=\emptyset$. The following lemma is elementary, and will be needed later.
\begin{lemma}\label{i series all}
For integers $1\leq a<r$, $I(r,a)=[0,r]$ if and only if $a=r-1$.
\end{lemma}
For cyclic quotient singularities $G=\frac{1}{r}(1,a)$, consider
\[
S_{t}:=\{ f\in\KK[x,y] \mid \sigma\cdot f=\varepsilon^t f \},
\]
for $t\in [0,r]$, and note that $S_0\cong S_r$. Further, for $k$ with $0\leq k\leq r-1$, we say that a monomial $x^my^n$ has \emph{weight $k$} if $m+an=k$ mod $r$, that is $x^my^n\in S_k$. It is the $i$-series that determines which CM $S^G$-modules are special.
\begin{thm}\label{Wunram specials thm}
For $G=\frac{1}{r}(1,a)$,
\begin{enumerate}
\item \cite{Herzog} $\CM S^G=\add \{ S_t\mid t\in [0,r] \}$.
\item \cite{WunramCyclic} $\SCM S^G=\add \{ S_{t}\mid t\in I(r,a) \}$.
\end{enumerate}
\end{thm}
\begin{proof}
Both results are usually stated in the complete case, with no pseudoreflections, so since we are working more generally, we give the proof. Since $S^G$ has a unique singular point, by \ref{Check locally gen} (and its counterpart in the $\CM S^G$ case) it suffices to prove both results in the complete local setting. In this case, when $(r,a)=1$, part (1) is \cite{Herzog} and part (2) is \cite{WunramCyclic}. When $(r,a)\neq1$, the result is still true since $\frac{1}{r}(1,a)=\frac{1}{r/h}(1,a/h)$ for $h:=(r,a)$.
\end{proof}
In what follows, we will require a different characterization of members of the $i$-series, by reinterpreting of a result of Ito \cite[3.7]{Ito}. As notation, if $(r,a)=1$ then the \emph{$G$-basis} $B(G)$ is defined to be the set of monomials which are not divisible by any $G$-invariant monomial. We usually draw $B(G)$ in a $2\times 2$ grid.
\begin{example}
Consider $G=\frac{1}{17}(1,10)$. Then $B(G)$ is
\[
\begin{tikzpicture}[xscale=0.8,yscale=0.4]
\node at (0,0) {$\scriptstyle 1$};
\node at (0,-1) {$\scriptstyle x$};
\node at (0,-2) {$\scriptstyle x^2$};
\node at (0,-3) {$\scriptstyle x^3$};
\node at (0,-4) {$\scriptstyle x^4$};
\node at (0,-5) {$\scriptstyle x^5$};
\node at (0,-6) {$\scriptstyle x^6$};
\node at (0,-7) {$\scriptstyle x^7$};
\node at (0,-8) {$\scriptstyle \vdots$};
\node at (0,-9) {$\scriptstyle x^{16}$};
\node at (1,0) {$\scriptstyle y^{\phantom 2}$};
\node at (1,-1) {$\scriptstyle xy^{\phantom 2}$};
\node at (1,-2) {$\scriptstyle x^2y$};
\node at (1,-3) {$\scriptstyle x^3y$};
\node at (1,-4) {$\scriptstyle x^4y$};
\node at (1,-5) {$\scriptstyle x^5y$};
\node at (1,-6) {$\scriptstyle x^6y$};
\node at (2,0) {$\scriptstyle y^2$};
\node at (2,-1) {$\scriptstyle xy^2$};
\node at (2,-2) {$\scriptstyle x^2y^2$};
\node at (2,-3) {$\scriptstyle x^3y^2$};
\node at (2,-4) {$\scriptstyle x^4y^2$};
\node at (2,-5) {$\scriptstyle x^5y^2$};
\node at (2,-6) {$\scriptstyle x^6y^2$};
\node at (3,0) {$\scriptstyle y^3$};
\node at (3,-1) {$\scriptstyle xy^3$};
\node at (3,-2) {$\scriptstyle x^2y^3$};
\node at (3,-3) {$\scriptstyle x^3y^3$};
\node at (4,0) {$\scriptstyle y^4$};
\node at (4,-1) {$\scriptstyle xy^4$};
\node at (4,-2) {$\scriptstyle x^2y^4$};
\node at (4,-3) {$\scriptstyle x^3y^4$};
\node at (5,0) {$\scriptstyle y^5$};
\node at (6,0) {$\scriptstyle \hdots$};
\node at (7,0) {$\scriptstyle y^{16}$};
\draw (-0.5,0.5) -- (7.5,0.5) -- (7.5,-0.5) -- (4.5,-0.5) -- (4.5,-3.5) -- (2.5,-3.5)--(2.5,-6.5)--(0.5,-6.5)--(0.5,-9.5)--(-0.5,-9.5)--cycle;
\end{tikzpicture}
\]
\end{example}
For $G=\frac{1}{r}(1,a)$ with $(r,a)=1$, recall that the \emph{$L$-space} $L(G)$ is defined to be
\[
L(G):=\{1,x,\hdots,x^{r-1},y,\hdots,y^{r-1}\},
\]
so called since in the $2\times 2$ grid the shape of $L(G)$ looks like the letter L.
\begin{thm}[{=\cite[3.7]{Ito}}]\label{Ito thm}
When $(r,a)=1$, the elements of $I(r,a)$ are precisely those numbers in $[0,r]$ that do not appear as weights of monomials in the region $B(G)\backslash L(G)$.
\end{thm}
\begin{example}
Consider $G=\frac{1}{17}(1,10)$. Then $B(G)\backslash L(G)$ is the region
\[
\begin{tikzpicture}[xscale=0.8,yscale=0.4]
\node at (0,0) {$\scriptstyle 1$};
\node at (0,-1) {$\scriptstyle x$};
\node at (0,-2) {$\scriptstyle x^2$};
\node at (0,-3) {$\scriptstyle x^3$};
\node at (0,-4) {$\scriptstyle x^4$};
\node at (0,-5) {$\scriptstyle x^5$};
\node at (0,-6) {$\scriptstyle x^6$};
\node at (0,-7) {$\scriptstyle x^7$};
\node at (0,-8) {$\scriptstyle \vdots$};
\node at (0,-9) {$\scriptstyle x^{16}$};
\node at (1,0) {$\scriptstyle y^{\phantom 2}$};
\node at (1,-1) {$\scriptstyle xy^{\phantom 2}$};
\node at (1,-2) {$\scriptstyle x^2y$};
\node at (1,-3) {$\scriptstyle x^3y$};
\node at (1,-4) {$\scriptstyle x^4y$};
\node at (1,-5) {$\scriptstyle x^5y$};
\node at (1,-6) {$\scriptstyle x^6y$};
\node at (2,0) {$\scriptstyle y^2$};
\node at (2,-1) {$\scriptstyle xy^2$};
\node at (2,-2) {$\scriptstyle x^2y^2$};
\node at (2,-3) {$\scriptstyle x^3y^2$};
\node at (2,-4) {$\scriptstyle x^4y^2$};
\node at (2,-5) {$\scriptstyle x^5y^2$};
\node at (2,-6) {$\scriptstyle x^6y^2$};
\node at (3,0) {$\scriptstyle y^3$};
\node at (3,-1) {$\scriptstyle xy^3$};
\node at (3,-2) {$\scriptstyle x^2y^3$};
\node at (3,-3) {$\scriptstyle x^3y^3$};
\node at (4,0) {$\scriptstyle y^4$};
\node at (4,-1) {$\scriptstyle xy^4$};
\node at (4,-2) {$\scriptstyle x^2y^4$};
\node at (4,-3) {$\scriptstyle x^3y^4$};
\node at (5,0) {$\scriptstyle y^5$};
\node at (6,0) {$\scriptstyle \hdots$};
\node at (7,0) {$\scriptstyle y^{16}$};
\draw (0.5,-0.5) -- (4.5,-0.5) -- (4.5,-3.5) -- (2.5,-3.5)--(2.5,-6.5)--(0.5,-6.5)--cycle;
\end{tikzpicture}
\]
Replacing the monomials in the above region by their corresponding weights gives
\[
\begin{tikzpicture}[xscale=0.8,yscale=0.4]
\node at (1,-1) {$\scriptstyle 11$};
\node at (1,-2) {$\scriptstyle 12$};
\node at (1,-3) {$\scriptstyle 13$};
\node at (1,-4) {$\scriptstyle 14$};
\node at (1,-5) {$\scriptstyle 15$};
\node at (1,-6) {$\scriptstyle 16$};
\node at (2,-1) {$\scriptstyle 4$};
\node at (2,-2) {$\scriptstyle 5$};
\node at (2,-3) {$\scriptstyle 6$};
\node at (2,-4) {$\scriptstyle 7$};
\node at (2,-5) {$\scriptstyle 8$};
\node at (2,-6) {$\scriptstyle 9$};
\node at (3,-1) {$\scriptstyle 14$};
\node at (3,-2) {$\scriptstyle 15$};
\node at (3,-3) {$\scriptstyle 16$};
\node at (4,-1) {$\scriptstyle 7$};
\node at (4,-2) {$\scriptstyle 8$};
\node at (4,-3) {$\scriptstyle 9$};
\draw (0.5,-0.5) -- (4.5,-0.5) -- (4.5,-3.5) -- (2.5,-3.5)--(2.5,-6.5)--(0.5,-6.5)--cycle;
\end{tikzpicture}
\]
and so by \ref{Ito thm}, the $i$-series consists of those numbers that do not appear in the above region, which are precisely the numbers $0$, $1$, $2$, $3$, ${10}$ and $17$. Indeed, in this example $\frac{17}{10}=[2,4,2,2]$ and the $i$-series is
\[
i_0=17>i_1=10>i_2=3>i_3=2>i_4=1>i_5=0.
\]
\end{example}
The following lemma, which we use later, is an extension of \ref{Ito thm}. For integers $r>0$ and $k$, write $[k]_r$ for the unique integer $k'$ satisfying $0\le k'\le r-1$ and $k-k'\in r\bZ$.
\begin{lemma}\label{i-series region}
Assume $(r,a)=1$. For $0\le u\le r-1$, the following are equivalent.
\begin{enumerate}
\item $u\in I(r,r-a)$.
\item $u$ does not appear in $B(G)\backslash L(G)$ for $G:=\frac{1}{r}(1,-a)$.
\item For every integer $\ell\ge1$, there exists an integer $m\in [1,\ell]$ such that $[u+\ell a-1]_r\ge[ma-1]_r$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1)$\Leftrightarrow$(2) This is \ref{Ito thm}.\\
(2)$\Leftrightarrow$(3) We will establish the following claim: $u$ does not appear in column $\ell$ of $B(G)\backslash L(G)$ if and only if there exists an integer $m$ satisfying $1\le m\le\ell$ and $[u+\ell a-1]_r\ge[ma-1]_r$.
The first row of $B(G)$ is $0,-a,-2a,-3a,\ldots$. Now for each $m$ with $1\le m\le \ell$, we find the first occurrence of weight $0$ in column $m$, and use this to draw the following diagram:
\[
\begin{tikzpicture}[xscale=0.8,yscale=0.4]
\node at (1,-1) {$\scriptstyle -ma$};
\node at (1,-2) {$\scriptstyle 1-ma$};
\node at (1,-3) {$\scriptstyle 2-ma$};
\node at (1,-4) {$\scriptstyle \vdots$};
\node at (1,-5) {$\scriptstyle -2$};
\node at (1,-6) {$\scriptstyle -1$};
\node at (1,-7) {$\scriptstyle 0$};
\node at (2,-1) {$\scriptstyle\cdots$};
\node at (2,-2) {$\scriptstyle\cdots$};
\node at (2,-3) {$\scriptstyle\cdots$};
\node at (2,-4) {$\scriptstyle\ddots$};
\node at (2,-5) {$\scriptstyle\cdots$};
\node at (2,-6) {$\scriptstyle\cdots$};
\node at (2,-7) {$\scriptstyle\cdots$};
\node at (3.5,-1) {$\scriptstyle -\ell a$};
\node at (3.5,-2) {$\scriptstyle 1-\ell a$};
\node at (3.5,-3) {$\scriptstyle 2-\ell a$};
\node at (3.5,-4) {$\scriptstyle \vdots$};
\node at (3.5,-5) {$\scriptstyle -2+(m-\ell)a$};
\node at (3.5,-6) {$\scriptstyle -1+(m-\ell)a$};
\node at (3.5,-7) {$\scriptstyle (m-\ell)a$};
\draw (0.25,-1.5) -- (4.75,-1.5) -- (4.75,-6.5) --(0.25,-6.5)--cycle;
\draw[densely dotted] (2.5,-1.6) -- (4.65,-1.6) -- (4.65,-6.4) --(2.5,-6.4)--cycle;
\end{tikzpicture}
\]
The column $\ell$ of $B(G)\backslash L(G)$ is the intersection, over all $m$ with $1\le m\le \ell$, of the above dotted regions.
It is clear that $u$ does not appear in the dotted region in the above diagram if and only if $[u+\ell a-1]_r\geq[ma-1]_r$.
The claim follows.
\end{proof}
\begin{notation}\label{notationremark}
Throughout the remainder of the paper, to aid readability we will use the following simplified notation.
\begin{equation*}
\begin{tabular}{*3c}
\toprule
{\bf Notation}&{\bf Meaning}&{\bf Simplified Notation}\\
\midrule
$\mathbb{X}_{\bp,\bl}$&Weighted projective line & $\mathbb{X}$\\
$S_{\bp,\bl}$ & Defining ring of $\mathbb{X}_{\bp,\bl}$& $S$\\
$\Lambda_{\bp,\bl}$&Canonical algebra&$\Lambda$\\
$S_{\bp,\bl}^{\ox}$ & Veronese of $\mathbb{S}_{\bp,\bl}$ with respect to $\ox\in\bL$& $S^{\ox}$\\
${Y}^{\ox}_{\bp,\bl}$&Resolution of $\Spec S^{\ox}_{\bp,\bl}$ in \eqref{stack diagram 1} & $Y^{\ox}$\\
\bottomrule\\
\end{tabular}
\end{equation*}
Throughout it will be implicit that we are working generally, with parameters $(\bp,\bl)$.
\end{notation}
\section{The Total Space $\bT$}\label{stack}
Throughout this section we work with arbitrary parameters $(\bp,\bl)$ and use the simplified notation of \ref{notationremark}. As in the introduction, we choose $0\neq\ox\in\bL_+$, and consider the Veronese $S^{\ox}$, and the total space stack
\[
\bT^{\ox}=\bTot(\cO_{\bX}(-\ox)):=[(\Spec S\backslash 0\times\Spec\KK[t])/\Spec \KK\bL],
\]
where $\bL$ acts on $t$ with weight $-\ox$. There is a natural projection $q\colon\bTot(\cO_{\bX}(-\ox))\to\bX$, and a natural map $g\colon\bTot(\cO_{\bX}(-\ox))\to T^{\ox}$ to its coarse moduli space. There is also a map $f\colon\bX\to X$ from $\bX$ to its coarse moduli space, and there is an obvious morphism $p\colon T^{\ox}\to X$. Also, either simply by construction of $T^{\ox}$, or by an easy \v{C}ech calculation, it follows that
\[
H^0(\cO_{T^{\ox}})=\bigoplus_{k\geq 0}H^0(\bX,\cO_{\bX}(k\ox))t^k\cong S^{\ox}
\]
and so there is a natural map $T^{\ox}\to\Spec S^{\ox}$, which we denote by $\psi$. Furthermore, since $0\neq\ox\in\bL_+$, necessarily $\dim S^{\ox}=2$ and by inspection the morphism $\psi$ is projective birational.
We let $\varphi\colon Y^{\ox}\to T^{\ox}$ denote the minimal resolution of $T^{\ox}$, and consider the composition $\pi\colon Y^{\ox}\to T^{\ox}\to \Spec S^{\ox}$. We remark that this composition need not be the minimal resolution of $\Spec S^{\ox}$, and indeed later we give a precise criterion for when it is. Nevertheless, as in the introduction, we summarize the above information in the following commutative diagram
\begin{equation}
\begin{array}{c}
\begin{tikzpicture}
\node (top 1) at (0,0) {$\bT^{\ox}=\bTot(\cO_{\bX}(-\ox))$};
\node (top 2) at (2.5,0) {$\bX$};
\node (bottom 1) at (0,-1.5) {$T^{\ox}$};
\node (bottom 2) at (2.5,-1.5) {$X\cong\mathbb{P}^1$};
\node (min) at (-1.5,-1) {$Y^{\ox}$};
\node (V) at (0,-2.5) {$\Spec S^{\ox}$};
\draw[->] (top 1) -- node[left] {$\scriptstyle g$} (bottom 1);
\draw[->] (top 2) -- node[right] {$\scriptstyle f$} (bottom 2);
\draw[->] (top 1) -- node[above] {$\scriptstyle q$} (top 2);
\draw[->] (bottom 1) -- node[above] {$\scriptstyle p$} (bottom 2);
\draw[->] (min) -- node[above] {$\scriptstyle \varphi$} (bottom 1);
\draw[->] (bottom 1) -- node[right] {$\scriptstyle \psi$} (V);
\draw[->,densely dotted] (min) --node[below left] {$\scriptstyle \pi$} (V);
\end{tikzpicture}
\end{array}\label{stack diagram 2}
\end{equation}
\subsection{Tilting on $\bT$ and $T$}
Write $\cV:=\cO_{\bP^1}\oplus\cO_{\bP^1}(1)\in\coh\bP^1$, and $\cE:=\bigoplus_{\oy\in[0,\oc\,]}\cO_{\bX}(\oy)\in\coh\bX$. The following result is well known.
\begin{thm}\label{tilting on P1 and X}
The following statements hold.
\begin{enumerate}
\item $\cV$ is a tilting bundle on $\mathbb{P}^1$.
\item $\cE$ is a tilting bundle on $\bX$.
\end{enumerate}
\end{thm}
\begin{proof}
Part (1) is \cite{B83} and part (2) is \cite{GL1}.
\end{proof}
In this subsection we lift these tilting bundles to tilting bundles on both $T^{\ox}$ and $\bT^{\ox}$, and from this we deduce that $S^{\ox}$ is a rational surface singularity.
\begin{thm}\label{tilting on stack T}
If $0\neq\ox\in\bL_+$, then $q^*\cE$ is a tilting bundle on $\bT^{\ox}$ such that
\[
\begin{tikzpicture}[xscale=1]
\node (a1) at (0,0) {$\Db(\Qcoh\bT^{\ox})$};
\node (a2) at (5,0) {$\Db(\Mod\End_{\bT^{\ox}}(q^*\cE))$};
\node (b1) at (0,-1.5) {$\Db(\Qcoh\bX)$};
\node (b2) at (5,-1.5) {$\Db(\Mod\Lambda)$};
\draw[->] (a1) -- node[above] {$\scriptstyle \RHom_{\bT^{\ox}}(q^*\cE,-)$} node[below] {$\scriptstyle \sim$} (a2);
\draw[->] (b1) -- node[above] {$\scriptstyle \RHom_{\bX}(\cE,-)$} node[below] {$\scriptstyle \sim$} (b2);
\draw[->] (a1) -- node[left] {$\scriptstyle \Rq_*$} (b1);
\draw[->] (a2) -- node[right] {{\scriptsize\rm res}} (b2);
\end{tikzpicture}
\]
commutes, where $\Lambda$ is the canonical algebra.
\end{thm}
\begin{proof}
To simplify, we drop all $\ox$ from the notation and set $\bT:=\bTot(\cO_{\bX}(-\ox))$. The generation argument is standard, as in \cite[4.1]{TarigUeda} and \cite[4.1]{Bridgelandt}, namely if $M\in\D(\Qcoh \bT)$ with $\Hom_{\D(\bT)}(q^*\cE,M[i])=0$ for all $i$, then $\Hom_{\D(\bX)}(\cE,q_*M[i])=0$ for all $i$, so since $\cE$ generates $\bX$, $q_*M=0$ and so since $q$ is affine $M=0$. Hence $q^*\cE$ generates $\D(\Qcoh \bT)$, so since $\D(\Qcoh \bT)$ is compactly generated, $q^*\cE$ is a classical generator of $\Perf(\bT)$.
For Ext vanishing,
\begin{align*}
\Ext^1_{\bT}(q^*\cE,q^*\cE)&\cong\Ext^1_{\bX}(\cE,q_*q^*\cE)\tag{by adjunction}\\
&\cong\Ext^1_{\bX}(\cE,\bigoplus_{k\geq 0}\cE\otimes\cO_{\bX}(k\ox))\tag{by projection formula}\\
&\cong\bigoplus_{k\geq 0}\Ext^1_{\bX}(\cE,\cE\otimes\cO_{\bX}(k\ox))\\
&\cong\bigoplus_{k\geq 0}\bigoplus_{i\in[0,\oc\,]}\bigoplus_{j\in[0,\oc\,]}H^1(\bX,\cO_{\bX}(i-j+k\ox))\\
&\cong\bigoplus_{k\geq 0}\bigoplus_{i\in[0,\oc\,]}\bigoplus_{j\in[0,\oc\,]}H^0(\bX,\cO_{\bX}(\vec{\omega}-i+j-k\ox))^*\tag{by Serre duality}
\end{align*}
It is well known that $H^0(\bX,\cO_{\bX}(\oy))=S_{\oy}=0$ if $\oy\notin \bL_+$, so it suffices to check that $\vec{\omega}-i+j-k\ox\notin\bL_+$ for all $k\geq 0$ and all $i,j\in[0,\oc\,]$. Since $0\neq\ox\in\bL_+$, clearly if suffices to check $i=k=0$ and $j=\oc$, this being the most positive case. But $\vec{\omega}=(n-2)\oc-\sum_{t=1}^n\ox_t$, and so $\vec{\omega}+\oc=(n-1)\oc-\sum_{t=1}^n\ox_t\notin\bL_+$,
as required. Since $\bX$ is hereditary, the above proof also shows that higher Exts also vanish.
The commutativity is just adjunction $\RHom_{\bX}(\cE,\Rq_*(-))\cong\RHom_{\bT}(q^*\cE,-)$.
\end{proof}
\begin{thm}\label{tilting on coarse T}
If $0\neq\ox\in\bL_+$, then $p^*\cV$ is a tilting bundle on $T^{\ox}$. \end{thm}
\begin{proof}
As above, when possible we drop all $\ox$ from the notation. The generation argument is identical to the argument in \ref{tilting on stack T}. The Ext-vanishing is also similar, namely if we denote $\cF:=\cO_{\bX}\oplus\cO_{\bX}(\oc)$ then,
\begin{align*}
\Ext^i_T(p^*\cV,p^*\cV)&\cong\Ext^i_T(p^*\cV,g_*\cO_{\bT}\otimes_T p^*\cV)\tag{$g_*\cO_{\bT}=\cO_T$}\\
&\cong\Ext^i_T(p^*\cV,g_*g^*p^*\cV)\tag{projection formula}\\
&\cong\Ext^i_{\bT}(g^*p^*\cV,g^*p^*\cV)\tag{adjunction}\\
&\cong\Ext^i_{\bT}(q^*f^*\cV,q^*f^*\cV)\tag{commutativity of \eqref{stack diagram 2}}\\
&\cong\Ext^i_{\bT}(q^*\cF,q^*\cF)
\end{align*}
which is zero by \ref{tilting on stack T} since $q^*\cF$ is a summand of $q^*\cE$.
\end{proof}
The following two results give precise information regarding the singularities in $T^{\ox}$.
\begin{prop}\label{sings on T2}
$T^{\ox}$ is a surface containing the coarse module $X\cong \bP^1$ of $\bX$.
Moreover $T^{\ox}$ is normal, and all its singularities are isolated and lie on $X$.
\end{prop}
\begin{proof}
We use the presentation of $S$ given in \ref{notation throughout}:
\[
S=\frac{\KK[x_1,\hdots,x_n]}{(x_i^{p_i}+x_1^{p_1}-\lambda_ix_2^{p_2}\mid 3\leq i\leq n)}.
\]
The open cover $\Spec S\backslash0=U_1\cup U_2$ with $U_i:=\Spec S_{x_i}$ induces an open cover $(\Spec S\backslash0)\times\Spec k[t]=U'_1\cup U'_2$ with $U'_i:=U_i\times\Spec k[t]=\Spec S[t]_{x_i}$ for $i=1,2$.
Thus $T^\ox$ has an open cover $T^{\ox}=V_1\cup V_2$ where $V_i:=\Spec (S[t]_{x_i})_0$ and $(S[t]_{x_i})_0$ is the degree zero part of $S[t]_{x_i}$. As in \ref{notation throughout}\eqref{notation throughout 2b}, the curve $X_i:=\Spec (S_{x_i})_0$ in $V_i$ for $i=1,2$ gives the coarse moduli $X=X_1\cup X_2\cong\mathbb{P}^1$ of $\bX$.
Fix $i=1,2$ and let $A:=S[t]_{x_i}$ and $B:=(S[t]_{x_i})_0$ so that $V_i=\Spec B$. We first claim that $B$ is a normal domain.
Since $S$ is an $\bL$-factorial $\bL$-domain by \ref{L-factorial}, so are $S[t]$ and $A$.
Thus the degree zero part $B$ of $A$ is a domain.
To prove that $B$ is normal, assume that an element $x$ in the quotient field of $B$ satisfies an equality $x^m+b_1x^{m-1}+\ldots+b_m=0$ for some $b_i\in B$.
Write $x=y/z$ for homogeneous elements $y$ and $z$ in $A$ which do not have a common $\bL$-prime factor.
Since $y^m=-z(b_1y^{m-1}+\ldots+b_mz^{m-1})$ holds, $z$ must be a unit in $A$. Thus we have $x\in A$ and $x\in B$, and the assertion follows.
Consider next the $\bZ$-grading on $S[t]$ defined by $\deg t=1$ and $\deg x=0$ for any $x\in S$. This gives a $\bZ$-grading on $B$ such that $B=\bigoplus_{i\ge0}B_i$ and $B_0=(S_{x_i})_0$.
Since $B$ is a $\bZ$-graded finitely generated $\KK$-algebra, by the Jacobian criterion, there is a $\bZ$-graded ideal $I$ of $B$ such that $\Sing B=\Spec(B/I)$.
Since $B$ is a normal domain of dimension two, $\Sing B$ consists of finitely many closed points and hence $\dim_k(B/I)<\infty$ holds. Since $I$ is $\bZ$-graded, it contains $\bigoplus_{i>\ell}B_i$ for $\ell\gg0$ and hence $\sqrt{I}$ contains $\bigoplus_{i>0}B_i$. Consequently, $\Sing B$ is contained in $\Spec B_0\subset X$.
\end{proof}
Next we prove the following.
\begin{prop}\label{sings on T}
On $X\cong\bP^1$, complete locally the singularities of $T^{\ox}$ are of the form
\begin{equation}\label{T picture 2}
\begin{array}{c}
\begin{tikzpicture}[scale=1.5]
\draw (-0.2,0) to [bend left=5] node[pos=0.75,above] {$\scriptstyle \bP^1$} (4.2,0);
\filldraw [black] (0.0,0.01) circle (1pt);
\filldraw [black] (1,0.08) circle (1pt);
\filldraw [black] (2,0.11) circle (1pt);
\filldraw [black] (4,0.01) circle (1pt);
\node at (0,-0.15) {$\scriptstyle \frac{1}{p_1}(1,-a_1)$};
\node at (0,0.2) {$\scriptstyle \lambda_1$};
\node at (1,-0.1) {$\scriptstyle \frac{1}{p_2}(1,-a_2)$};
\node at (1,0.275) {$\scriptstyle \lambda_2$};
\node at (2,-0.075) {$\scriptstyle \frac{1}{p_3}(1,-a_3)$};
\node at (2,0.3) {$\scriptstyle \lambda_3$};
\node at (3,-0.1) {$\scriptstyle \hdots$};
\node at (4,-0.15) {$\scriptstyle \frac{1}{p_n}(1,-a_n)$};
\node at (4,0.2) {$\scriptstyle \lambda_n$};
\end{tikzpicture}
\end{array}
\end{equation}
\end{prop}
\begin{proof}
We use the open cover $T^{\ox}=V_1\cup V_2$ given in the proof of \ref{sings on T2}, and we will show that $\widehat{\cO}_{T^{\ox},\lambda_i}$ is the completion of $\frac{1}{p_i}(1,-a_i)$. By symmetry, we only have to consider the case $i=1$.
For the polynomial ring $\KK[x_1,\ldots,x_n,t]$ and the formal power series ring $\KK[[\sx_1,\st]]$, consider the morphism
\[
f\colon\KK[x_1,\ldots,x_n,t]\to P:=\KK[[\sx_1,\st]]
\]
of $\KK$-algebras defined by $f(t)=\st$, $f(x_1)=\sx_1$, $f(x_2)=1$ and $f(x_i)=(\lambda_i-\sx_1^{p_1})^{1/p_i}$ for $3\le i\le n$, where a $p_i$-th root of $\lambda_i-\sx_1^{p_1}$ exists since $\KK$ is an algebraically closed field of characteristic zero.
Since $f$ sends $x_i^{p_i}+x_1^{p_1}-\lambda_i x_2^{p_i}$ to zero for all $3\le i\le n$, it induces a morphism of $\KK$-algebras
$f\colon S[t]\to P$, and further since $f(x_2)=1$ this induces a morphism of $\KK$-algebras
\begin{equation}\label{invert x_2}
f\colon S[t]_{x_2}\to P.
\end{equation}
Let $C:=\frac{1}{p_1}(1,-a_1)=\langle\zeta\rangle$ be the cyclic group acting on $P$ by $\zeta\sx_1=\ve\sx_1$ and $\zeta\st=\ve^{-a_1}\st$ for a primitive $p_1$-th root $\ve$ of unity. Certainly $f(x_i)$ with $2\le i\le n$ belongs to $k[[\sx_1^{p_1}]]\subset P^C$. Now we claim that \eqref{invert x_2} induces a morphism of $\KK$-algebras
\begin{equation}\label{then degree zero}
f\colon (S[t]_{x_2})_0\to P^C.
\end{equation}
If a monomial $X=x_1^{\ell_1}\ldots x_n^{\ell_n}t^\ell\in S[t]_{x_2}$ has degree zero, then $\ell_1\ox_1+\ldots+\ell_n\ox_n+\ell\ox=0$ holds. Looking at the coefficients of $\ox_1$, we have $\ell_1+\ell a_1\in p_1\bZ$. Thus $f(X)=\sx_1^{\ell_1}\st^\ell\prod_{i=2}^nf(x_i)^{\ell_i}$ belongs to $P^C$, and the assertion follows.
Now let $\mathfrak{m}$ be the maximal ideal of $(S[t]_{x_2})_0$ corresponding to $\lambda_1$, and $\mathfrak{n}$ be the maximal ideal of $P^C$. Then $\widehat{\cO}_{T^{\ox},\lambda_1}$ is the completion of $(S[t]_{x_2})_0$ at $\mathfrak{m}$.
Moreover $f(\mathfrak{m})\subset\mathfrak{n}$ holds since $\mathfrak{m}$ is generated by monomials $x_1^{\ell_1}\cdots x_n^{\ell_n}t^\ell$ with $\ell\ge1$ and $x_1^{p_1}/x_2^{p_2}$.
Thus \eqref{then degree zero} induces a morphism
\begin{equation}\label{then complete}
f\colon \widehat{\cO}_{T^{\ox},\lambda_1}\to P^C.
\end{equation}
To prove surjectivity, it suffices to show that $f$ gives a surjective map $\mathfrak{m}\to\mathfrak{n}/\mathfrak{n}^2$.
Since the $\KK$-vector space $\mathfrak{n}/\mathfrak{n}^2$ is spanned by monomials in $P^C$, it suffices to show that any monomial $\sx_1^{\ell_1}\st^\ell$ in $P^C$ belongs to $\Im f+\mathfrak{n}^2$.
Since $\sx_1^{\ell_1}\st^\ell$ is invariant under the action of $C$, the coefficient of $\ox_1$ in the normal form of $\ell_1\ox_1+\ell\ox$ is zero.
Thus there exist $\ell_2\in\bZ$ and $\ell_3,\ldots,\ell_n\in\bZ_{\ge0}$ such that $\ell_1\ox_1+\ldots+\ell_n\ox_n+\ell\ox=0$.
Now $X:=x_1^{\ell_1}\cdots x_n^{\ell_n}t^\ell\in (S[t]_{x_2})_0$ satisfies
\[
f(\alpha X)\equiv\sx_1^{\ell_1}\st^\ell\ \mod\mathfrak{n}^2\ \mbox{ for }\ \alpha:=\prod_{i=3}^n\lambda_i^{-\ell_i/p_i}.
\]
Hence \eqref{then complete} is surjective. Since $(S[t]_{x_2})_0$ is an algebraic normal domain by the proof of \ref{sings on T2}, its completion is also a normal domain by Zariski. Thus \eqref{then complete} is a surjective morphism between two-dimensional domains, and so is necessarily an isomorphism.
\end{proof}
The following is now a corollary of \ref{tilting on coarse T} and \ref{sings on T}.
\begin{cor}
If $0\neq\ox\in\bL_+$, then $S^{\ox}$ is a rational surface singularity.
\end{cor}
\begin{proof}
By \ref{sings on T} all the singularities on $T^{\ox}$ are rational, hence there exists a resolution $f\colon Y\to T^{\ox}$ such that $\Rf_*\cO_{Y}=\cO_{T^{\ox}}$. Since by \ref{tilting on coarse T} $T^{\ox}$ has a tilting bundle with summand $\cO_{T^{\ox}}$, necessarily $H^i(\cO_{T^{\ox}})=\Ext^i_{T^{\ox}}(\cO_{T^{\ox}},\cO_{T^{\ox}})=0$ for all $i>0$. By construction, we already know that $\psi_*\cO_{T^{\ox}}=\cO_{S^{\ox}}$, so this implies that $\RDerived \psi_*\cO_{T^{\ox}}=\cO_{S^{\ox}}$. Composing, we see that $\RDerived( \psi\circ f)_*\cO_{Y}=\cO_{S^{\ox}}$, and so $\Spec S^{\ox}$ is rational.
\end{proof}
\begin{cor}\label{fund is reduced}
If $0\neq\ox\in\bL_+$, then the fundamental cycle associated to the morphism $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ is reduced.
\end{cor}
\begin{proof}
Since $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ is a resolution of a rational surface singularity, the fundamental cycle exists. Resolving the singularities in \eqref{T picture 2} it is clear that the dual graph of $\pi$ is star shaped, with the middle curve of this star corresponding to the $\mathbb{P}^1$ in $T^{\ox}$. By \ref{tilting on coarse T} the line bundle $\cL:=p^{*}\cO_{\mathbb{P}^1}(1)$ on $T^{\ox}$ satisfies $\Ext^1_{T^{\ox}}(\cL,\cO_{T^{\ox}})=0$. It clearly has degree one on the exceptional curve. Then $\cL_Y:=\varphi^*\cL=\mathbf{L}\varphi^*\cL$ is a line bundle on $Y^{\ox}$, with degree one on the middle curve and degree zero on all other curves. Furthermore
\begin{align*}
H^1(\cL_Y^{-1})&\cong\Ext^1_{Y^{\ox}}(\cL_Y,\cO_{Y^{\ox}})\\
&\cong\Hom_{\Db(Y^{\ox})}(\mathbf{L}\varphi^*\cL,\cO_{Y^{\ox}}[1])\\
&\cong\Hom_{\Db(T^{\ox})}(\cL,\mathbf{R}\varphi_*\cO_{Y^{\ox}}[1])\\
&\cong\Ext^1_{T^{\ox}}(\cL,\cO_{T^{\ox}})\\
&=0.
\end{align*}
Since $\cL_Y$ has rank one, as in \cite{Wunram} (see also \cite[3.5.4]{VdB1d}), this implies that in the fundamental cycle of $\pi$, the middle curve has number one. Since the dual graph is star shaped, and all other self-intersection numbers are $\leq -2$, by \eqref{Zf dot Ei} this then implies that the whole fundamental cycle is reduced.
\end{proof}
In the sequel, we require the following description of some degenerate cases.
\begin{lemma}\label{degen lemma}
Let $0\neq\ox\in\bL_+$ and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ with $a\ge0$ in normal form.
\begin{enumerate}
\item\label{degen lemma 1} If all $a_i$=0 (so necessarily $a>0$), then $Y^{\ox}=T^{\ox}=\cO_{\mathbb{P}^1}(-a)$ and $S^{\ox}=\KK[x,y]^{\frac{1}{a}(1,1)}$.
\item\label{degen lemma 2} If $a_i\neq 0$ and $a_j=0$ for all $j\neq i$, then the dual graph of $\pi$ in \eqref{stack diagram 2} is
\[
\begin{tikzpicture}[xscale=1.2]
\node (1) at (1,0) [vertex] {};
\node (2) at (2,0) [vertex] {};
\node (3) at (4,0) [vertex] {};
\node (4) at (5,0)[vertex] {};
\node at (3,0) {$\cdots$};
\node (1a) at (0.9,-0.25) {$\scriptstyle -1-a$};
\node (2a) at (1.9,-0.25) {$\scriptstyle - \alpha_{i1}$};
\node (3a) at (3.9,-0.25) {$\scriptstyle -\alpha_{im_i-1}$};
\node (4a) at (4.9,-0.25) {$\scriptstyle - \alpha_{im_i}$};
\draw [-] (1) -- (2);
\draw [-] (2) -- (2.6,0);
\draw [-] (3.4,0) -- (3);
\draw [-] (3) -- (4);
\end{tikzpicture}
\]
where $\frac{p_i}{p_i-a_i}=[\alpha_{i1},\hdots,\alpha_{im_i}]$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) It is well known that $S^{\oc}=\KK[t_0,t_1]$, and hence $S^{a\oc}$ is the $a$-th Veronese of $\KK[t_0,t_1]$, which is $\KK[x,y]^{\frac{1}{a}(1,1)}$. There is only a single curve above the origin in the smooth modification $\pi$, which necessarily must have negative self-intersection. Since the contraction of a $(-b)$ curve is always analytically isomorphic to $\KK[[x,y]]^{\frac{1}{b}(1,1)}$, this then implies that $a=b$ and so $Y^{\ox}=T^{\ox}=\cO_{\mathbb{P}^1}(-a)$. \\
(2) There is only one singularity in \eqref{T picture 2}, which implies that the dual graph has the above Type $A$ form. It is standard that $\alpha_{ij}$ from $\frac{p_i}{p_i-a_i}=[\alpha_{i1},\hdots,\alpha_{im_i}]$ resolves $\frac{1}{p_i}(1,-a_i)$, and thus the only thing still to be verified is the self-intersection number $-1-a$. There are two ways of doing this: since the fundamental cycle of $\pi$ is reduced by \ref{fund is reduced}, the reconstruction algebra is easy to calculate and it can be directly verified that its quiver has the form given by intersection rules in \ref{GL2 for R} (which, by \cite{WemGL2}, hold for non-minimal resolutions too). Alternatively, the number $-1-a$ can be determined by an explicit gluing calculation on $T^{\ox}$; in both cases we suppress the details.
\end{proof}
\subsection{Special CM Modules and the Dual Graph}
Choose $0\neq\ox\in\bL_+$. In this subsection we first give a precise criterion for when $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ in \eqref{stack diagram 2} is the minimal resolution, then we use the results of the previous subsections to determine the indecomposable special CM $S^{\ox}$-modules.
\begin{lemma}\label{when Y min res}
Let $0\neq\ox\in\bL_+$. Then $\pi$ is the minimal resolution if and only if $\ox\notin[0,\oc\,]$.
\end{lemma}
\begin{proof}
Write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. Since $0\neq\ox\in\bL_+$, necessarily $a\geq 0$. Note that, by construction, the only curve in the star-shaped dual graph of $\pi$ that might be a ($-1$)-curve is the middle one. \\
($\Leftarrow$) Suppose that $\ox\notin[0,\oc\,]$. If all $a_i=0$ then necessarily $a\geq 2$, and so \ref{degen lemma}\eqref{degen lemma 1} shows that $\pi$ is the minimal resolution. Similarly, if $a_i\neq 0$ but $a_j=0$ for all $j\neq i$, then the assumption $\ox\notin[0,\oc\,]$ forces $a\geq 1$, and \ref{degen lemma}\eqref{degen lemma 2} then shows that $\pi$ is minimal.
Hence we can assume that $\ox\notin[0,\oc\,]$ with at least two of the $a_i$ being non-zero. This being the case, there are at least two singular points in \eqref{T picture 2}. By \ref{fund is reduced}, since the fundamental cycle is reduced, the calculation \eqref{Zf dot Ei} shows that middle curve then cannot be a ($-1$)-curve, hence the resolution is minimal.\\
($\Rightarrow$) By contrapositive, suppose that $0\neq \ox\in[0,\oc\,]$, say $\ox=a_i\ox_i$ for some $i$ and some $0<a_i<p_i$. Since $a=0$, by \ref{degen lemma}\eqref{degen lemma 2} the resolution $\pi$ is not minimal.
\end{proof}
Hence if $x\in\bL_+$ with $x\notin[0,\oc\,]$, it follows that the dual graph of the minimal resolution $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ is \eqref{key dual graph}, except that we have not yet determined the precise value of $\beta$. We will do this later in \ref{middle SI number}, since for the moment this value is not needed. As notation, for $\oy\in\bL$ write $S(\oy)^{\ox}:=\bigoplus_{i\in\bZ}S_{\oy+i\ox}$.
\begin{thm}\label{specials determined thm}
Suppose that $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ with $a\ge0$ in normal form.
Then
\[
\SCM S^{\ox}=\add\{S(u\ox_j)^{\ox}\mid j\in [1,n], u\in I(p_j,p_j-a_j) \}.
\]
\end{thm}
\begin{proof}
The ring $S^{\ox}$ has a unique singular point corresponding to the graded maximal ideal, since it is two dimensional normal and positively graded (see e.g.\ \cite[p1]{Pinkham}). Thus, by \ref{Check locally gen} we complete $S^{\ox}$ at this point and pass to the formal fibre, which is still the minimal resolution. However, to aid readability, we do not add $\widehat{(-)}$ to the notation.
Consider the bundle $q^*\cE$ on $\bT$, and its pushdown $g_*q^*\cE$ on $T^{\ox}$. At the point $\lambda_1$ of $T^{\ox}$, which is the singularity $\frac{1}{p_1}(1,-a_1)$ by \ref{sings on T}, the sheaves
\begin{equation}\label{CM at 1}
g_*q^*\cO,g_*q^*\cO(\ox_1),\hdots,g_*q^*\cO((p_1-1)\ox_1)
\end{equation}
are all locally free away from the point $\lambda_1$, since at any other singular point $\lambda_i$, multiplication by $x_1$ is invertible. Further, at the point $\lambda_1$, \eqref{CM at 1} is a full list of the CM modules, indexed by the characters of $\bZ_{p_1}=\frac{1}{p_1}(1,-a_1)$ in the obvious way. Hence by \ref{Wunram specials thm}, which does not require any coprime assumption, the torsion-free pullbacks under $\varphi$ of
\[
\{ g_*q^*\cO(u\ox_1)\mid u\in I(p_1,p_1-a_1)\}
\]
are precisely the line bundles on $Y^{\ox}$ corresponding to the curves in arm $1$ of the dual graph. By \ref{fund is reduced} they are the special bundles on $Y^{\ox}$ corresponding to the curves in arm $1$ of the dual graph, hence their pushdown (via $\pi$) to $S^{\ox}$ are the special CM $S^{\ox}$-modules corresponding to arm $1$. Since the pushdown under $\varphi$ of the torsion-free pullback of $\varphi$ is the identity, the pushdown to $S^{\ox}$ gives the modules
\[
\{ \psi_*g_*q^*\cO(u\ox_1)\mid u\in I(p_1,p_1-a_1)\},
\]
which are precisely the modules $S(u\ox_1)^{\ox}$. The argument for the other arms is identical. The argument for $S(\oc)^{\ox}$ follows again by \ref{fund is reduced}.
\end{proof}
\begin{example}\label{Ex3.10}
Consider the example $(p_1,p_2,p_3)=(3,5,5)$ and $\ox=2\ox_1+2\ox_2+3\ox_3$. The continued fractions for $\frac{p_i}{p_i-a_i}$, and the corresponding $i$-series are given by:
\[
\begin{array}{ll}
\frac{3}{3-2}=[3] & 3>1>0\\
\frac{5}{5-2}=[2,3] & 5>3>1>0\\
\frac{5}{5-3}=[3,2] & 5>2>1>0
\end{array}
\]
It follows from \ref{specials determined thm} that an additive generator of $\SCM R$ is given by the direct sum of the following circled modules:
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1.3,yscale=1.3]
\node (0) at (0,0) {$\scriptstyle S(\oc)^{\ox}$};
\node (A1) at (-1.5,1) {$\scriptstyle S(2\ox_1)^{\ox}$};
\node (A4) at (-1.5,4) {$\scriptstyle S(\ox_1)^{\ox}$};
\node (B1) at (0,1) {$\scriptstyle S(4\ox_2)^{\ox}$};
\node (B2) at (0,2) {$\scriptstyle S(3\ox_2)^{\ox}$};
\node (B3) at (0,3) {$\scriptstyle S(2\ox_2)^{\ox}$};
\node (B4) at (0,4) {$\scriptstyle S(\ox_2)^{\ox}$};
\node (C1) at (1.5,1) {$\scriptstyle S(4\ox_3)^{\ox}$};
\node (C2) at (1.5,2) {$\scriptstyle S(3\ox_3)^{\ox}$};
\node (C3) at (1.5,3) {$\scriptstyle S(2\ox_3)^{\ox}$};
\node (C4) at (1.5,4) {$\scriptstyle S(\ox_3)^{\ox}$};
\node (T) at (0,5) {$\scriptstyle S^{\ox}$};
\draw [->] (A1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(0);
\draw [->] (B1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(0);
\draw [->] (C1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(0);
\draw [->] (A4) -- node[right] {$\scriptstyle x_1$} (A1);
\draw [->] (B2) -- node[right] {$\scriptstyle x_2$}(B1);
\draw [->] (C2) -- node[right] {$\scriptstyle x_3$}(C1);
\draw [->] (B3) -- node[right] {$\scriptstyle x_2$}(B2);
\draw [->] (C3) -- node[right] {$\scriptstyle x_3$}(C2);
\draw [->] (B4) -- node[right] {$\scriptstyle x_2$}(B3);
\draw [->] (C4) -- node[right] {$\scriptstyle x_3$}(C3);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(A4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(B4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(C4);
\draw[blue] (T) circle (11pt);
\draw[blue] (A4) circle (11pt);
\draw[blue] (B4) circle (11pt);
\draw[blue] (B2) circle (11pt);
\draw[blue] (C4) circle (11pt);
\draw[blue] (C3) circle (11pt);
\draw[blue] (0) circle (11pt);
\end{tikzpicture}
\end{array}
\]
\end{example}
\begin{remark}\label{positions forced}
It is in fact possible to assign each special CM $S^{\ox}$-module to its vertex in the dual graph of the minimal resolution. As in \ref{Ex3.10} above there are obvious irreducible morphisms between the special CM $R$-modules, so they must appear in the quiver of the reconstruction algebra. By the intersection theory in \ref{GL2 for R}, we conclude that $S(\oc)^{\ox}$ corresponds to the middle vertex, and this forces the positions of the other special CM modules relative to the dual graph.
\end{remark}
\begin{cor}\label{stackminres}
If $0\neq\ox\in\bL_+$, then the following statements hold:
\begin{enumerate}
\item\label{stackminres 1} There is an idempotent $e\in\End_{\bT}(q^*\cE)$ such that $e\End_{\bT}(q^*\cE)e\cong\End_{Y^{\ox}}(\cM)$.
\item\label{stackminres 2} There is a fully faithful embedding
\[
\Db(\coh Y^{\ox})\hookrightarrow \Db(\coh\bT^{\ox}).
\]
\end{enumerate}
\end{cor}
\begin{proof}
(1) Consider the tilting bundle $\cM$ on $Y^{\ox}$, generated by global sections, constructed in \cite[3.5.4]{VdB1d}. Even although $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ need not be the minimal resolution, it is still true by \cite[4.3]{DW2} that
\[
\End_{Y^{\ox}}(\cM)\cong\End_{S^{\ox}}(\pi_*\cM)
\]
On the other hand,
\[
\End_{\bT}(q^*\cE)\cong\End_{S^{\ox}}(\bigoplus_{\oy\in[0,\oc\,]}S(\oy)^{\ox}).
\]
As in the proof of \ref{specials determined thm}, $\pi_*\cM$ consists of summands of $\bigoplus_{\oy\in[0,\oc\,]}S(\oy)^{\ox}$, hence there is an idempotent $e\in\End_{\bT}(q^*\cE)$ such that $e\End_{\bT}(q^*\cE)e\cong\End_{Y^{\ox}}(\cM)$.\\
(2) By \eqref{stackminres 1}, writing $A:=\End_{\bT}(q^*\cE)$ then $\End_{Y^{\ox}}(\cM)=eAe$, thus there is an obvious embedding of derived categories
\[
\RHom_{eAe}(Ae,-)\colon\D(\Mod\End_{Y^{\ox}}(\cM))\hookrightarrow\D(\Mod\End_{\bT}(q^*\cE)),
\]
and also an embedding given by $-\otimes^{\bf L}_{eAe}eA$.
Regardless, since $\gl eAe<\infty$, the above induces an embedding
\[
\Db(\mod\End_{Y^{\ox}}(\cM))\hookrightarrow\Db(\mod\End_{\bT}(q^*\cE)).
\]
The left hand side is derived equivalent to $Y^{\ox}$, and the right hand side is derived equivalent to $\bT^{\ox}:=\bTot(\cO_{\bX}(-\ox)$ by \ref{tilting on stack T}, so the result follows.
\end{proof}
We give a simple criterion for when the above is an equivalence later in \ref{stackminres 2A}. Note that the above result is formally very similar to the case of quotient singularities, where the reconstruction algebra embeds into the quotient stack $[\KK^2/G]$, but this embedding is also very rarely an equivalence.
\section{Categorical Equivalences}\label{qgr Veronese section}
In this section we investigate the conditions on $\ox$ under which
\[
\coh\bX\simeq\qgr^{\bZ}\!S^{\ox}\simeq\qgr^{\bZ}\!\Gamma_{\!\ox}
\]
holds. This allows us, in \ref{stackminres 2A}, to give a precise criterion for when the embedding in \ref{stackminres}\eqref{stackminres 2} is an equivalence, and further it allows us in \ref{middle SI number} to determine the middle self-intersection number in \eqref{key dual graph}. Throughout many results in this section, a coprime condition $(p_i,a_i)=1$ naturally appears, and in \S\ref{changing parameters} we show that we can always change parameters so that this coprime condition holds.
\subsection{General Results on Categorical Equivalences}
To simplify the notation, in this subsection we first produce categorical equivalences in a very general setting, before specialising in the next subsection to the case of the weighted projective line.
We start with a basic observation. Let $G$ be an abelian group and $A$ a noetherian $G$-graded $k$-algebra. For an idempotent $e\in A_0$, $B:=eAe$ is a noetherian $G$-graded $k$-algebra. The functor
\[
E:=e(-)\colon\mod^G\!A\to\mod^G\!B
\]
has a left adjoint functor $E_\lambda$ and a right adjoint functor $E_\rho$ given by
\begin{align*}
E_\lambda:=Ae\otimes_B-&\colon\mod^G\!B\to\mod^G\!A\\
E_\rho:=\Hom_B(eA,-)&\colon\mod^G\!B\to\mod^G\!A.
\end{align*}
Moreover $EE_\lambda={\rm id}_{\mod^G\!B}=EE_\rho$ holds, and for the natural morphism $m:Ae\otimes_BeA\to A$, the counit $\eta:E_\lambda E\to {\rm id}_{\mod^G\!A}$ is given by $m\otimes_{A}-$ and the unit $\varepsilon:{\rm id}_{\mod^G\!A}\to E_\rho E$ is given by $\Hom_A(m,-)$.
The following basic observation is a prototype of our results in this subsection.
\begin{prop}\label{comm to noncomm}
If $\dim_{\KK}A/(e)<\infty$, then $E$ induces an equivalence $\qgr^G\!A\simeq \qgr^G\!B$.
\end{prop}
\begin{proof}
Clearly $E_\lambda$ and $E$ induce an adjoint pair $E_\lambda:\qgr^G\!A\to \qgr^G\!B$ and $E:\qgr^G\!B\to\qgr^G\!A$. For any $X\in\mod^G\!A$, both the kernel and cokernel of $m\otimes_{A}X:E_\lambda EX\to X$ are finite dimensional since they are finitely generated $(A/(e))$-modules. Therefore $E_\lambda$ and $E$ give the desired equivalences.
\end{proof}
In the rest of this subsection, let $G$ be an abelian group and $H$ a subgroup of $G$ of finite index. Assume that $A$ is a noetherian $G$-graded $k$-algebra, and let $B:=A^H=\bigoplus_{g\in H}A_g$ be the $H$-Veronese subring of $A$.
\begin{lemma}\label{Veron is fg2}
$B$ is a noetherian $k$-algebra and $A$ is a finitely generated $B$-module.
\end{lemma}
\begin{proof}
There is a finite direct sum decomposition $A=\bigoplus_{g\in G/H}A(g)^H$ as $B$-modules. For any submodule $M$ of $A(g)^H$, it is easy to check that the ideal $AM$ of $A$ satisfies $AM\cap A(g)^H=M$. Therefore $A(g)^H$ is a noetherian $B$-module, since $A$ is a noetherian ring. The assertion follows.
\end{proof}
We say that $X\in\mod^G\!A$ has \emph{depth at least two} if $\Ext^i_A(Y,X)=0$ for any $i=0,1$ and $Y\in\mod^G\!A$ with $\dim_kY<\infty$. We denote by $\mod_2^G\!A$ the full subcategory of $\mod^G\!A$ consisting of modules with depth at least two. We define $\mod^H_{2}B$ similarly.
\begin{thm}\label{WPL as qgrZ2}
Let $G$ be an abelian group, $H$ a subgroup of $G$ of finite index, $A$ a noetherian $G$-graded $k$-algebra, and $B:=A^H$. Then the following conditions are equivalent.
\begin{enumerate}
\item\label{WPL as qgrZ 2-1} The natural functor $(-)^{H}\colon\qgr^G\!A\to\qgr^H\!B$ is an equivalence.
\item\label{WPL as qgrZ 3-1} For any $i\in G$, the ideal $I^{i}:=A(i)^{H}\cdot A(-i)^{H}$ of $B$ satisfies $\dim_k(B/I^{i})<\infty$.
\end{enumerate}
If $A$ belongs to $\mod_2^G\!A$, then the following condition is also equivalent.
\begin{enumerate}
\item[(3)]\label{WPL as qgrZ 1-1} The natural functor $(-)^{H}\colon\mod_2^G\!A\to\mod_2^H\!B$ is an equivalence.
\end{enumerate}
\end{thm}
\begin{proof}
Consider the matrix algebra
\[
C=(A(i-j)^H)_{i,j\in G/H}
\]
whose rows and columns are indexed by $G/H$, and the product is given by the matrix multiplication together with the product in $A$:
\[
(s_{i,j})\cdot(t_{i,j}):=(\sum_{k\in G/H}s_{i,k}\cdot t_{k,j}).
\]
Now we fix a complete set $I$ of representatives of $G/H$ in $G$. Then $C$ has an $H$-grading given by
\[
C_h:=(A_{i-j+h})_{i,j\in I}.
\]
By \cite[Theorem 3.1]{IL} there is an equivalence
\begin{equation}\label{f functor}
F\colon\mod^G\!A\simeq \mod^H\!C
\end{equation}
sending $M=\bigoplus_{i\in G}M_i$ to $F(M)=\bigoplus_{h\in H}F(M)_h$, where $F(M)_h$ is defined by
\[
F(M)_h:=(M_{i+h})_{i\in I}
\]
and the $C$-module structure is given by
\[
(s_{i,j})_{i,j\in I}\cdot(m_i)_{i\in I}:=(\sum_{j\in I}s_{i,j}\cdot m_j)_{i\in I}.
\]
On the other hand, let $e\in C$ be the idempotent corresponding to $0\in G/H$.
Since $eCe=B$ holds, there is an exact functor
\begin{equation}\label{e functor}
E:=e(-)\colon\mod^H\!C\to\mod^H\!B
\end{equation}
such that the following diagram commutes:
\[
\begin{tikzpicture}
\node (A) at (0,0) {$\mod^G\!A$};
\node (B) at (4,0) {$\mod^H\!B$};
\node (C) at (2,-1) {$\mod^H\!C$};
\draw[->] (A) -- node[above] {$\scriptstyle (-)^H$} (B);
\draw[->] (A) -- node[below] {$\scriptstyle F$} node[above] {$\scriptstyle\sim$} (C);
\draw[->] (C) --node[below] {$\scriptstyle E$} (B);
\end{tikzpicture}
\]
The functor \eqref{e functor} has a left adjoint functor $E_\lambda:=Ce\otimes_B-\colon\mod^H\!B\to\mod^H\!C$ and a right adjoint functor $E_\rho:=\Hom_B(eC,-)\colon\mod^H\!B\to\mod^H\!C$.\\
(1)$\Leftrightarrow$(2) The functors $F$ and $E$ induce an equivalence $F\colon\qgr^G\!A\simeq\qgr^H\!C$ and a functor
\begin{equation}\label{e functor2}
E\colon\qgr^H\!C\to\qgr^H\!B
\end{equation}
respectively, which make the following diagram commutative:
\[
\begin{tikzpicture}
\node (A) at (0,0) {$\qgr^G\!A$};
\node (B) at (4,0) {$\qgr^H\!B$};
\node (C) at (2,-1) {$\qgr^H\!C$};
\draw[->] (A) -- node[above] {$\scriptstyle (-)^H$} (B);
\draw[->] (A) -- node[below] {$\scriptstyle F$} node[above] {$\scriptstyle\sim$} (C);
\draw[->] (C) --node[below] {$\scriptstyle E$} (B);
\end{tikzpicture}
\]
Thus the functor $(-)^H\colon\qgr^G\!A\to\qgr^H\!B$ is an equivalence if and only if the functor \eqref{e functor2} is an equivalence.
The functor $E_\lambda\colon\mod^H\!B\to\mod^H\!C$ induces a left adjoint functor $E_\lambda\colon\qgr^H\!B\to\qgr^H\!C$ of \eqref{e functor2}. Clearly $E E_\lambda={\rm id}_{\qgr^H\!B}$ holds, and the counit $\eta:E_\lambda E\to {\rm id}_{\qgr^H\!C}$ is given by $m\otimes_{C}-$, where $m$ is the natural morphism
\begin{equation}\label{m morphism}
m:Ce\otimes_BeC\to C.
\end{equation}
Thus the condition (1) holds if and only if $\eta$ is an isomorphism of functors if and only if $m$ is an isomorphism in $\qgr^H\!C$. On the other hand, the cokernel of $m$ is $C/(e)$, where $(e)$ is the two-sided ideal of $C$ generated by $e$, and the kernel of $m$ is a finitely generated $C/(e)$-module. Therefore (1) holds if and only if the factor algebra $C/(e)$ of $C$ is finite dimensional if and only if (2) holds, by the following observation.
\begin{lemma}\label{fin dim and (3)}
$\dim_{\KK}C/(e)<\infty$ if and only if the condition (2) holds.
\end{lemma}
\begin{proof}
Since
\[
C/(e)=(A(i-j)^H/(A(i)^H\cdot A(-j)^H))_{i,j\in I}
\]
holds, $C/(e)$ is finite dimensional if and only if $A(i-j)^H/(A(i)^H\cdot A(-j)^H)$ is finite dimensional for any $i,j\in I$.
This implies the condition (2) by considering the case $i=j$.
Conversely assume that (2) holds. Since there is a surjective map
\[
A(i-j)^H\otimes_B\frac{B}{A(j)^H\cdot A(-j)^H}=\frac{A(i-j)^H}{A(i-j)^H\cdot A(j)^H\cdot A(-j)^H}\to
\frac{A(i-j)^H}{A(i)^H\cdot A(-j)^H}
\]
whose domain is finite dimensional, the target is also finite dimensional.
Thus the assertion holds.
\end{proof}
\noindent
(2)$\Leftrightarrow$(3) Assume that $A\in\mod_2^G\!A$. Clearly the equivalence \eqref{f functor} induces equivalences
\[
F\colon\mod_0^G\!A\simeq \mod_0^H\!C\ \mbox{ and }\
F\colon\mod_2^G\!A\simeq \mod_2^H\!C.
\]
The remainder of the proof requires the following general lemma.
\begin{lemma}\label{depth 2 properties}
With the setup as above,
\begin{enumerate}
\item\label{depth 2 properties 1} The functor \eqref{e functor} induces a functor
\begin{equation}\label{e functor3}
E\colon\mod_2^H\!C\to\mod_2^H\!B.
\end{equation}
\item\label{depth 2 properties 2} The functor $E_\rho\colon\mod^H\!B\to\mod^H\!C$ induces a functor $E_\rho\colon\mod_2^H\!B\to\mod_2^H\!C$.
\item\label{depth 2 properties 3} $X\in\mod^H\!C$ belongs to $\mod_0^H\!C$ if and only if $\Ext^i_{C}(X,\mod_2^H\!C)=0$ for $i=0,1$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) Let $X\in\mod_2^H\!C$, $Y\in\mod_0^H\!B$ and ${\bf E}_\lambda Y:=Ce\Ltimes_BY$.
Since $H^i({\bf E}_\lambda Y)$ is zero for any $i>0$ and belongs to $\mod_0^H\!C$ for any $i\le0$, we have $\Hom_{\Db(\mod C)}({\bf E}_\lambda Y,X[i])=0$ for $i=0,1$.
Using $\RHom_B(Y,EX)=\RHom_C({\bf E}_\lambda Y,X)$, we have $\Ext^i_B(Y,EX)=0$ for $i=0,1$.\\
(2) Let $X\in\mod_2^H\!B$, $Y\in\mod_0^H\!C$ and ${\bf E}_\rho X:=\RHom_B(eC,X)$. Since $EY\in\mod_0^H\!B$ and $\RHom_C(Y,{\bf E}_\rho X)=\RHom_B(EY,X)$ hold, we have $\Hom_{\Db(\mod C)}(Y,{\bf E}_\rho X[i])=0$ for $i=0,1$.
There is a triangle $E_\rho X\to{\bf E}_\rho X\to Z\to E_\rho X[1]$ satisfying $H^i(Z)=0$ for all $i\le 1$. Applying $\Hom_{\Db(\mod C)}(Y,-)$ gives $\Ext^i_C(Y,E_\rho X)=0$ for $i=0,1$.\\
(3) Our assumption $A\in\mod_2^G\!A$ implies $C=\bigoplus_{i\in I}FA(i)\in\mod_2^H\!C$.
Let $0\to T\to X\to F\to0$ and $0\to\Omega F\to P\to F\to0$ be exact sequences in $\mod^H\!C$ such that $T$ is the largest submodule of $X$ which belongs to $\mod_0^H\!C$ and $P$ is an $H$-graded projective $C$-module. Then $\Omega F$ belongs to $\mod_2^H\!C$ since $C\in\mod_2^H\!C$. Applying $\Hom_C(-,\Omega F)$ to the first sequence gives an exact sequence
\[
0=\Hom_C(T,\Omega F)\to\Ext^1_C(F,\Omega F)\to\Ext^1_C(X,\Omega F)=0.
\]
Thus $\Ext^1_C(F,\Omega F)=0$ holds, and $F$ is projective in $\mod^H\!C$. Hence $X=T\oplus F$, and so $\Hom_C(X,C)=0$ implies that $F=0$. Therefore $X=T$ belongs to $\mod_0^H\!C$.
\end{proof}
It follows from \ref{depth 2 properties} that there is a commutative diagram
\[
\begin{tikzpicture}
\node (A) at (0,0) {$\mod^G_{2}A$};
\node (B) at (4,0) {$\mod^H_{2}B$};
\node (C) at (2,-1) {$\mod^H_{2}C$};
\draw[->] (A) -- node[above] {$\scriptstyle (-)^{H}$} (B);
\draw[->] (A) -- node[below] {$\scriptstyle F$} node[above] {$\scriptstyle\sim$} (C);
\draw[->] (C) --node[below] {$\scriptstyle E$} (B);
\end{tikzpicture}
\]
Thus the functor $(-)^H\colon\mod_2^G\!A\to\mod_2^H\!B$ is an equivalence if and only if the functor \eqref{e functor3} is an equivalence. By \ref{depth 2 properties}\eqref{depth 2 properties 2}, there is a right adjoint functor $E_\rho\colon\mod_2^H\!B\to\mod_2^H\!C$ of \eqref{e functor3}. Clearly $E E_\rho={\rm id}_{\mod_2^H\!B}$ holds, and the unit $\varepsilon:{\rm id}_{\mod_2^H\!C}\to E_\rho E$ is given by $\Hom_{C}(m,-)$, where $m$ is the morphism \eqref{m morphism}.
Thus the condition (3) holds if and only if $\varepsilon=\Hom_{C}(m,-)$ is an isomorphism of functors.
Now fix $X\in\mod_2^H\!C$ and apply $\Hom_{C}(-,X)$ to exact sequences $0\to (e)\to C\to C/(e)\to0$ and $0\to\Ker m\to Ce\otimes_BeC\to (e)\to0$. This gives exact sequences
\begin{eqnarray*}
&0\to\Hom_{C}(C/(e),X)\to X\to\Hom_{C}((e),X)\to\Ext^1_{C}(C/(e),X)\to0&\\
&0\to\Hom_{C}((e),X)\to\Hom_{C}(Ce\otimes_BeC,X)\to\Hom_{C}(\Ker m,X).&
\end{eqnarray*}
Therefore, if $C/(e)$ is finite dimensional, then so is $\Ker m$ and hence $\varepsilon$ is an isomorphism. Conversely, if $\varepsilon$ is an isomorphism, then $\Ext^i_{C}(C/(e),X)=0$ for $i=0,1$ for any $X\in\mod_2^H\!C$ and hence $C/(e)$ is finite dimensional by \ref{depth 2 properties}\eqref{depth 2 properties 3}. Consequently (3) is equivalent to (2), again by \ref{fin dim and (3)}.
\end{proof}
Later we need the following observation.
\begin{lemma}\label{slightly stronger}
In the setting of \ref{WPL as qgrZ2}, assume that the condition \eqref{WPL as qgrZ 3-1} is satisfied. Then for any $X\in\mod^G\!A$ and $Y\in\mod_2^G\!A$, there is an isomorphism
\[
\Hom_B(X^H,Y^H)\cong\Hom_A(X,Y)^H
\]
of $H$-graded $k$-modules.
\end{lemma}
\begin{proof}
Clearly $\Hom_B(X^H,Y^H)=\Hom_B(EFX,EFY)=\Hom_C(E_\lambda EFX,FY)$. This is isomorphic to $\Hom_C(FX,FY)=\Hom_A(X,Y)^H$ since the kernel and the cokernel of $\eta_X\colon E_\lambda EX\to X$ are finite dimensional by our assumptions.
\end{proof}
\subsection{Categorical Equivalences for Weighted Projective Lines} In this subsection, we apply the general results of the previous subsection to describe the precise conditions on $\ox\in\bL$ for which $\qgr^\bZ\! S^{\ox}\simeq\coh\bX$ holds. As before, write $S(\oy)^{\ox}:=\bigoplus_{i\in\bZ}S_{\oy+i\ox}$. This subsection does not require the condition that $\ox$ belongs to $\bL_+$. Instead we assume that $\ox$ is not torsion. Then $\bZ\ox$ is a subgroup of $\bL$ of finite index, and the following observation holds by \ref{Veron is fg2}.
\begin{lemma}\label{Veron is fg}
If $\ox\in\bL$ is not torsion, then $S$ is a finitely generated $S^{\ox}$-module.
\end{lemma}
The following is the main result in this subsection, where a special case $\ox=\ow$ was given in \cite{GL91}. Another approach can be found in \cite{H}.
\begin{thm}\label{WPL as qgrZ}
Suppose that $\ox=\sum_{i=1}^na_i\ox_i+a\oc\in\bL$ is not torsion, and denote $R:=S^{\ox}$. Then the following conditions are equivalent.
\begin{enumerate}
\item\label{WPL as qgrZ 1} The natural functor $(-)^{\ox}\colon\CM^\bL\!S\to\CM^\bZ\!R$ is an equivalence.
\item\label{WPL as qgrZ 2} The natural functor $(-)^{\ox}\colon\qgr^\bL\!S\to\qgr^\bZ\!R$ is an equivalence.
\item\label{WPL as qgrZ 3} For any $\oz\in\bL$, the ideal $I^{\oz}:=S(\oz)^{\ox}\cdot S(-\oz)^{\ox}$ of $R$ satisfies $\dim_k(R/I^{\oz})<\infty$.
\item\label{WPL as qgrZ 4} $(p_i,a_i)=1$ for all $1\le i\le n$.
\end{enumerate}
\end{thm}
\begin{proof}
(1)$\Leftrightarrow$(2)$\Leftrightarrow$(3) These are shown in \ref{WPL as qgrZ2} since $\CM^\bL\!S=\mod_2^\bL\!S$ and $\CM^\bZ\!R=\mod_2^\bZ\!R$.\\
(3)$\Rightarrow$(4). By contrapositive, assume that $a_1$ and $p_1$ are not coprime. Then the normal form of any element in $\ox_1+\bZ\ox$ (respectively, $-\ox_1+\bZ\ox$) contains a positive multiple of $\ox_1$. Thus we have
\[
I^{\ox_1}\subset Sx_1\cdot Sx_1=Sx_1^2.
\]
Therefore the condition (3) implies that the algebra $R/(R\cap Sx_1^2)$ is finite dimensional. Since $S/Sx_1^2$ is a finitely generated $R/(R\cap Sx_1^2)$-module by \ref{Veron is fg}, it is also finite dimensional. This is a contradiction since $S$ has Krull dimension two.\\
(4)$\Rightarrow$(3). Assume that $(p_i,a_i)=1$ for all $i$. If $R/I^{\oy}$ and $R/I^{\oz}$ are finite dimensional, then so is $R/I^{\oy+\oz}$ since $I^{\oy}\cdot I^{\oz}\subset I^{\oy+\oz}$ holds.
Thus we only have to show that $R/I^{\ox_i}$ is finite dimensional for each $i$ with $1\le i\le n$. We will show that $I^{\ox_i}$ contains a certain power $A$ of $x_i$ and a certain monomial $B$ of $x_j$'s with $j\neq i$. Then it is easy to check that $S/(SA+SB)$ is finite dimensional, and hence $R/(RA+RB)=(S/(SA+SB))^{\ox}$ and $R/I^{\ox_i}$ are also finite dimensional.
For the least common multiple $p$ of $p_1,\ldots,p_n$, we have $p\ox=q\oc$ for some $q>0$. Then
\[
I^{\ox_i}= S(\ox_i)^{\ox}\cdot S(-\ox_i)^{\ox}
\supset S_{\ox_i}\cdot S_{-\ox_i+p\ox}\ni x_i\cdot x_i^{p_iq-1}=x_i^{p_iq}.
\]
Thus $I^{\ox_i}$ contains a power of $x_i$.
On the other hand, since $a_i$ and $p_i$ are coprime, there exist integers $\ell$ and $m$ such that $a_i\ell+1=p_i m$.
Then the normal form of $\ox_i+\ell\ox$ does not contain a positive multiple of $\ox_i$, and hence $S(\ox_i)^{\ox}\supset S_{\ox_i+\ell\ox}$ contains a monomial of $x_j$'s with $j\neq i$.
Applying a similar argument to $S(-\ox_i)^{\ox}$, we have that $I^{\ox_i}=S(\ox_i)^{\ox}\cdot S(-\ox_i)^{\ox}$ contains a monomial of $x_j$'s with $j\neq i$. Thus the assertion follows.
\end{proof}
The following is a geometric corollary of the results in this subsection.
\begin{cor}\label{stackminres 2A}
Suppose that $0\neq\ox\in\bL_+$ and write $\ox=\sum_{i=1}^{n}a_i\ox_i+a\oc$ in normal form. If $n\ge1$ and $(p_i,a_i)=1$ for all $1\leq i\leq n$, then the fully faithful embedding
\[
\Db(\coh Y^{\ox})\hookrightarrow \Db(\coh\bT^{\ox})
\]
in \ref{stackminres} is an equivalence if and only if every $a_i=1$, that is $\ox=\sum_{i=1}^nx_i+a\oc$.
\end{cor}
\begin{proof}
We use the notation from the proof of \ref{stackminres}. Note that from the assumption $(p_i,a_i)=1$ for every $1\leq i\leq n$, necessarily each $a_i$ is non-zero. Next, the indecomposable summands of $\pi_*\cM$ are pairwise non-isomorphic by combining \cite[3.5.3]{VdB1d} and \cite[4.3]{DW2}, and the summands of $\bigoplus_{\oy\in[0,\oc\,]}S(\oy)^{\ox}$ are pairwise non-isomorphic by \ref{WPL as qgrZ}\eqref{WPL as qgrZ 1}.
The embedding in \ref{stackminres} is induced from idempotents using the observation that $\pi_*\cM$ is a summand of $\bigoplus_{\oy\in[0,\oc\,]}S(\oy)^{\ox}$. It follows that the embedding is an equivalence if and only for all $t=1,\hdots,n$, the $i$-series on arm $t$ has maximum length. By \ref{i series all} this holds if and only if every $a_i=1$.\end{proof}
\subsection{Changing Parameters}\label{changing parameters}
Our next main result, \ref{change parameters so coprime}, shows that up to a change of parameters, we can always assume the condition $(p_i,a_i)=1$ for all $1\le i\le n$ that appears in both \ref{WPL as qgrZ}\eqref{WPL as qgrZ 4} and \ref{stackminres 2A}. This requires the following lemma.
\begin{lemma}\label{basic observation}
Let $\ox=\sum_{i=1}^na_i\ox+a\oc$ be in normal form.
\begin{enumerate}
\item\label{basic observation 1} We have $S_{\ox}=(\prod_{i=1}^nx_i^{a_i})S_{a\oc}$ and $S_{\ox+m\oc}=S_{\ox}\cdot S_{m\oc}$ for all $m\geq 0$.
\item\label{basic observation 2} $S_{a\oc}$ is an $(a+1)$-dimensional vector space,
and for any $j\neq k$ a basis of $S_{a\oc}$ is given by
$t_0^\ell t_1^{a-\ell}$ with $0\le\ell\le a$.
\end{enumerate}
\end{lemma}
\begin{proof}
Both assertions are elementary, see \cite{GL1}.
\end{proof}
We now fix notation. Let $S:=S_{\bp,\bl}$, and fix a subset $I$ of $\{1,\ldots,n\}$. For each $i\in I$, choose a divisor $d_i$ of $p_i$. Let $\sp_i:=p_i/d_i$, $\bp'\!\!:=(\sp_i\mid i\in I)$, $\bl':=(\lambda_i\mid i\in I)$,
\[
S':=S_{\bp'\!,\bl'}=\frac{\KK[\st_0,\st_1,\sx_i\mid i\in I]}{(\sx_i^{\sp_i}-\ell_i(\st_0,\st_1)\mid i\in I)}
\]
and $\bL^{\! \prime}_{\phantom\prime}:=\bL(\sp_i\mid i\in I)=\langle\osx_i,\osc\mid i\in I\rangle/
(\sp_i\osx_i-\osc\mid i\in I)$.
\begin{prop}\label{Veronese trick}
With notation as above,
\begin{enumerate}
\item\label{Veronese trick 1} There is a monomorphism $\iota\colon\bL^{\! \prime}_{\phantom\prime}\to\bL$ of groups sending
$\osx_i$ to $d_i\ox_i$ for each $i\in I$ and $\osc$ to $\oc$.
\item\label{Veronese trick 2} There is a monomorphism $S'\to S$ of $k$-algebras
sending $\sx_i$ to $x_i^{d_i}$ for each $i\in I$ and $\st_j$ to $t_j$ for $j=0,1$, which induces an isomorphism
$S'\simeq \bigoplus_{\osx\in\bL^{\! \prime}_{\phantom\prime}}S_{\iota(\osx)}$.
\item\label{Veronese trick 3} Let $\ox\in\bL$ be an element with normal form
$\ox=\sum_{i\in I}a_i\ox_i+a\oc$ such that $a_i$ is a multiple of $d_i$. For $\sa_i:=a_i/d_i$ and $\osx:=\sum_{i\in I}\sa_i\osx_i+a\osc\in\bL^{\! \prime}_{\phantom\prime}$, we have $(S')^{\osx}=S^{\ox}$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Clearly $\iota$ is well-defined. Assume that $\osx\in\bL^{\! \prime}_{\phantom\prime}$ with
normal form $\osx=\sum_{i\in I}a_i\osx_i+a\osc$ belongs to the kernel of $\iota$. Then $0=\iota(\osx)=
\sum_{i\in I}a_id_i\ox_i+a\oc$, where the right
hand side is a normal form in $\bL$, and so $a_i=0=a$
for all $i$. Hence $\osx=0$.\\
(2) Take any element $\osx\in\bL^{\! \prime}_{\phantom\prime}$ with
a normal form $\osx=\sum_{i\in I}a_i\osx_i+a\osc$.
Then by \ref{basic observation}\eqref{basic observation 1}\eqref{basic observation 2}, $S'_{\osx}$ has a $\KK$-basis
\[
\st_0^{j}\st_1^{a-j}\prod_{i\in I}\sx_i^{a_i}\ \ \ 0\le j\le a.
\]
Since $\iota(\osx)$ has a normal form $\sum_{i\in I}a_id_i\ox_i+a\oc$,
it follows from \ref{basic observation}\eqref{basic observation 1}\eqref{basic observation 2} that $S_{\iota(\osx)}$ has a $\KK$-basis $t_0^{j}t_1^{a-j}\prod_{i\in I}x_i^{a_id_i}$ for $0\le j\le a$. The assertion follows.\\
(3) Immediate from \eqref{Veronese trick 2}.
\end{proof}
\begin{prop}\label{change parameters so coprime}
Suppose that $\ox\in\bL$ is not torsion, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc\in\bL$ in normal form. Let $I:=\{1\le i\le n\mid a_i\neq0\}$, and we consider the parameters $(\bp'\!,\bl')$ defined by $\bp'\!:=(\sp_i\mid i\in I)$ for $\sp_i:=p_i/(a_i,p_i)$ and $\bl':=(\lambda_i\mid i\in I)$. As above, set $\osx:=\sum_{i\in I}\sa_i\osx_i+a\osc\in\bL^{\! \prime}_{\phantom\prime}$, then the following statements hold.
\begin{enumerate}
\item\label{change parameters so coprime 1} There is an isomorphism $S_{\bp,\bl}^{\ox}\cong S_{\bp'\!,\bl'}^{\osx}$ as $\bZ$-graded $\KK$-algebras.
\item\label{change parameters so coprime 2} There are equivalences $\CM^\bZ\!S_{\bp,\bl}^{\ox}\simeq \CM^\bZ\! S_{\bp'\!,\bl'}^{\osx}\simeq \CM^\bL\! S_{\bp'\!,\bl'}$.
\item\label{change parameters so coprime 3} There are equivalences $\qgr^\bZ\!S_{\bp,\bl}^{\ox}\simeq \qgr^\bZ\!S_{\bp'\!,\bl'}^{\osx}\simeq \coh \bX_{\bp'\!,\bl'}$.
\end{enumerate}
\end{prop}
\begin{proof}
Part \eqref{change parameters so coprime 1} follows directly from \ref{Veronese trick}\eqref{Veronese trick 3}. Certainly this induces the left equivalences in \eqref{change parameters so coprime 2} and \eqref{change parameters so coprime 3}. Applying \ref{WPL as qgrZ} to $S_{\bp'\!,\bl'}^{\osx}$ gives the right equivalences in \eqref{change parameters so coprime 2} and \eqref{change parameters so coprime 3}.
\end{proof}
Thus we can always replace $\bX_{\bp,\bl}$ by some equivalent $\bX_{\bp'\!,\bl'}$ for which the coprime assumptions in both \ref{WPL as qgrZ}\eqref{WPL as qgrZ 4} and \ref{stackminres 2A} hold. Note also that the above implies that $\qgr$ of Veronese subrings (with $0\neq\ox\in\bL_+$) of weighted projective lines always give weighted projective lines, but maybe with different parameters.
\subsection{Algebraic Approach to Special CM Modules}
In this subsection we give an algebraic treatment of the special CM $S^{\ox}$-modules, and show how to determine the rank one special CM modules without assuming any of the geometry. Hence this subsection is independent of \S\ref{stack}, and the techniques developed will be used later to obtain geometric corollaries. Note however that the geometry is required to deduce that there are no higher rank indecomposable special CM modules; this algebraic approach seems only to be able to deal with the rank one specials.
Consider $\bX_{\bp,\bl}$ and let $\ox\in\bL$ be an element with normal form $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ with $a\ge0$. By \ref{change parameters so coprime} we can assume, by changing parameters if necessary, that $(a_i,p_i)=1$ for all $1\le i\le n$. As before, let $R:=S^{\ox}$, then by \ref{WPL as qgrZ} there is an equivalence
\[
(-)^{\ox}\colon\CM^\bL\!S\to\CM^\bZ\!R.
\]
Below we will often use the identification
\begin{equation}\label{Hom between Ss}
S_{\oy-\ox}\cong\Hom_S^{\bL}(S(\ox),S(\oy))
\end{equation}
for any $\ox,\oy\in\bL$. Recall that the \emph{AR translation functor} of $R$ is given by
\[
\tau_R:=\Hom_R(-,\omega_R)\circ\Hom_R(-,R)\colon\CM^{\bZ}\!R\to\CM^{\bZ}\!R,
\]
where $\omega_R$ is the $\bZ$-graded canonical module of $R$.
\begin{prop}\label{AR prep}
With the setup as above, the following statements hold.
\begin{enumerate}
\item\label{AR prep 1} There is an isomorphism $\omega_R\cong S(\ow)^{\ox}$.
\item\label{AR prep 2} There is a commutative diagram
\begin{equation}\label{Veronese equivalence}
\xymatrix{
\CM^\bL\!S\ar[r]^{(-)^{\ox}}\ar[d]_{\tau_S=(\ow)}&\CM^\bZ\!R\ar[d]^{\tau_R}\\
\CM^\bL\!S\ar[r]^{(-)^{\ox}}&\CM^\bZ\!R.}
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
(1) Using \ref{slightly stronger}, we have $\Ext^i_R(k,S(\ow)^{\ox})=\Ext^i_S(k,S(\ow))^{\ox}$, which is $k$ for $i=2$ and zero for $i\neq2$. Thus $S(\ow)^{\ox}$ is the $\bZ$-graded canonical module of $R$.\\
(2) Let $X\in\CM^{\bL}\!S$. Using \eqref{AR prep 1} and \ref{slightly stronger},
\[
\tau_R(X^{\ox})=\Hom_R(\Hom_R(X^{\ox},S^{\ox}),S(\ow)^{\ox})=\Hom_S((X,S),S(\ow))^{\ox}=X(\ow)^{\ox}.\qedhere
\]
\end{proof}
The following gives an algebraic criterion for certain CM $R$-modules to be special.
\begin{lemma}\label{criterion for special}
For $\oy\in\bL$, the CM $R$-module $S(\oy)^{\ox}$ is special
if and only if
\begin{eqnarray}
S_{\oy+\ow+\ell\ox}=\sum_{m\in\bZ}S_{\ow+m\ox}\cdot S_{\oy+(\ell-m)\ox}\label{specials equation}
\end{eqnarray}
holds for all $\ell\in\bZ$.
\end{lemma}
\begin{proof}
As above write $\tau_R\colon\CM^\bZ\!R\simeq\CM^\bZ\!R$ for the AR-translation. This yields AR duality
\begin{equation}\label{AR duality}
D\Ext^1_{\mod^{\bZ}\!R}(X,Y)\simeq\Hom_{\overline{\CM}^{\bZ}\!R}(Y,\tau_RX)
\end{equation}
for any $X,Y\in\CM^\bZ\!R$, where $\overline{\CM}^{\bZ}\!R$ is the quotient category of $\CM^\bZ\!R$ by the ideal generated by $\{\omega_R(\ell)\mid \ell\in\bZ\}$. By \ref{AR prep}\eqref{AR prep 1}, $S(\ow+\ell\ox)^{\ox}=\omega_R(\ell)$ holds, and hence there is an induced equivalence
\begin{equation}\label{Veronese equivalence2}
(-)^{\ox}:(\CM^\bL\!S)/I\simeq\overline{\CM}^{\bZ}\!R
\end{equation}
for the ideal $I$ of the category $\CM^\bL\!S$ generated by $\add\{S(\ow+\ell\ox)\mid\ell\in\bZ\}$. It follows that
\begin{eqnarray*}
D\Ext^1_R(S(\oy)^{\ox},R)&=&
\bigoplus_{\ell\in\bZ}D\Ext^1_{\mod^{\bZ}\!R}(S(\oy)^{\ox},R(\ell))\\
&\stackrel{\eqref{AR duality}}{\simeq}&
\bigoplus_{\ell\in\bZ}\Hom_{\overline{\CM}^{\bZ}\!R}(R,(\tau_R(S(\oy)^{\ox}))(\ell))\\
&\stackrel{\eqref{Veronese equivalence}\eqref{Veronese equivalence2}}{\simeq}&
\bigoplus_{\ell\in\bZ}\frac{\Hom_{\CM^\bL\!S}(S,S(\oy+\ow+\ell\ox))}{I(S,S(\oy+\ow+\ell\ox))}.
\end{eqnarray*}
Thus $S(\oy)^{\ox}$ is special if and only if
$\Hom_{\CM^\bL\!S}(S,S(\oy+\ow+\ell\ox))=I(S,S(\oy+\ow+\ell\ox))$ holds for all $\ell\in\bZ$.
Since $\Hom_{\CM^\bL\!S}(S,S(\oy+\ow+\ell\ox))=S_{\oy+\ow+\ell\ox}$
and $I(S,S(\oy+\ow+\ell\ox))=\sum_{m\in\bZ}S_{\ow+m\ox}\cdot S_{\oy+(\ell-m)\ox}$ hold by \eqref{Hom between Ss}, the assertion follows.
\end{proof}
We also require the following, which is much more elementary.
\begin{lemma}[{\cite{GL1}}]\label{basic observation B}
Suppose that $x\in\bL$.
\begin{enumerate}
\item\label{basic observation 3} If $\oy\in\bL$ with $\ox-\oy\in\bL_+$, write $\oy=\sum_{i=1}^nb_i\ox_i+b\oc$
in normal form. Then for $I:=\{1\le i\le n\mid a_i<b_i\}$,
\[
\ox\ge|I|\oc\quad\mbox{ and }\quad S_{\oy}\cdot S_{\ox-\oy}=(\prod_{i\in I}x_i^{p_i})S_{\ox-|I|\oc}.
\]
\item\label{basic observation 4}
Let $X,Y$ be a basis of $S_{\oc}$. If $\ox\ge i\oc\ge0$, then
\[
S_{\ox}=XS_{\ox-\oc}+f(X,Y)S_{\ox-i\oc}
\]
for any $f(X,Y)\in S_{i\oc}$ which is not a multiple of $X$.
\end{enumerate}
\end{lemma}
Before proving the main result \ref{specials lag approach thm}, we first illustrate a special case.
\begin{example}
Let $\os_a=\sum_{i=1}^n\ox_i+a\oc$ with $a\ge0$ and $n+a\ge2$ (since $a\geq 0$, the last condition is equivalent to $\os_a\notin[0,\oc\,]$). Then $S(\oy)^{\os_a}$ is a special CM $S^{\os_a}$-module for all $\oy\in[0,\oc\,]$.
\end{example}
\begin{proof}
We use \ref{criterion for special}. When $\ell\le0$, both sides of \eqref{specials equation} are zero. When $\ell>0$, since $\ow+\os_a=(n-2+a)\oc$ we have
\[
S_{\oy+\ow+\ell\os_a}=S_{\oy+(\ell-1)\os_a+(n-2+a)\oc}
\stackrel{\mbox{\scriptsize\ref{basic observation}\eqref{basic observation 1}}}{=}S_{(n-2+a)\oc}\cdot S_{\oy+(\ell-1)\os_a}=S_{\ow+\os_a}\cdot S_{\oy+(\ell-1)\os_a}
\]
and so \eqref{specials equation} holds.
\end{proof}
The following is the main result in this section, and does not assume any of the geometry from \S\ref{stack}. The geometry is required to obtain the $\supseteq$ statement in part \eqref{specials lag approach thm 2}, as the algebra only gives the inclusion $\subseteq$. However, the algebraic method of proof developed below feeds back into the geometry, and allows us to extract the middle self-intersection number in \ref{middle SI number}. As notation, we denote those special CM $R$-modules that are $\mathbb{Z}$-graded by $\SCM^{\bZ}\! R$.
\begin{thm}\label{specials lag approach thm}
Let $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$ and $R:=S^{\ox}$. Write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form, then the following statements hold.
\begin{enumerate}
\item\label{specials lag approach thm 1} Up to degree shift, the objects of rank one in $\SCM^{\bZ}\!R$ are precisely $S(u\ox_j)^{\ox}$ for $1\le j\le n$ and $u$ appearing in the
$i$-series for $\frac{1}{p_j}(1,-a_j)$.
\item\label{specials lag approach thm 2} Forgetting the grading, $\add\{S(u\ox_j)^{\ox}\mid j\in [1,n], u\in I(p_j,p_j-a_j) \}\subseteq \SCM R$.
\end{enumerate}
In particular, $R$, $S(\oc)^{\ox}$ and $S((p_j-a_j)\ox_j)^{\ox}$ for all $j\in[1,n]$ are always special.
\end{thm}
\begin{proof}
We only prove \eqref{specials lag approach thm 1}, since the other statements follow immediately. By \ref{change parameters so coprime}\eqref{change parameters so coprime 1} we can assume that $(a_i,p_i)=1$ for all $1\le i\le n$.
\noindent
(a) We first claim that, up to degree shift, $\bZ$-graded special CM $R$-modules of rank one must have the form $S(u\ox_j)^{\ox}$ for some $1\le j\le n$ and $0\le u\le p_j$. By \cite[1.3]{GL1} $S$ is an $\bL$-graded factorial domain, so all rank one objects in $\CM^{\bL}\!S$ have the form $S(\oy)$ for some $\oy\in\bL$. Under the rank preserving equivalence \ref{WPL as qgrZ}\eqref{WPL as qgrZ 1}, it follows that all rank one graded CM $R$-modules have the form $S(\oy)^{\ox}$ for some $\oy\in\bL$. Since we are working up to degree shift, and $\ox\ge0$, we can assume without loss of generality that $\oy\ge0$ and $\oy\ {\not\ge}\ \ox$, by if necessary replacing $\oy$ by $\oy-\ell\ox$ for some $\ell\in\bZ$.
Hence we can assume that our rank one special CM module has the form $S(\oy)^{\ox}$ with $\oy\geq 0$ and $\oy\ {\not\ge}\ \ox$.
Now assume that $\oy$ can not be written as $u\ox_j$ for some $1\le j\le n$ and $0\le u\le p_j$. Then there exists $j\neq k$ such that $\oy\ge\ox_j+\ox_k$. By applying \ref{criterion for special} for $\ell=0$, it follows that
\[
S_{\oy+\ow}=\sum_{m\in\bZ}S_{\ow+m\ox}\cdot S_{\oy-m\ox}.
\]
Now $S_{\oy+\ow}\neq0$ by our assumption $\oy\ge\ox_j+\ox_k$, hence there exists $m\in\bZ$ such that $S_{\ow+m\ox}\neq0$ and $S_{\oy-m\ox}\neq0$. On one hand, since $\ow\ {\not\ge}\ 0$, this implies that $m>0$. On the other hand, since $\oy\ {\not\ge}\ \ox$, this implies that $m\leq 0$, a contradiction. Thus the rank 1 special CM modules have the claimed form.
\noindent
(b) Let $1\le j\le n$ and $0\le u\le p_j$. We now show that $S(u\ox_j)^{\ox}$ is a special CM $R$-module if and only
if $u$ appears in the $i$-series for $\frac{1}{p_j}(1,-a_j)$. By \ref{criterion for special}, the CM $R$-module $S(u\ox_j)^{\ox}$ is special if and only if
\begin{equation}\label{special for ux_j}
S_{u\ox_j+\ow+\ell\ox}=
\sum_{m\in\bZ}S_{\ow+m\ox}\cdot S_{u\ox_j+(\ell-m)\ox}
\end{equation}
holds for all $\ell\in\bZ$, or equivalently, for all $\ell>0$ since
the left hand side vanishes for $\ell\le0$ by $u\ox_j+\ow\le \oc+\ow\ {\not\ge}\ 0$. Thus in what follows, we fix an arbitrary $\ell> 0$.
Note first that $\supseteq$ in \eqref{special for ux_j} is clear since the weight of each product in the right hand side is $u\ox_j+\ow+\ell\ox$. Hence equality holds in \eqref{special for ux_j} if and only if $\subseteq$ holds.
Also note that the term $S_{\ow+m\ox}\cdot S_{u\ox_j+(\ell-m)\ox}$ in the right hand side of \eqref{special for ux_j} is non-zero only if $1\le m\le\ell$ since $\ow\ {\not\ge}\ 0$ and $u\ox_j-\ox\le\oc-\ox\ {\not\ge}\ 0$; here we have used the assumption that $\ox\notin[0,\oc\,]$. Thus, to simplify notation, for $1\leq m\leq \ell$ write
\begin{eqnarray*}
\begin{array}{l}
\osx:=u\ox_j+\ow+\ell \ox\\
\osym:=\ow+m\ox
\end{array}
\end{eqnarray*}
then \eqref{special for ux_j} holds if and only if
\begin{equation}\label{special for ux_j easy}
S_{\osx}\subseteq
\sum_{m=1}^{\ell}S_{\osym}\cdot S_{\osx-\osym}
\end{equation}
holds. Note that $\osx$ and $\osym$ can be written more explicitly as
\begin{eqnarray}
\left.\begin{array}{l}
\osx=\left( \sum_{i\neq j}(\ell a_i-1)\ox_i \right)+(u+\ell a_j-1)\ox_j+(n-2+a\ell)\oc\\
\osym=\sum_{i=1}^n(m a_i-1)\ox_i +(n-2+am)\oc.
\end{array}\right\}\label{forms of x and y}
\end{eqnarray}
Now by \ref{basic observation B}\eqref{basic observation 3}, for each $1\leq m\leq \ell$ we have
\begin{eqnarray}
S_{\osym}\cdot S_{\osx-\osym}=
(\prod_{i\in I_{m}}x_i^{p_i})S_{\osx-|I_{m}|\oc}\label{simple product}
\end{eqnarray}
where $I_m$ is the set in \ref{basic observation B}\eqref{basic observation 3}.
As before, for an integer $k$, we denote by $[k]_{p_i}$ the integer $k'$ satisfying $0\le k'\le p_i-1$ and $k-k'\in p_i\bZ$. Simply writing out $\osx$ and $\osym$ into normal form, from \eqref{forms of x and y} we see that
\begin{equation}\label{I set}
I_{m}=\{1\le i\le n\mid [u_i+\ell a_i-1]_{p_i}<[ma_i-1]_{p_i}\}
\end{equation}
where $u_i:=u$ if $i=j$ and $u_i:=0$ otherwise.
For the case $m=\ell$, it is clear that $I_{\ell}\subseteq\{j\}$. Hence we see that
\begin{eqnarray}
S_{\osyl}\cdot S_{\osx-\osyl}\stackrel{\eqref{simple product}}{=}
(\prod_{i\in I_{\ell}}x_i^{p_i})S_{\osx-|I_{\ell}|\oc}
\supseteq x_j^{p_j}S_{\osx-\oc}.\label{ell case product}
\end{eqnarray}
Now we claim that \eqref{special for ux_j easy} holds
if and only if $j\notin I_m$ for some $1\le m\le\ell$.\\
($\Rightarrow$) Assume that \eqref{special for ux_j easy} holds. If further $j\in I_m$ for all $1\le m\le\ell$, then using
\[
S_{\osx}
\stackrel{\mbox{\scriptsize{\eqref{special for ux_j easy}}}}{=}
\sum_{m=1}^{\ell}S_{\osym}\cdot S_{\osx-\osym}
\stackrel{\mbox{\scriptsize{\eqref{simple product}}}}{=}
\sum_{m=1}^{\ell}(\prod_{i\in I_{m}}x_i^{p_i})S_{\osx-|I_{m}|\oc}
\]
we see that $x_j^{p_j}$ divides every element in $S_{\osx}$. This gives a contradiction, since we can use the normal form of $\osx$ to obtain elements of $S_{\osx}$ which are not divisible by $x_j^{p_j}$ .\\
($\Leftarrow$) Suppose that $j\notin I_m$ for some $1\le m\le\ell$. Since $\ox\ge|I_m|\oc\ge0$ holds by \ref{basic observation B}\eqref{basic observation 3}, we have
\[
S_{\osx}=x_j^{p_j}S_{\osx-\oc}+(\prod_{i\in I_{m}}x_i^{p_i})S_{\osx-|I_{m}|\oc}.
\]
by choosing $X:=x_j^{p_j}$ and $f(X,Y):=\prod_{i\in I_{m}}x_i^{p_i}$ in \ref{basic observation B}\eqref{basic observation 4}.
Finally, using \eqref{simple product} and \eqref{ell case product} this gives
\[
S_{\osx}\subseteq S_{\osyl}\cdot S_{\osx-\osyl}+S_{\osym}\cdot S_{\osx-\osym},
\]
which clearly implies \eqref{special for ux_j easy}.
Consequently, \eqref{special for ux_j easy} holds if and only if
$j\notin I_m$ for some $1\leq m\leq \ell$, which by \eqref{I set} holds if and only if $[u+\ell a_j-1]_{p_j}\ge[ma_j-1]_{p_j}$ for some $1\le m\le\ell$. By \ref{i-series region}, this holds if and only if $u$ appears in the $i$-series for $\frac{1}{p_j}(1,-a_j)$.
\end{proof}
\subsection{The Middle Self-Intersection Number}
In this subsection we use the techniques of the previous subsections to determine the middle self-intersection number in \eqref{key dual graph}. This requires the following two elementary but technical lemmas.
\begin{lemma}\label{basis of Sn-1}
Let $\ell_1,\ldots,\ell_m$ be elements in $S_{\oc}$ such that any two elements are linearly independent. Then $\prod_{j\neq1}\ell_j,\ldots,\prod_{j\neq m}\ell_j$ is a basis of $S_{(m-1)\oc}$.
\end{lemma}
\begin{proof}
Assume that the assertion holds for $m-1$. Then
$\prod_{j\neq1}\ell_j,\ldots,\prod_{j\neq m-1}\ell_j$ gives a basis of $\ell_mS_{(m-2)\oc}$.
Since $S_{(m-1)\oc}=\ell_mS_{(m-2)\oc}+\KK\prod_{j\neq m}\ell_j$ holds, the assertion also holds for $m$.
\end{proof}
The following lemma is general, and does not require $n>0$.
\begin{lemma}\label{decompose Stx-c}
Let $\ox\in\bL_+$, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form.
If $t\ge2$, then every morphism in $\Hom_S^{\bL}(S(\oc),S(t\ox))$ factors through $\add\{S(\ox+(p_i-a_i)\ox_i)\mid1\le i\le n\}$.
\end{lemma}
\begin{proof}
It suffices to show that
\[
S_{t\ox-\oc}\subset\sum_{i=1}^nS_{\ox-a_i\ox_i}\cdot S_{(t-1)\ox-(p_i-a_i)\ox_i}.
\]
For each $i$ with $1\le i\le n$, take $m_i\ge0$ and $\varepsilon_i\in\{0,1\}$ such that
\[
(t-1)a_i=[(t-1)a_i]_{p_i}+m_ip_i\ \mbox{ and }\ ta_i=[ta_i]_{p_i}+(m_i+\varepsilon_i)p_i.
\]
Let $m:=\sum_{i=1}^nm_i$ and $\varepsilon:=\sum_{i=1}^n\varepsilon_i$. Then the equality
\[
t\ox-\oc=\sum_{j=1}^n[ta_j]_{p_j}\ox_j+(m+\varepsilon-1+ta)\oc
\]
implies that
\begin{equation}\label{tx-c}
S_{t\ox-\oc}=(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})S_{(m+\varepsilon-1+ta)\oc}.
\end{equation}
Similarly the equality
\[
(t-1)\ox-(p_i-a_i)\ox_i=[ta_i]_{p_i}\ox_i+\sum_{j\neq i}[(t-1)a_j]_{p_j}\ox_j+(m+\varepsilon_i-1+(t-1)a)\oc
\]
implies that
\[
S_{(t-1)\ox-(p_i-a_i)\ox_i}=x_i^{[ta_i]_{p_i}}(\prod_{j\neq i}x_j^{[(t-1)a_j]_{p_j}})S_{(m+\varepsilon_i-1+(t-1)a)\oc}.
\]
Multiplying $S_{\ox-a_i\ox_i}=(\prod_{j\neq i}x_j^{a_j})S_{a\oc}$ and using $[(t-1)a_j]_{p_j}+a_j=[ta_j]_{p_j}+\varepsilon_jp_j$ gives
\begin{equation}\label{product of S}
S_{\ox-a_i\ox_i}\cdot S_{(t-1)\ox-(p_i-a_i)\ox_i}=(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})(\prod_{j\neq i}x_j^{\varepsilon_jp_j})S_{a\oc}\cdot S_{(m+\varepsilon_i-1+(t-1)a)\oc}.
\end{equation}
Now set $I:=\{1\le i\le n\mid \varepsilon_i=1\}$. Clearly $|I|=\varepsilon$ holds.
First we assume $I\neq\emptyset$. By \ref{basis of Sn-1} we have $\sum_{i\in I}\KK\prod_{j\neq i}x_j^{\varepsilon_jp_j}=S_{(\varepsilon-1)\oc}$ and thus
\begin{eqnarray*}
\sum_{i\in I}S_{\ox-a_i\ox_i}\cdot S_{(t-1)\ox-(p_i-a_i)\ox_i}&\stackrel{\eqref{product of S}}{=}&(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})S_{(\varepsilon-1)\oc}\cdot S_{a\oc}\cdot S_{(m+(t-1)a)\oc}\\
&=&(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})S_{(m+\varepsilon-1+ta)\oc}\\
&\stackrel{\eqref{tx-c}}{=}&S_{t\ox-\oc},
\end{eqnarray*}
as desired.
Next we assume $I=\emptyset$.
If further $m-1+(t-1)a\ge0$, then \eqref{product of S} is equal to
\[
(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})S_{a\oc}\cdot S_{(m-1+(t-1)a)\oc}=(\prod_{j=1}^nx_j^{[ta_j]_{p_j}})S_{(m-1+ta)\oc}\stackrel{\eqref{tx-c}}{=}S_{t\ox-\oc},
\]
as desired, so we can assume that $m-1+(t-1)a<0$. But $a\ge0$ since $\ox\in\bL_+$, and $t\ge2$ by assumption, so necessarily $m=0=a$.
Then $m-1+ta<0$ holds, so $S_{t\ox-\oc}=0$ by \eqref{tx-c}, which implies the assertion.
\end{proof}
The following is the main algebraic result of this subsection; the main point is that the manipulations above involving the combinatorics of the weighted projective line give the geometric corollary in \ref{middle SI number} below.
\begin{thm}\label{number of arrows}
Let $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. Set $R:=S^{\ox}$ and $N:=S(\oc)^{\ox}$, and consider their completions $\mathfrak{R}$ and $\widehat{N}$. Then in the quiver of the reconstruction algebra of $\mathfrak{R}$, the number of arrows from $\widehat{N}$ to $\mathfrak{R}$ is $a$.
\end{thm}
\begin{proof}
By \ref{change parameters so coprime}\eqref{change parameters so coprime 2}, $S_{\bp,\bl}^{\ox}\cong S_{\bp'\!,\bl'}^{\osx}$ as $\bZ$-graded algebras, where $\osx:=\sum_{i\in I}\sa_i\osx_i+a\osc\in\bL^{\! \prime}_{\phantom\prime}$ satisfies the hypothesis in \ref{WPL as qgrZ}\eqref{WPL as qgrZ 4}. Note that this change in parameters has not changed the value $a$ on $\oc$, hence in what follows, we can assume that $\CM^\bL\!S\simeq\CM^\bZ\!R$ holds, via the functor $(-)^{\ox}$.
Let $\cC$ be the full subcategory of $\CM^{\bL}\!S$ corresponding to $\SCM^\bZ\!R$ via the functor $(-)^{\ox}$.
Then the number of arrows from $\widehat{N}$ to $\mathfrak{R}$ is equal to the dimension of the $\KK$-vector space
\[
\frac{{\rm rad}_{\SCM\mathfrak{R}}(\widehat{N},\mathfrak{R})}{{\rm rad}^2_{\SCM\mathfrak{R}}(\widehat{N},\mathfrak{R})}
\cong\prod_{t\in\bZ}\frac{\Hom_R^{\bZ}(N,R(t))}{{\rm rad}^2_{\SCM^{\bZ}\!R}(N,R(t))}\cong\prod_{t\in\bZ}\frac{\Hom_S^{\bL}(S(\oc),S(t\ox))}{{\rm rad}^2_{\cC}(S(\oc),S(t\ox))}.
\]
By \ref{specials lag approach thm}, $\cC$ is the additive closure of $S(u\ox_j+s\ox)$, where $s\in\bZ$, $1\le j\le n$ and $u$ appears in the $i$-series for $\frac{1}{p_j}(1,-a_j)$. We split into three cases.
\begin{enumerate}
\item If $t\le 0$, then $\Hom_S^{\bL}(S(\oc),S(t\ox))=0$.
\item If $t\ge2$, then since $S(\ox+(p_i-a_i)\ox_i)$ belongs to $\cC$ and is not isomorphic to both $S(\oc)$ and $S(t\ox)$ (since $\ox\notin[0,\oc\,]$), we have $\Hom_S^{\bL}(S(\oc),S(t\ox))={\rm rad}^2_{\cC}(S(\oc),S(t\ox))$ by \ref{decompose Stx-c}.
\item Suppose that $t=1$. By definition any morphism in ${\rm rad}^2_{\cC}(S(\oc),S(\ox))$ can be written as
a sum of compositions $S(\oc)\to S(u\ox_j+s\ox)\to S(\ox)$.
If $s\le0$, then
\[
\Hom_S^{\bL}(S(\oc),S(u\ox_j+s\ox))\cong S_{u\ox_j+s\ox-\oc}=0,
\]
and if $s\ge1$, then
\[
\Hom_S^{\bL}(S(u\ox_j+s\ox),S(\ox))\cong S_{(1-s)\ox-u\ox_j}=0.
\]
It follows that ${\rm rad}^2_{\cC}(S(\oc),S(t\ox))=0$ in this case.
\end{enumerate}
Combining all cases, the desired number is thus
\[
\sum_{t\in\bZ}\dim_{\KK}\left(\frac{\Hom_S^{\bL}(S(\oc),S(t\ox))}{{\rm rad}^2_{\cC}(S(\oc),S(t\ox))}\right)=\dim_{\KK}\Hom_S^{\bL}(S(\oc),S(\ox))=\dim_{\KK}S_{\ox-\oc}
\stackrel{\scriptsize\mbox{\ref{basic observation}}}{=}a.\qedhere
\]
\end{proof}
This allows us to finally complete the proof of \ref{dual graph general intro} from the introduction.
\begin{cor}\label{middle SI number}
Let $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, and write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. Then $\pi\colon Y^{\ox}\to\Spec S^{\ox}$ is the minimal resolution, and its dual graph is precisely \eqref{key dual graph} with $\beta=a+v=a+\#\{i\mid a_i\neq0\}$.
\end{cor}
\begin{proof}
We know from \ref{when Y min res} that $\pi$ is the minimal resolution, and we know from construction of $Y^{\ox}$ that all the self-intersection numbers are determined by the continued fraction expansions, except the middle curve $E_i$ corresponding to the special CM module $S(\oc)^{\ox}$. The dual graph does not change under completion. By \ref{number of arrows} the number of arrows in the reconstruction algebra from the middle vertex to the vertex $\circ$ is $a$. Thus the calculation \eqref{Zf dot Ei} combined with \ref{GL2 for R} shows that $a=-E_i\cdot Z_f=\beta-v$.
\end{proof}
\subsection{The Reconstruction Algebra and its $\qgr$}
Using the above subsections, we next describe the quiver of the reconstruction algebra and determine the associated $\qgr$ category. Consider the dual graph \eqref{key dual graph}, then with the convention that we only draw the arms that are non-empty, we see from \eqref{Zf dot Ei} and $Z_K\cdot E_i=E_i^2+2$ that
\begin{equation}\label{ZfZk dot Ei}
\begin{array}{cc}
((Z_K-Z_f)\cdot E_i)_{i}=&
\begin{array}{c}
\begin{tikzpicture}[xscale=0.75,yscale=0.75]
\node (0) at (0,0) {$\scriptstyle 2-v$};
\node (A1) at (-3,1) {$\scriptstyle 0$};
\node (A2) at (-3,2) {$\scriptstyle 0$};
\node (A3) at (-3,3) {$\scriptstyle 0$};
\node (A4) at (-3,4) {$\scriptstyle 1$};
\node (B1) at (-1.5,1) {$\scriptstyle 0$};
\node (B2) at (-1.5,2) {$\scriptstyle 0$};
\node (B3) at (-1.5,3) {$\scriptstyle 0$};
\node (B4) at (-1.5,4) {$\scriptstyle 1$};
\node (C1) at (0,1) {$\scriptstyle 0$};
\node (C2) at (0,2) {$\scriptstyle 0$};
\node (C3) at (0,3) {$\scriptstyle 0$};
\node (C4) at (0,4) {$\scriptstyle 1$};
\node (n1) at (2,1) {$\scriptstyle 0$};
\node (n2) at (2,2) {$\scriptstyle 0$};
\node (n3) at (2,3) {$\scriptstyle 0$};
\node (n4) at (2,4) {$\scriptstyle 1$};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,4.25) {};
\draw (A1) -- (0);
\draw (B1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (B2) -- (B1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (B4) -- (B3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\end{tikzpicture}
\end{array}
\end{array}
\end{equation}
Note that the cases $v=0$ and $v=1$ are degenerate, and are already well understood \cite{WemA}. Therefore in the next result, we only consider the case $v\geq 2$.
Inspecting the list of special CM $S^{\ox}$-modules in \ref{specials determined thm}, the conditions in \ref{Gamma convention remark} are satisfied, so we consider the particular choice of reconstruction algebra
\[
\Gamma_{\ox}:=\End_{S^{\ox}}\left(S^{\ox}\oplus(\bigoplus_{j\in [1,n], u} S(u\ox_j)^{\ox})\oplus S^{\ox}(\oc)\right),
\]
where $u$ in the middle direct sum ranges over $I(p_j,p_j-a_j)\backslash \{0,p_j\}$. Since the above $S^{\ox}$-modules are clearly $\bZ$-graded, this induces a $\bZ$ grading on $\Gamma_{\ox}$.
\begin{cor}\label{recon quiver and number relations}
For $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, write $\ox=\sum_{i=1}^na_i\ox_i+a\oc$ in normal form. For each $i$ with $a_i\neq 0$, as before $m_i$ is defined via $\frac{p_i}{p_i-a_i}=[\alpha_{i1},\hdots,\alpha_{im_i}]$, and if $a_i=0$ set $m_i=0$. Suppose that $v=\#\{i\mid a_i\neq 0\}$ satisfies $v\geq 2$. Then the reconstruction algebra $\Gamma_{\!\ox}$ can be presented as a quiver with relations, where the relations are homogeneous with respect to the natural grading, and the quiver is the following: we first consider the double quiver of the dual graph \eqref{key dual graph} and add an extending vertex (denoted $\begin{tikzpicture} \node at (0,0) [cvertex] {};\end{tikzpicture}$) as follows:
\begin{equation}\label{recon quiver}
\begin{array}{c}
\\
\begin{tikzpicture}[xscale=1,yscale=1,bend angle=30, looseness=1]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-3,1) [vertex] {};
\node (A2) at (-3,2) [vertex] {};
\node (A3) at (-3,3) [vertex] {};
\node (A4) at (-3,4) [vertex] {};
\node (B1) at (-1.5,1) [vertex] {};
\node (B2) at (-1.5,2) [vertex] {};
\node (B3) at (-1.5,3) [vertex] {};
\node (B4) at (-1.5,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1.5,2.5) {$\hdots$};
\node (T) at (0,5) [cvertex] {};
\draw [->] (A1)+(-30:4.5pt) -- ($(0) + (190:4.5pt)$);
\draw [->] (B1) --(0);
\draw [->] (C1) --(0);
\draw [->] (n1) --(0);
\draw [->] (A2) -- (A1);
\draw [->] (B2) --(B1);
\draw [->] (C2) --(C1);
\draw [->] (n2) -- (n1);
\draw [->] (A4) -- (A3);
\draw [->] (B4) --(B3);
\draw [->] (C4) --(C3);
\draw [->] (n4) -- (n3);
\draw [->] (T)+(-190:4.5pt) -- ($(A4) + (30:4.5pt)$);
\draw [->] (T) -- (B4);
\draw [->] (T) -- (C4);
\draw [->] (T) -- (n4);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (A1)+(-50:4.5pt) to ($(0) + (200:4.5pt)$);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (B1)+(-50:4.5pt) to ($(0) + (160:4.5pt)$);
\draw [bend right, <-, red](C1) to (0);
\draw [bend right, bend angle=10, looseness=0.5, <-, red](n1) to (0);
\draw [bend right, <-, red] (A2) to (A1);
\draw [bend right, <-, red] (B2) to (B1);
\draw [bend right, <-, red] (C2) to (C1);
\draw [bend right, <-, red] (n2) to (n1);
\draw [bend right, <-, red] (A4) to (A3);
\draw [bend right, <-, red] (B4) to (B3);
\draw [bend right, <-, red] (C4) to (C3);
\draw [bend right, <-, red] (n4) to (n3);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (T)+(-200:4.5pt) to ($(A4) + (50:4.5pt)$);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (T)+(-155:4.5pt) to ($(B4) + (60:4.5pt)$);
\draw [bend right, <-, red] (T) to (C4);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (T) to (n4);
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=4pt,yshift=0pt]
(2,1) -- (2,4) node [black,midway,xshift=0.55cm]
{$\scriptstyle m_n$};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=4pt,yshift=0pt]
(0,1) -- (0,4) node [black,midway,xshift=0.55cm]
{$\scriptstyle m_3$};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=4pt,yshift=0pt]
(-1.5,1) -- (-1.5,4) node [black,midway,xshift=0.55cm]
{$\scriptstyle m_2$};
\draw [decorate,decoration={brace,amplitude=5pt,mirror},xshift=4pt,yshift=0pt]
(-3,1) -- (-3,4) node [black,midway,xshift=0.55cm]
{$\scriptstyle m_1$};
\end{tikzpicture}
\end{array}
\end{equation}
where by convention if $m_i=0$ the $i$th arm does not exist.
Further, we add extra arrows subject to the following rules:
\begin{enumerate}
\item If some $\alpha_{ij}>2$, add $\alpha_{ij}-2$ extra arrows from that vertex to the top vertex.
\item Add further $a$ arrows from the bottom vertex to the top vertex.
\end{enumerate}
\end{cor}
\begin{proof}
As in \cite[\S4]{D1}, we first work on the completion, which is naturally filtered, then go back to the graded setting by taking the associated graded ring. Doing this, the result is then immediate from \eqref{Zf dot Ei}, \eqref{ZfZk dot Ei} and \ref{GL2 for R}.
\end{proof}
It is possible to describe the relations too in this level of generality, but for notational ease we will only do this for the $0$-Wahl Veronese in \S\ref{Hilb series and Veronese} below. However, in full generality, we do have the following:
\begin{prop}\label{deg 0 general prop}
Suppose that $0\neq \ox=\sum_{i=1}^na_i\ox_i+a\oc\in\bL_+$ with $\ox\notin[0,\oc\,]$. Then, with notation as in \ref{recon quiver and number relations}, $(\Gamma_{\!\ox})_0$, the degree zero part of the reconstruction algebra $\Gamma_{\!\ox}$, is isomorphic to the canonical algebra $\Lambda_{\bq,\bm}$, where $I:=\{i\in[1,n]\mid a_i\neq 0\}$, $\bq:=(m_i+1)_{i\in I}$ and $\bm:=(\lambda_i)_{i\in I}$.
\end{prop}
\begin{proof}
By \ref{change parameters so coprime} we can change parameters to assume that the coprime assumption of \ref{WPL as qgrZ} holds. Thus we have $\CM^{\bZ}\!R\simeq\CM^{\bL}\!S$ and hence
\[
(\Gamma_{\!\ox})_0\cong\End^{\bL}_S(\bigoplus_u S(u\ox_i)),
\]
where $u$ ranges over the respective $i$-series. But by \cite{GL1} it is well known that the ring $\End^{\bL}_S(\bigoplus_{\oy\in[0,\oc\,]} S(\oy)^{\ox})$ is isomorphic to a canonical algebra, and thus $(\Gamma_{\!\ox})_0$ is obtained from this by composing arrows through the vertices which do not appear in the $i$-series. It is clear that this gives the canonical algebra $\Lambda_{\bq,\bm}$.\end{proof}
When $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, we will next show in \ref{qgrR via qgrLambda} that $\qgr^\bZ\! S^{\ox}\simeq \qgr^\bZ\!\Gamma_{\!\ox}$, since this then allows us to interpret the weighted projective line as a `noncommutative scheme' over the canonical algebra. This result requires two lemmas. As notation, set $R:=S^{\ox}$, let $N:=\bigoplus_uS(u\ox_i)^{\ox}$ be the sum of all the indecomposable special CM $R$-modules.
\begin{lemma}\label{canonicalEnd}
$\End_{\qgr^\bZ\! R}(N)\cong \End_{\gr^\bZ\! R}(N)\cong (\Gamma_{\!\ox})_0\cong \Lambda_{\bq,\bm}$.
\end{lemma}
\begin{proof}
Since $N\in\mod_2^\bZ\!R$ the first isomorphism is \cite[2.1]{GL91}. The second one is clear. The final isomorphism is \ref{deg 0 general prop}.
\end{proof}
\begin{prop}\label{qgrR via qgrLambda}
For $\ox\in\bL_+$ with $\ox\notin[0,\oc\,]$, there is an equivalence
\[
\qgr^\bZ\! S^{\ox}\simeq\qgr^\bZ\!\Gamma_{\!\ox},
\]
where $\Gamma_{\!\ox}$ is a $\bZ$-graded $k$-algebra such that $(\Gamma_{\!\ox})_i$ is $\Lambda_{\bq,\bm}$ for $i=0$ and zero for $i<0$.
\end{prop}
\begin{proof}
We apply \ref{comm to noncomm}, with the last statement simply being \ref{canonicalEnd}. Let $B:=\Gamma=\End_{A}(N)$ and let $e$ be the idempotent corresponding to the summand $R$. Clearly $A:=eBe\cong R$. Note that $\dim_k(B/\langle e\rangle)<\infty$ since the normality of $R$ implies that $\add N_{\p}=\add R_{\p}$ for any non-maximal ideal $\p$ of $R$. This shows that $\End_{R_{\p}}(N_{\p})$ is a matrix algebra over $R_{\p}$ for all non-maximal $\p$, so $(B/\langle e\rangle)_{\p}=0$.
\end{proof}
\section{The $0$-Wahl Veronese}\label{Hilb series and Veronese}
Throughout this section we work with an arbitrary $\bX_{\bp,\bl}$ with $n\ge3$, and consider the $0$-Wahl Veronese from the introduction, namely $S^{\os}$, where $\os=\sum_{i=1}^n\ox_i$. It is not too hard, but more notationally complicated, to extend to cover the case $\os_a=\os+a\oc$, but we shall not do this here. We investigate the more general $S^{\os_a}$ for Dynkin type in \S\ref{domestic section}.
\subsection{Presenting the $0$-Wahl Veronese}
The aim of this subsection is to give a presentation of the $0$-Wahl Veronese subring $S^{\os}$ of $S$ by constructing an isomorphism $S^{\os}\cong R_{\bp,\bl}$.
We define elements of $S^{\os}$ as follows:
\begin{eqnarray*}
\mathsf{u}_i&:=&\left\{\begin{array}{ll}x_1^{p_1+p_2}x_3^{p_2}\hdots x_n^{p_2}&i=1,\\
x_2^{p_1+p_2}x_3^{p_1}\hdots x_n^{p_1}&i=2,\\
-x_1^{p_i}x_2^{p_2+p_i}x_3^{p_i}\hdots \widehat{x_i}\hdots x_n^{p_i}&3\leq i\leq n,\end{array}\right.\\
\mathsf{v}&:=&x_1x_2\hdots x_n,
\end{eqnarray*}
where we write $\widehat{x_i}$ to mean `omit $x_i$'. Then $\mathsf{v}$ is homogeneous of degree one, and $\mathsf{u}_i$ is homogeneous of degree $p_2$ if $i=1$, $p_1$ if $i=2$ and $p_i$ if $3\le i\le n$.
To construct an isomorphism between $R_{\bp,\bl}$ and $S^{\os}$, we first construct a morphism of graded algebras.
\begin{lemma}\label{check relations}
The morphism $\KK[u_1,\ldots,u_n,v]\to S^{\os}$ of graded algebras given by $u_i\mapsto\mathsf{u}_i$ for $1\le i\le n$ and $v\mapsto\mathsf{v}$ indues a morphism $R_{\bp,\bl}\to S^{\os}$ of graded algebras.
\end{lemma}
\begin{proof}
It suffices to show that all $2\times2$ minors of the following matrix has determinant zero.
\begin{equation}\label{R matrix}
\left(
\begin{array}{ccccc}
{\mathsf u}_2&{\mathsf u}_3&\hdots&{\mathsf u}_{n}&{\mathsf v}^{p_2}\\
{\mathsf v}^{p_1}&\lambda_3{\mathsf u}_3+{\mathsf v}^{p_3}&\hdots&\lambda_n{\mathsf u}_n+{\mathsf v}^{p_n}&{\mathsf u}_1
\end{array}
\right)
\end{equation}
Since $S^{\os}$ is a domain, it suffices to show that all $2\times2$ minors containing the last column has determinant zero. The outer $2\times 2$ minor has determinant
\[
\mathsf{u}_1\mathsf{u}_2-\mathsf{v}^{p_1+p_2}=(x_1\hdots x_n)^{p_1+p_2}-(x_1\hdots x_n)^{p_1+p_2}=0.
\]
Further for any $i\geq 3$, using the relation $x_1^{p_1}=\lambda_ix_2^{p_2}-x_i^{p_i}$ it follows that
\begin{align*}
\mathsf{u}_1\mathsf{u}_i&=-x_1^{p_1+p_2+p_i}x_2^{p_2+p_i}x_3^{p_2+p_i}\hdots x_i^{p_2}\hdots x_n^{p_2+p_i}\\
&=-(\lambda_ix_2^{p_2}-x_i^{p_i})x_1^{p_2+p_i}x_2^{p_2+p_i}x_3^{p_2+p_i}\hdots x_i^{p_2}\hdots x_n^{p_2+p_i}\\
&=x_1^{p_2}x_2^{p_2}\hdots x_n^{p_2}(-\lambda_i x_1^{p_i}x_2^{p_2+p_i}x_3^{p_i}\hdots \widehat{x_i}\hdots x_n^{p_i}+x_1^{p_i}x_2^{p_i}\hdots x_n^{p_i}) \\
&=\mathsf{v}^{p_2}(\lambda_i\mathsf{u}_i+\mathsf{v}^{p_i}).
\end{align*}
Thus the $2\times 2$ minor consisting of $i$th column and the last one has determinant zero.
\end{proof}
The following calculation is elementary.
\begin{prop}\label{S is generated}
\begin{enumerate}
\item The $\KK$-algebra $S^{\os}$ is generated by $\mathsf{v}$ and $\mathsf{u}_i$ with $1\le i\le n$.
\item The $\KK$-vector space $S^{\os}/\mathsf{v}S^{\os}$ is generated by $\mathsf{u}_i^\ell$ with $1\le i\le n$ and $\ell\ge0$.
\end{enumerate}
\end{prop}
\begin{proof}
It is enough to prove (2). Let $V$ be a subspace of $S^{\os}/\mathsf{v}S^{\os}$ generated by $\mathsf{u}_i^\ell$ with $1\le i\le n$ and $\ell\ge0$. Take any monomial $X:=x_1^{a_1}\hdots x_n^{a_n}$ in $S_{N\os}$ with $N>0$, then
\[
a_1\ox_1+\hdots+a_n\ox_n=N\ox_1+\hdots +N\ox_n.
\]
For each $1\le i\le n$, there exists $\ell_i\in\bZ$ such that $a_i=N+\ell_ip_i$. Then $\sum_{i=1}^n\ell_i=0$ holds.\\
(i) We show that $X$ belongs to $V$ if there exists $1\le i\le n$ satisfying $\ell_i\le0$, $\ell_j\ge0$ and $\ell_k=0$ for all $k\neq i,j$, where $j$ is defined by $j:=2$ if $i\neq2$ and $j:=1$ if $i=2$.
If $a_i\neq0$, then $X$ belongs to $\mathsf{v}S^{\os}$.
Assume $a_i=0$. Then $N=-\ell_ip_i$ and $a_j=N+\ell_jp_j=-\ell_i(p_i+p_j)$ hold. We have
\[X=x_j^{a_j}\prod_{k\neq i,j}x_k^N=(x_j^{p_i+p_j}\prod_{k\neq i,j}x_k^{p_i})^{-\ell_i}=\pm\mathsf{u}_{i'}^{-\ell_i},\]
where $i':=2$ if $i=1$, $i':=1$ if $i=2$ and $i'=i$ if $i\ge 3$.
Thus the assertion follows.\\
(ii) We consider the general case.
Using induction on $\ell(X):=\sum_{1\le i\le n,\ \ell_i>0}\ell_i$, we show that $X$ belongs to $V$.
Assume $\ell(X)=0$. Then $X=\mathsf{v}^N$ holds, and hence $X$ belongs to $V$.
Assume that there exist $1\le i\neq j\le n$ such that $\ell_i<0$ and $\ell_j<0$.
Take $1\le k\le n$ such that $\ell_k>0$. Using the relation $x_k^{p_k}=\lambda'x_i^{p_i}+\lambda''x_j^{p_j}$ with $\lambda',\lambda''\in\KK$, we have $X=\lambda'X'+\lambda''X''$ for some monomials $X',X''$ satisfying $\ell(X')<\ell(X)$ and $\ell(X'')<\ell(X)$.
Since $Y$ and $Y'$ belongs to $V$, so does $X$.
In the rest, assume that there exists unique $1\le i\le n$ satisfying $\ell_i<0$. Define $j$ by $j:=2$ if $i\neq2$ and $j:=1$ if $i=2$.
Using the relation $x_k^{p_k}=\lambda'_kx_i^{p_i}+\lambda''_kx_j^{p_j}$ with $\lambda'_k,\lambda''_k\in\KK$, we have
\[X=x_i^{a_i}x_j^{a_j}\prod_{k\neq i,j}x_k^N(\lambda'_kx_i^{p_i}+\lambda''_kx_j^{p_j})^{\ell_k}.\]
This is a linear combination of monomials $Y=x_i^{b_i}x_j^{b_j}\prod_{k\neq i,j}x_k^N$ which satisfies the condition in (i).
Thus $X$ belongs to $V$.
\end{proof}
The above leads to the following, which is the main result of this subsection.
\begin{thm}\label{Veronese result}
There is an isomorphism $R_{\bp,\bl}\cong S^{\os}$ of graded algebras given by $u_i\mapsto\mathsf{u}_i$ for $1\le i\le n$ and $v\mapsto\mathsf{v}$.
\end{thm}
\begin{proof}
Combining \ref{check relations} and \ref{S is generated}, there is a surjective graded ring homomorphism $\vartheta\colon R_{\bp,\bl}\twoheadrightarrow S^{\os}$. But now $R_{\bp,\bl}$, being a rational surface singularity, is automatically a domain. Since $S^{\os}$ is two-dimensional, $\vartheta$ must be an isomorphism.
\end{proof}
\subsection{Special CM $S^{\os}$-Modules and the Reconstruction Algebra}\label{Sect 5.2}
The benefit of our Veronese construction of $R_{\bp,\bl}$ is that it also produces the special CM modules, and we now describe these explicitly as $2$-generated ideals. We first do this in the notation of $S$, then translate into the co-ordinates $\mathsf{u}_1,\hdots,\mathsf{u}_n,\mathsf{v}$.
\begin{prop}\label{2 gen as S modules}
The following are, up to degree shift, precisely the $\bZ$-graded special CM $S^{\os}$-modules. Moreover, they have the following generators and degrees:
\[
\begin{tabular}{*3c}
\toprule
Module & Generators & Degree of generators\\
\midrule
$S(q\ox_1)^{\os}$& $x_2^{p_2}(x_2x_3\hdots x_n)^{p_1-q}$ and $ x_1^{q}$ & $p_1-q$ and $0$\\
$S(q\ox_2)^{\os}$& $x_1^{p_1}(x_1x_3\hdots x_n)^{p_2-q}$ and $x_2^{q}$ & $p_2-q$ and $0$\\
$S(q\ox_i)^{\os}$& $x_2^{p_2}(x_1\hdots\widehat{x_i}\hdots x_n)^{p_i-q}$ and $x_i^{q}$& $p_i-q$ and $0$\\
$S(\oc)^{\os}$ & $x_1^{p_1}$ and $x_2^{p_2}$& $0$ and $0$\\
\bottomrule
\end{tabular}
\]
where in row one $q\in[1, p_1]$, in row two $q\in [1,p_2]$, and in row three $i\in [3,n], \,q\in [1,p_i]$.
\end{prop}
\begin{proof}
The first statement is \ref{specials lag approach thm}\eqref{specials lag approach thm 1}. We only prove the generators (and their degrees) for $S(q\ox_1)^{\os}$ since all other cases are similar. Let $M$ be a submodule of $S(q\ox_1)^{\os}$ generated by $g_1:=x_1^q$ and $g_2:=x_2^{p_1+p_2-q}x_3^{p_1-q}\hdots x_n^{p_1-q}$. To prove $M=S(q\ox_1)^{\os}$, it suffices to show that any monomial $X=x_1^{a_1}\hdots x_n^{a_n}\in S(q\ox_1)^{\os}$ of degree $N\ge0$ has either $g_1$ or $g_2$ as a factor. Since
\[
a_1\ox_1+\hdots+a_n\ox_n=(N+q)\ox_1+N\ox_2+\hdots +N\ox_n.
\]
holds, there exists $\ell_i\in\bZ$ for each $1\le i\le n$ such that $a_1=N+q+\ell_1p_1$ and $a_i=N+\ell_ip_i$ for $i\ge2$.
Then $\sum_{i=1}^n\ell_i=0$ holds.\\
(i) If $a_1\ge q$, then $X$ belongs to $M$ since $X$ has $g_1=x_1^q$ as a factor.\\
(ii) We show that $X$ belongs to $M$ if $\ell_3=\hdots=\ell_n=0$.
By (i), we can assume that $a_1<q$ and hence $\ell_1<0$. Then $N=a_1-q-\ell_1p_1\ge p_1-q$ holds. Since $\ell_2=-\ell_1>0$, we have $a_2=N+\ell_2p_2\ge p_1+p_2-q$, which implies that $X=x_1^{a_1}x_2^{a_2}x_3^N\hdots x_n^N$ has $g_2=x_2^{p_1+p_2-q}x_3^{p_1-q}\hdots x_n^{p_1-q}$ as a factor.\\
\noindent(iii) We show that $X$ belongs to $M$ if all $\ell_3, \hdots,\ell_n$ are non-positive. By (i), we can assume that $a_1<q$ and hence $\ell_1<0$.
Using $\ell_2=-\sum_{i\neq 2}\ell_i$ and the relation $x_2^{p_2}=\lambda'_ix_1^{p_1}+\lambda''_ix_i^{p_i}$, it follows that
\[
X=x_1^{a_1}x_2^{N-\sum_{i\neq 2}\ell_ip_2}x_3^{a_3}\hdots x_n^{a_n}=x_1^{a_1}x_2^{N-\ell_1p_2}\prod_{i\ge3}x_i^{a_i}(\lambda'_ix_1^{p_1}+\lambda''_ix_i^{p_i})^{-\ell_i}.
\]
This is a linear combination of monomials which have $g_1=x_1^q$ as a factor and a monomial $x_1^{a_1}x_2^{N-\ell_1p_2}\prod_{i\ge3}x_i^{a_i-\ell_ip_i}=x_1^{a_1}x_2^{N-\ell_1p_2}x_3^N\hdots x_n^N$ satisfying (ii). Thus $X$ belongs to $M$.\\
(iv) We show that $X$ belongs to $M$ in general. Let $\ell_i^+=\max\{\ell_i,0\}$ and $\ell_i^-=\min\{\ell_i,0\}$, then $\ell_i=\ell_i^++\ell_i^-$. Further, using the relation $x_i^{p_i}=-x_1^{p_1}-\lambda_ix_2^{p_2}$,
\[
X=x_1^{a_1}x_2^{a_2}\prod_{i\ge3}x_i^{N+\ell_ip_i}=-x_1^{a_1}x_2^{a_2}\prod_{i\ge3}x_i^{N+\ell_i^-p_i}(x_1^{p_1}+\lambda_ix_2^{p_2})^{\ell^+_i}.
\]
This is a linear combination of monomials satisfying (iii), so $X$ belongs to $M$.
\end{proof}
Using \ref{S is generated} we now translate the modules in \ref{2 gen as S modules} into ideals.
\begin{prop}\label{dual graph assignment}
With notation in \ref{Veronese result}, up to degree shift, the indecomposable objects in $\SCM^{\bZ}\!S^{\os}$ are precisely the following ideals of $S^{\os}$, and furthermore they correspond to the dual graph of the minimal resolution of $\Spec S^{\os}$ \eqref{s Veron dual graph} in the following way:
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=1,bend angle=30, looseness=1]
\node (0) at (0,0) {$\scriptstyle (\mathsf{v}^{p_2},\mathsf{u}_1)$};
\node (A1) at (-3.25,1) {$\scriptstyle (\mathsf{v}^{p_2+1},\mathsf{u}_1)$};
\node (A2) at (-3.25,2) {$\scriptstyle (\mathsf{v}^{p_2+2},\mathsf{u}_1)$};
\node (A3) at (-3.25,3) {$\scriptstyle (\mathsf{v}^{p_2+p_1-2},\mathsf{u}_1)$};
\node (A4) at (-3.25,4) {$\scriptstyle (\mathsf{v}^{p_2+p_1-1},\mathsf{u}_1)$};
\node (B1) at (-1.5,1) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-1})$};
\node (B2) at (-1.5,2) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-2})$};
\node (B3) at (-1.5,3) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^2)$};
\node (B4) at (-1.5,4) {$\scriptstyle (\mathsf{u}_1,\mathsf{v})$};
\node (C1) at (0,1) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-1})$};
\node (C2) at (0,2) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-2})$};
\node (C3) at (0,3) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{2})$};
\node (C4) at (0,4) {$\scriptstyle (\mathsf{u}_3,\mathsf{v})$};
\node (n1) at (2,1) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-1})$};
\node (n2) at (2,2) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-2})$};
\node (n3) at (2,3) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{2})$};
\node (n4) at (2,4) {$\scriptstyle (\mathsf{u}_n,\mathsf{v})$};
\node at (-3.25,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,2.5) {$\hdots$};
\draw [-] (A1)+(-30:8.5pt) -- ($(0) + (160:15pt)$);
\draw [-] (B1) --(0);
\draw [-] (C1) --(0);
\draw [-] (n1) --(0);
\draw [-] (A2) -- (A1);
\draw [-] (B2) --(B1);
\draw [-] (C2) --(C1);
\draw [-] (n2) --(n1);
\draw [-] (A4) -- (A3);
\draw [-] (B4) --(B3);
\draw [-] (C4) --(C3);
\draw [-] (n4) -- (n3);
\end{tikzpicture}
\end{array}
\]
\end{prop}
\begin{proof}
We first claim that $S(\ox_1)^{\os}\cong (\mathsf{v}^{p_1+p_2-1},\mathsf{u}_1)$. Indeed, since $S$ is a graded domain, multiplication by any homogeneous element $S\to S$ is injective. Thus, multiplying by $x_2\hdots x_n$, we see that $S(\ox_1)^{\os}$ is isomorphic to the $S^{\os}$-submodule of $S$ generated by $x_2^{p_1+p_2}x_3^{p_1}\hdots x_n^{p_1}$ and $x_1\hdots x_n$, that is generated by $\mathsf{u}_2$ and $\mathsf{v}$. But then
\[
\mathsf{u}_1(\mathsf{u}_2,\mathsf{v})=(\mathsf{u}_1\mathsf{u}_2,\mathsf{u}_1\mathsf{v})
\stackrel{{\scriptstyle\ref{Veronese result}}}{\cong}
(\mathsf{v}^{p_1+p_2},\mathsf{u}_1\mathsf{v})=(\mathsf{v}^{p_1+p_2-1},\mathsf{u}_1)\mathsf{v},
\]
which shows that $S(\ox_1)^{\os}\cong(\mathsf{v}^{p_1+p_2-1},\mathsf{u}_1)$. The other cases are similar. Next,
\begin{equation}\label{irred s dual}
\End^{\bL}_{S}(\bigoplus_{\oy\in[0,\oc\,]}S(\oy))
\cong
\begin{array}{c}
\begin{tikzpicture}[xscale=1.3,yscale=1]
\node (0) at (0,0) {$\scriptstyle S(\oc)$};
\node (A1) at (-3,1) {$\scriptstyle S((p_1-1)\ox_1)$};
\node (A2) at (-3,2) {$\scriptstyle S((p_1-2)\ox_1)$};
\node (A3) at (-3,3) {$\scriptstyle S(2\ox_1)$};
\node (A4) at (-3,4) {$\scriptstyle S(\ox_1)$};
\node (B1) at (-1.5,1) {$\scriptstyle S((p_2-1)\ox_2)$};
\node (B2) at (-1.5,2) {$\scriptstyle S((p_2-2)\ox_2)$};
\node (B3) at (-1.5,3) {$\scriptstyle S(2\ox_2)$};
\node (B4) at (-1.5,4) {$\scriptstyle S(\ox_2)$};
\node (C1) at (0,1) {$\scriptstyle S((p_3-1)\ox_3)$};
\node (C2) at (0,2) {$\scriptstyle S((p_3-2)\ox_3)$};
\node (C3) at (0,3) {$\scriptstyle S(2\ox_3)$};
\node (C4) at (0,4) {$\scriptstyle S(\ox_3)$};
\node (n1) at (2,1) {$\scriptstyle S((p_n-1)\ox_n)$};
\node (n2) at (2,2) {$\scriptstyle S((p_n-2)\ox_n)$};
\node (n3) at (2,3) {$\scriptstyle S(2\ox_n)$};
\node (n4) at (2,4) {$\scriptstyle S(\ox_n)$};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,3.5) {$\hdots$};
\node at (1,1.5) {$\hdots$};
\node (T) at (0,5) {$\scriptstyle S$};
\draw [->] (A1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(0);
\draw [->] (B1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(0);
\draw [->] (C1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(0);
\draw [->] (n1) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_n$}(0);
\draw [->] (A2) -- node[right] {$\scriptstyle x_1$} (A1);
\draw [->] (B2) -- node[right] {$\scriptstyle x_2$}(B1);
\draw [->] (C2) -- node[right] {$\scriptstyle x_3$}(C1);
\draw [->] (n2) -- node[right] {$\scriptstyle x_n$}(n1);
\draw [->] (A4) -- node[right] {$\scriptstyle x_1$} (A3);
\draw [->] (B4) -- node[right] {$\scriptstyle x_2$}(B3);
\draw [->] (C4) -- node[right] {$\scriptstyle x_3$}(C3);
\draw [->] (n4) -- node[right] {$\scriptstyle x_n$}(n3);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_1$}(A4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_2$}(B4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_3$}(C4);
\draw [->] (T) -- node[fill=white,inner sep=1pt]{$\scriptstyle x_n$}(n4);
\end{tikzpicture}
\end{array}
\end{equation}
by \cite{GL1}, which, after passing through the categorical equivalence $\CM^{\bL}\!S\simeq \CM^{\bZ}\!R$ of \ref{WPL as qgrZ} gives the degree zero part of $\Gamma_{\ox}$. After passing to the completion, as in \ref{recon quiver and number relations} these arrows remain in the quiver of the reconstruction algebra, which by \ref{GL2 for R} forces the positions.
\end{proof}
\begin{prop}\label{all arrows}
The reconstruction algebra $\Gamma_{\os}$ is given by the following quiver, where the arrows correspond to the following morphisms
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1.5,yscale=1.3,bend angle=30, looseness=1]
\node (0) at (-0.75,0) {$\scriptstyle (v^{p_2},u_1)$};
\node (A1) at (-2.75,1) {$\scriptstyle (v^{p_2+1},u_1)$};
\node (A3) at (-2.75,2) {$\scriptstyle (v^{p_2+p_1-2},u_1)$};
\node (A4) at (-2.75,3) {$\scriptstyle (v^{p_2+p_1-1},u_1)$};
\node (B1) at (-1.5,1) {$\scriptstyle (u_1,v^{p_2-1})$};
\node (B3) at (-1.5,2) {$\scriptstyle (u_1,v^2)$};
\node (B4) at (-1.5,3) {$\scriptstyle (u_1,v)$};
\node (C1) at (0,1) {$\scriptstyle (u_3,v^{p_3-1})$};
\node (C3) at (0,2) {$\scriptstyle (u_3,v^{2})$};
\node (C4) at (0,3) {$\scriptstyle (u_3,v)$};
\node (n1) at (2,1) {$\scriptstyle (u_n,v^{p_n-1})$};
\node (n3) at (2,2) {$\scriptstyle (u_n,v^{2})$};
\node (n4) at (2,3) {$\scriptstyle (u_n,v)$};
\node at (-3,1.6) {$\vdots$};
\node at (-1.5,1.6) {$\vdots$};
\node at (0,1.6) {$\vdots$};
\node at (2,1.6) {$\vdots$};
\node at (1,2) {$\hdots$};
\node (T) at (-0.75,4) {$\scriptstyle R$};
\draw [->] (A1)+(-30:8.5pt) -- node[gap] {$\scriptstyle \inc$} ($(0) + (180:11.5pt)$);
\draw [->] (B1) --node[gap] {$\scriptstyle v$}(0);
\draw [->] (C1) --node[right=-0.05] {$\scriptstyle \frac{v^{p_2+1}}{u_3}$}(0);
\draw [->] (n1) --node[above=-0.05] {$\scriptstyle \frac{v^{p_2+1}}{u_n}$}(0);
\draw [->] (A4) -- node[gap] {$\scriptstyle \inc$} (A3);
\draw [->] (B4) --node[gap] {$\scriptstyle v$}(B3);
\draw [->] (C4) --node[gap] {$\scriptstyle v$}(C3);
\draw [->] (n4) -- node[gap] {$\scriptstyle v$} (n3);
\draw [->] (T)+(-190:4.5pt) -- node[gap] {$\scriptstyle u_1$} ($(A4) + (30:7.5pt)$);
\draw [->] (T) -- node[gap] {$\scriptstyle v$} (B4);
\draw [->] (T) -- node[gap] {$\scriptstyle v$} (C4);
\draw [->] (T) -- node[gap] {$\scriptstyle v$} (n4);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (A1)+(-50:6.5pt) to node[gap] {$\scriptstyle v$} ($(0) + (190:11.5pt)$);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (B1)+(-70:5.5pt) to node[gap,xshift=-3pt] {$\scriptstyle \inc$} ($(0) + (150:8pt)$);
\draw [bend right, <-, red] ($(C1)+(-140:7pt)$) to node[gap] {$\scriptstyle \frac{u_3}{v^{p_2}}$} (0);
\draw [bend left, bend angle=10, looseness=0.5, <-, red] (n1)+(-140:8pt) to node[gap] {$\scriptstyle \frac{u_n}{v^{p_2}}$} ($(0)+(7.5:14pt)$);
\draw [bend right, <-, red] (A4) to node[left] {$\scriptstyle v$} (A3);
\draw [bend right, <-, red] (B4) to node[left] {$\scriptstyle \inc$}(B3);
\draw [bend right, <-, red] (C4) to node[left] {$\scriptstyle \inc$} (C3);
\draw [bend left, <-, red] (n4) to node[right] {$\scriptstyle \inc$}(n3);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (T)+(-200:4.5pt) to node[gap,yshift=2pt] {$\scriptstyle\frac{v}{u_1}$} ($(A4) + (50:6.5pt)$);
\draw [bend right, bend angle=10, looseness=0.5, <-, red] (T)+(-155:4.5pt) to node[gap,xshift=-2pt] {$\scriptstyle \inc$} ($(B4) + (70:5.5pt)$);
\draw [bend right, <-, red] (T) to node[gap,xshift=-2pt] {$\scriptstyle \inc$} (C4);
\draw [bend left, bend angle=10, looseness=0.5, <-, red] (T) to node[gap] {$\scriptstyle \inc$} ($(n4) + (150:10pt)$);
\end{tikzpicture}
\end{array}
\]
\end{prop}
\begin{proof}
Under the isomorphisms in \ref{Veronese result} and \ref{dual graph assignment}, the morphisms in \eqref{irred s dual} become
\begin{equation}\label{canonical in R coordinates}
\begin{array}{c}
\begin{tikzpicture}[xscale=1.3,yscale=1,bend angle=30, looseness=1]
\node (0) at (0,0) {$\scriptstyle (\mathsf{v}^{p_2},\mathsf{u}_1)$};
\node (A1) at (-3,1) {$\scriptstyle (\mathsf{v}^{p_2+1},\mathsf{u}_1)$};
\node (A2) at (-3,2) {$\scriptstyle (\mathsf{v}^{p_2+2},\mathsf{u}_1)$};
\node (A3) at (-3,3) {$\scriptstyle (\mathsf{v}^{p_2+p_1-2},\mathsf{u}_1)$};
\node (A4) at (-3,4) {$\scriptstyle (\mathsf{v}^{p_2+p_1-1},\mathsf{u}_1)$};
\node (B1) at (-1.5,1) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-1})$};
\node (B2) at (-1.5,2) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^{p_2-2})$};
\node (B3) at (-1.5,3) {$\scriptstyle (\mathsf{u}_1,\mathsf{v}^2)$};
\node (B4) at (-1.5,4) {$\scriptstyle (\mathsf{u}_1,\mathsf{v})$};
\node (C1) at (0,1) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-1})$};
\node (C2) at (0,2) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{p_3-2})$};
\node (C3) at (0,3) {$\scriptstyle (\mathsf{u}_3,\mathsf{v}^{2})$};
\node (C4) at (0,4) {$\scriptstyle (\mathsf{u}_3,\mathsf{v})$};
\node (n1) at (2,1) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-1})$};
\node (n2) at (2,2) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{p_n-2})$};
\node (n3) at (2,3) {$\scriptstyle (\mathsf{u}_n,\mathsf{v}^{2})$};
\node (n4) at (2,4) {$\scriptstyle (\mathsf{u}_n,\mathsf{v})$};
\node at (-3,2.6) {$\vdots$};
\node at (-1.5,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node at (1,2.5) {$\hdots$};
\node (T) at (0,5) {$\scriptstyle R$};
\draw [->] (A1)+(-30:8.5pt) -- node[gap] {$\scriptstyle \inc$} ($(0) + (160:11.5pt)$);
\draw [->] (B1) --node[gap] {$\scriptstyle \mathsf{v}$}(0);
\draw [->] (C1) --node[right=-0.1] {$\scriptstyle \frac{\mathsf{v}^{p_2+1}}{\mathsf{u}_3}$}(0);
\draw [->] (n1) --node[below,pos=0.2] {$\scriptstyle \frac{\mathsf{v}^{p_2+1}}{\mathsf{u}_n}$}(0);
\draw [->] (A2) -- node[right] {$\scriptstyle \inc$} (A1);
\draw [->] (B2) --node[right] {$\scriptstyle \mathsf{v}$}(B1);
\draw [->] (C2) --node[right] {$\scriptstyle \mathsf{v}$}(C1);
\draw [->] (n2) -- node[right] {$\scriptstyle \mathsf{v}$}(n1);
\draw [->] (A4) -- node[right] {$\scriptstyle \inc$} (A3);
\draw [->] (B4) --node[right] {$\scriptstyle \mathsf{v}$}(B3);
\draw [->] (C4) --node[right] {$\scriptstyle \mathsf{v}$}(C3);
\draw [->] (n4) -- node[right] {$\scriptstyle \mathsf{v}$} (n3);
\draw [->] (T)+(-160:7.5pt) -- node[gap] {$\scriptstyle \mathsf{u}_1$} ($(A4) + (30:9.5pt)$);
\draw [->] (T) -- node[gap] {$\scriptstyle \mathsf{v}$} (B4);
\draw [->] (T) -- node[right] {$\scriptstyle \mathsf{v}$} (C4);
\draw [->] (T) -- node[gap] {$\scriptstyle \mathsf{v}$} (n4);
\end{tikzpicture}
\end{array}
\end{equation}
From here, we first work on the completion, which is naturally filtered, then goes back to the graded setting by taking the associated graded ring. We know that the quiver of the reconstruction algebra from \eqref{recon quiver}, and we know that for every special CM module $X$, we must be able to hit the generators of $X$ by composing arrows starting at the vertex $R$ and ending at the vertex corresponding to $X$, without producing any cycles. Since the arrows in \eqref{canonical in R coordinates} are already forced to be arrows in the reconstruction algebra, it remains to choose a basis for the remaining red arrows. For example, the generator $v^{p_2+1}$ in $(v^{p_2+1},u_1)$ must come from a composition of arrows $R$ to $(v^{p_2},u_1)$, followed by the bottom left arrow. Since we can see $v^{p_2}$ as a composition of maps from $R$ to $(v^{p_2},u_1)$, this forces the bottom left red arrow to be $v$. The remaining arrows are similar.
\end{proof}
\begin{thm}\label{recon relations}
The reconstruction algebra $\Gamma_{\!\os}$ is isomorphic to the path algebra of the quiver $\overline{Q}_{\bp}$ subject to relations given by
\begin{enumerate}
\item The canonical algebra relations on the black arrows
\item At every vertex, all 2-cycles that exist at that vertex are equal.
\end{enumerate}
\end{thm}
\begin{proof}
This is very similar to \cite[4.11]{D1}. Denote the set of relations in the statement by $\cS^\prime$. Since everything is graded, we first work in the completed case (so we can use \cite[3.4]{BIRS}) and we prove that the completion of reconstruction algebra is given as the completion of $\KK Q$ (denoted $\KK\hat{Q}$) modulo the closure of the ideal $\langle \cS^\prime\rangle$ (denoted $\overline{\langle \cS^\prime\rangle}$). The non-completed version of the theorem then follows by simply taking the associated graded ring of both sides of the isomorphism.
Set $Q:=\overline{Q}_{\bp,\bl}$ (as in \eqref{recon quiver}), then by \ref{all arrows} there is a natural surjection $\psi\colon\KK\hat{Q}\rightarrow \hat{\Gamma}$ with $\cS^\prime\subseteq I:=\Ker\psi$. Denote the radical of $\KK\hat{Q}$ by $J$ and further let $V$ denote the set of vertices of $Q$. Below we show that the elements of $\cS^\prime$ are linearly independent in $I/ (IJ+JI)$, hence we may extend $\cS^\prime$ to a basis $\cS$ of $I/ (IJ+JI)$. Since $\cS$ is a basis, by \cite[3.4(a)]{BIRS} $I=\overline{\langle \cS\rangle}$, so it remains to show that $\cS=\cS^\prime$. But by \cite[3.4(b)]{BIRS}
\[
\# (e_{a}\KK\hat{Q}e_{b})\cap \cS=\dim \Ext^2_{\hat{\Gamma}}(S_a,S_b)
\]
for all $a,b\in V$, where $S_a$ is the simple module corresponding to vertex $a$. From \ref{recon quiver and number relations} (i.e.\ \cite{WemGL2}), this is equal to some number given by intersection theory. Simply inspecting our set $\cS^\prime$ and comparing to the numbers in \ref{recon quiver and number relations}, we see that
\[
\# (e_{a}\KK\hat{Q}e_{b})\cap \cS=\# (e_{a}\KK\hat{Q}e_{b})\cap \cS^\prime
\]
for all $a,b\in V$, proving that the number of elements in $\cS$ and $\cS^\prime$ are the same. Hence $\cS^\prime=\cS$ and so $I=\overline{\langle \cS^\prime\rangle}$, as required.
Thus it suffices to show that the elements of $\cS^\prime$ are linearly independent in $I/ (IJ+JI)$. This is identical to the proof
of \cite[4.12]{D1}, so we omit the details.
\end{proof}
Whilst thinking of the special CM modules as ideals makes everything much more explicit, doing this forgets the grading. Indeed, the reconstruction algebra $\Gamma_{\!\os}$ has a natural grading induced from the Veronese construction.
\begin{prop}\label{grading inherited}
The reconstruction algebra $\Gamma_{\!\os}$ is generated in degree one over its degree zero piece, which is the canonical algebra $\Lambda_{\bp,\bl}$.
\end{prop}
\begin{proof}
By \eqref{irred s dual} all the black arrows in the quiver in \ref{all arrows} have degree zero. It is easy to see that any red arrows in the reverse direction to an arrow labelled $x_i$ has label $x_1\hdots\widehat{x_i}\hdots x_n$, and it is easy to check that these all have degree one, using \ref{2 gen as S modules}. Hence the degree zero piece is the canonical algebra, and as an algebra $\Gamma_{\!\os}$ is generated in degree one over its degree zero piece.
\end{proof}
Note that for $a>0$, the reconstruction algebra $\Gamma_{\!\os_a}$ is not always generated in degree one over its degree zero piece.
\section{Domestic Case}\label{domestic section}
In this section we investigate the domestic case, that is when the dual graph is an ADE Dynkin diagram, and relate Ringel's work on the representation theory of the canonical algebra to the classification of the special CM modules for quotient singularities in \cite{IW}. This will explain the motivating coincidence from the introduction. Since this involves AR theory, typically in this section rings will be complete.
Throughout this section we consider $\bX=\bX_{\bp,\bl}$ and $S=S_{\bp,\bl}$ with $n=3$ and one of the triples $(p_1,p_2,p_3)=(2,3,3)$, $(2,3,4)$ or $(2,3,5)$. For $m\geq 3$, we consider the $(m-3)$-Wahl Veronese subring $R=S^{\os_{m-3}}$ and its completion $\mathfrak{R}$.
\begin{prop}\label{Veronese=quotient}
In the above setting, $\Spec R$ has the following dual graph:
\begin{equation}\label{dual graph 6}
\begin{array}{c}
\begin{tikzpicture}[xscale=1,yscale=0.7]
\node (0) at (0,0) [vertex] {};
\node (A1) at (-2,1) [vertex] {};
\node (A2) at (-2,2) [vertex] {};
\node (A3) at (-2,3) [vertex] {};
\node (A4) at (-2,4) [vertex] {};
\node (C1) at (0,1) [vertex] {};
\node (C2) at (0,2) [vertex] {};
\node (C3) at (0,3) [vertex] {};
\node (C4) at (0,4) [vertex] {};
\node (n1) at (2,1) [vertex] {};
\node (n2) at (2,2) [vertex] {};
\node (n3) at (2,3) [vertex] {};
\node (n4) at (2,4) [vertex] {};
\node at (-2,2.6) {$\vdots$};
\node at (0,2.6) {$\vdots$};
\node at (2,2.6) {$\vdots$};
\node (T) at (0,4.25) {};
\node at (0,-0.2) {$\scriptstyle -m$};
\node at (-1.7,1) {$\scriptstyle -2$};
\node at (-1.7,2) {$\scriptstyle -2$};
\node at (-1.7,3) {$\scriptstyle -2$};
\node at (-1.7,4) {$\scriptstyle -2$};
\node at (0.3,1) {$\scriptstyle -2$};
\node at (0.3,2) {$\scriptstyle -2$};
\node at (0.3,3) {$\scriptstyle -2$};
\node at (0.3,4) {$\scriptstyle -2$};
\node at (2.3,1) {$\scriptstyle -2$};
\node at (2.3,2) {$\scriptstyle -2$};
\node at (2.3,3) {$\scriptstyle -2$};
\node at (2.3,4) {$\scriptstyle -2$};
\draw (A1) -- (0);
\draw (C1) -- (0);
\draw (n1) -- (0);
\draw (A2) -- (A1);
\draw (C2) -- (C1);
\draw (n2) -- (n1);
\draw (A4) -- (A3);
\draw (C4) -- (C3);
\draw (n4) -- (n3);
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(2,1) -- (2,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_3-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(0,1) -- (0,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_2-1$};
\draw [decorate,decoration={brace,amplitude=5pt},xshift=-4pt,yshift=0pt]
(-2,1) -- (-2,4) node [black,midway,xshift=-0.55cm]
{$\scriptstyle p_1-1$};
\end{tikzpicture}
\end{array}
\end{equation}
Moreover $\mathfrak{R}$ is isomorphic to a quotient singularity $\KK[[x,y]]^G$ in the following list:
\[
\begin{tabular}{*3c}
\toprule
$(p_1,p_2,p_3)$ & Dual Graph & $G$\\
\midrule
$(2,3,3)$&
$\begin{array}{c}
\begin{tikzpicture}[xscale=0.7,yscale=0.8]
\node (-1) at (-1,0) [vertex] {};
\node (0) at (0,0) [vertex] {};
\node (1) at (1,0) [vertex] {};
\node (1b) at (1,0.75) [vertex] {};
\node (2) at (2,0) [vertex] {};
\node (3) at (3,0) [vertex] {};
\node (-1a) at (-1.2,-0.3) {$\scriptstyle - 2$};
\node (0a) at (-0.2,-0.3) {$\scriptstyle - 2$};
\node (1a) at (0.8,-0.3) {$\scriptstyle -m$};
\node (1ba) at (0.5,0.75) {$\scriptstyle - 2$};
\node (2a) at (1.8,-0.3) {$\scriptstyle - 2$};
\node (2a) at (2.8,-0.3) {$\scriptstyle - 2$};
\draw [-] (-1) -- (0);
\draw [-] (0) -- (1);
\draw [-] (1) -- (2);
\draw [-] (2) -- (3);
\draw [-] (1) -- (1b);
\end{tikzpicture}
\end{array}$
&
$\mathbb{T}_{6(m-2)+1}$\\
$(2,3,4)$ &
$\begin{array}{c}
\begin{tikzpicture}[xscale=0.7,yscale=0.8]
\node (-1) at (-1,0) [vertex] {};
\node (0) at (0,0) [vertex] {};
\node (1) at (1,0) [vertex] {};
\node (1b) at (1,0.75) [vertex] {};
\node (2) at (2,0) [vertex] {};
\node (3) at (3,0) [vertex] {};
\node (4) at (4,0) [vertex] {};
\node (-1a) at (-1.2,-0.3) {$\scriptstyle - 2$};
\node (0a) at (-0.2,-0.3) {$\scriptstyle - 2$};
\node (1a) at (0.8,-0.3) {$\scriptstyle -m$};
\node (1ba) at (0.5,0.75) {$\scriptstyle - 2$};
\node (2a) at (1.8,-0.3) {$\scriptstyle - 2$};
\node (2a) at (2.8,-0.3) {$\scriptstyle - 2$};
\node (4a) at (3.8,-0.3) {$\scriptstyle -2$};
\draw [-] (-1) -- (0);
\draw [-] (0) -- (1);
\draw [-] (1) -- (2);
\draw [-] (2) -- (3);
\draw [-] (3) -- (4);
\draw [-] (1) -- (1b);
\end{tikzpicture}
\end{array}$
&
$\mathbb{O}_{12(m-2)+1}$\\
$(2,3,5)$ &
$\begin{array}{c}
\begin{tikzpicture}[xscale=0.7,yscale=0.8]
\node (-1) at (-1,0) [vertex] {};
\node (0) at (0,0) [vertex] {};
\node (1) at (1,0) [vertex] {};
\node (1b) at (1,0.75) [vertex] {};
\node (2) at (2,0) [vertex] {};
\node (3) at (3,0) [vertex] {};
\node (4) at (4,0) [vertex] {};
\node (5) at (5,0)[vertex] {};
\node (-1a) at (-1.2,-0.3) {$\scriptstyle - 2$};
\node (0a) at (-0.2,-0.3) {$\scriptstyle - 2$};
\node (1a) at (0.8,-0.3) {$\scriptstyle -m$};
\node (1ba) at (0.5,0.75) {$\scriptstyle - 2$};
\node (2a) at (1.8,-0.3) {$\scriptstyle - 2$};
\node (2a) at (2.8,-0.3) {$\scriptstyle - 2$};
\node (4a) at (3.8,-0.3) {$\scriptstyle -2$};
\node (5a) at (4.8,-0.3) {$\scriptstyle - 2$};
\draw [-] (-1) -- (0);
\draw [-] (0) -- (1);
\draw [-] (1) -- (2);
\draw [-] (2) -- (3);
\draw [-] (3) -- (4);
\draw [-] (4) -- (5);
\draw [-] (1) -- (1b);
\end{tikzpicture}
\end{array}$
&
$\mathbb{I}_{30(m-2)+1}$\\
\bottomrule\\
\end{tabular}
\]
For the precise definition of the above subgroups of $\GL(2,\KK)$ we refer the reader to \cite{IW}.
\end{prop}
\begin{proof}
By \ref{middle SI number}, the dual graph of $R$ is known to be \eqref{s Veron dual graph}. On the other hand, the quotient singularity $\KK[[x,y]]^G$ has the same dual graph \cite[\S3]{Riemen}. Since the dual graphs \eqref{s Veron dual graph} for ADE triples are known to be taut \cite[Korollar 2.12]{Brieskorn}, the result follows.
\end{proof}
Let us finally explain why Ringel's picture \eqref{RingelPicture} in the introduction is the same as the ones found in \cite{IW} and \cite[\S4]{WemGL2}. For example, in the family of groups $\mathbb{O}_{12(m-2)+1}$ with $m\geq 3$ in \ref{Veronese=quotient}, by \cite{AR_McKayGraphs} the AR quiver of
$\mathfrak{R}\cong\KK[[x,y]]^{\mathbb{O}_m}$ is
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=1.25,yscale=1.25]
\draw[densely dotted] (0.5,0.5) -- (0.5,-2.5);
\draw[densely dotted] (1.5,0.5) -- (1.5,-2.5);
\node (R0) at (0.5,0.5) [gap] {$\scriptstyle R$};
\node (R1) at (1.5,0.5) [vertex] {};
\node (R2) at (2.5,0.5) [vertex] {};
\node (R11) at (3.5,0.5) [vertex] {};
\node (R12) at (4.5,0.5) [gap] {$\scriptstyle R$};
\node (A1) at (1,0) [gap] [vertex] {};
\node (A2) at (2,0) [gap] [vertex] {};
\node (A12) at (4,0) [vertex] {};
\node (B0) at (0.5,-0.5) [vertex] {};
\node (B1) at (1.5,-0.5) [vertex] {};
\node (B2) at (2.5,-0.5) [vertex] {};
\node (B11) at (3.5,-0.5) [vertex] {};
\node (B12) at (4.5,-0.5) [vertex] {};
\node (C1) at (0.5,-1.1) [vertex] {};
\node (C2) at (1,-1) [vertex] {};
\node (C3) at (1.5,-1.1) [vertex] {};
\node (C4) at (2,-1) [vertex] {};
\node (C5) at (2.5,-1.1) [vertex] {};
\node (C23) at (3.5,-1.1) [vertex] {};
\node (C24) at (4,-1) [vertex] {};
\node (C25) at (4.5,-1.1) [vertex] {};
\node (D0) at (0.5,-1.5) [vertex] {};
\node (D1) at (1.5,-1.5) [vertex] {};
\node (D2) at (2.5,-1.5) [vertex] {};
\node (D11) at (3.5,-1.5) [vertex] {};
\node (D12) at (4.5,-1.5) [vertex] {};
\node (E1) at (1,-2) [vertex] {};
\node (E2) at (2,-2) [vertex] {};
\node (E12) at (4,-2) [vertex] {};
\node (F0) at (0.5,-2.5) [vertex] {};
\node (F1) at (1.5,-2.5) [vertex] {};
\node (F2) at (2.5,-2.5) [vertex] {};
\node (F11) at (3.5,-2.5) [vertex] {};
\node (F12) at (4.5,-2.5) [vertex] {};
\draw[->] (R0) -- (A1);
\draw[->] (A1) -- (R1);
\draw[->] (R1) -- (A2);
\draw[->] (A2) -- (R2);
\draw[->] (R11) -- (A12);
\draw[->] (A12) -- (R12);
\draw[->] (B0) -- (A1);
\draw[->] (A1) -- (B1);
\draw[->] (B1) -- (A2);
\draw[->] (A2) -- (B2);
\draw[->] (B11) -- (A12);
\draw[->] (A12) -- (B12);
\draw[->] (B0) -- (C2);
\draw[->] (C2) -- (B1);
\draw[->] (B1) -- (C4);
\draw[->] (C4) -- (B2);
\draw[->] (B11) -- (C24);
\draw[->] (C24) -- (B12);
\draw[->] (C1) -- (C2);
\draw[->] (C2) -- (C3);
\draw[->] (C3) -- (C4);
\draw[->] (C4) -- (C5);
\draw[->] (C23) -- (C24);
\draw[->] (C24) -- (C25);
\draw[->] (D0) -- (C2);
\draw[->] (C2) -- (D1);
\draw[->] (D1) -- (C4);
\draw[->] (C4) -- (D2);
\draw[->] (D11) -- (C24);
\draw[->] (C24) -- (D12);
\draw[->] (D0) -- (E1);
\draw[->] (E1) -- (D1);
\draw[->] (D1) -- (E2);
\draw[->] (E2) -- (D2);
\draw[->] (D11) -- (E12);
\draw[->] (E12) -- (D12);
\draw[->] (F0) -- (E1);
\draw[->] (E1) -- (F1);
\draw[->] (F1) -- (E2);
\draw[->] (E2) -- (F2);
\draw[->] (F11) -- (E12);
\draw[->] (E12) -- (F12);
\node at (3,-1) {$\hdots$};
\end{tikzpicture}
\end{array}
\]
where there are precisely $m$ repetitions of the original $\tilde{E}_7$ shown in dotted lines. The left and right hand sides of the picture are identified, and there is no twist in this AR quiver.
Thus as $m$ increases (and the group $\mathbb{O}_{12(m-2)+1}$ changes), the AR quiver becomes longer.
Regardless of the length $m$, the special CM $\mathfrak{R}$-modules always have the following position in the AR quiver\\
\[
\begin{array}{c}
\begin{tikzpicture}[xscale=0.85,yscale=0.85]
\draw[gray] (0,-2) -- (0.5,-2.5);
\draw[gray] (0,-1) -- (1.5,-2.5);
\draw[gray] (0,0) -- (2.5,-2.5);
\draw[gray] (0.5,0.5) -- (3.5,-2.5);
\draw[gray] (1.5,0.5) -- (4.5,-2.5);
\draw[gray] (2.5,0.5) -- (5.5,-2.5);
\draw[gray] (3.5,0.5) -- (6.5,-2.5);
\draw[gray] (4.5,0.5) -- (7.5,-2.5);
\draw[gray] (5.5,0.5) -- (8.5,-2.5);
\draw[gray] (6.5,0.5) -- (9.5,-2.5);
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\end{array}
\]
In particular, comparing this to \eqref{RingelPicture}, we observe the following coincidences.
\begin{enumerate}
\item\label{ident} The AR quiver of $\CM\mathfrak{R}$ is the quotient of the AR quiver of $\vect\bX$ modulo $\tau^{12(m-2)+1}=((12(m-2)+1)\ow)$.
\item The canonical tilting bundle $\mathcal{E}$ on $\bX$ is given by the circled vertices in \eqref{RingelPicture}, and so under the identification in \eqref{ident}, this gives the additive generator of $\SCM\mathfrak{R}$.
\end{enumerate}
The same coincidence can also be observed for type $\mathbb{T}$ and $\mathbb{I}$ by replacing $12$ by $6$ and $30$ respectively.
To give a theoretical explanation to these observations, we need the following preparation, where recall that $\os=\sum_{i=1}^3\ox_i$ and $\ow=\oc-\os$ since $n=3$.
\begin{lemma}\label{h s lemma}
Define $h$ as follows
\[
\begin{tabular}{*2c}
\toprule
Type&$h$\\
\midrule
$\mathbb{T}$&$6$\\
$\mathbb{O}$&$12$\\
$\mathbb{I}$&$30$\\
\bottomrule
\end{tabular}
\]
Then $(h+1)\ow=-\os$ and $(h(m-2)+1)\ow=-\os_{m-3}$.
\end{lemma}
\begin{proof}
If $(p_1,p_2,p_3)=(2,3,3)$, then $6\ow=(6-3-2-2)\oc=-\oc$ and so $7\ow=-\os$. Similarly, if $(p_1,p_2,p_3)=(2,3,4)$ then $12\ow=(12-6-4-3)\oc=-\oc$, thus $13\ow=-\os$. Lastly, if $(p_1,p_2,p_3)=(2,3,5)$ then $30\ow=(30-15-10-6)\oc=-\oc$, hence $31\ow=-\os$.
Therefore $(h(m-2)+1)\ow=-(m-2)\os-(m-3)\ow=-\os-(m-3)\oc=-\os_{m-3}$.
\end{proof}
Let $\mathcal{C}$ be an additive category with an action by a cyclic group $G=\langle g\rangle\cong\bZ$.
Assume that, for any $X,Y\in\mathcal{C}$, $\Hom_{\mathcal{C}}(X,g^iY)=0$ holds for $i\gg0$.
The \emph{complete orbit category} $\mathcal{C}/G$ has the same object as $\mathcal{C}$ and the morphism sets are given by
\[
\Hom_{\mathcal{C}/G}(X,Y):=\prod_{i\in\bZ}\Hom_{\mathcal{C}}(X,g^iY)
\]
for $X,Y\in\mathcal{C}$, where the composition is defined in the obvious way.
\begin{thm}\label{equiv last}
Let $R$ be the $(m-3)$-Wahl Veronese subring associated with $(p_1,p_2,p_3)=(2,3,3)$, $(2,3,4)$ or $(2,3,5)$ and $m\ge3$, and $\mathfrak{R}$ its completion. Let $G\leq \bL$ be the infinite cyclic group generated by $(h(m-2)+1)\ow=-\os_{m-3}$. Then
\begin{enumerate}
\item\label{equiv last 1} There are equivalences $\vect\bX\simeq\CM^\bZ\!R$ and
\[
F\colon (\vect\bX)/G\xrightarrow{\sim}\CM\mathfrak{R}.
\]
\item\label{equiv last 2} For the canonical tilting bundle $\mathcal{E}$ on $\bX$, we have $\SCM\mathfrak{R}=\add F\mathcal{E}$.
\end{enumerate}
\end{thm}
\begin{proof}
Since $(h(m-2)+1)\ow=-\os_{m-3}$ is a non-zero element in $-\bL_+$, for any $X,Y\in\vect\bX$, necessarily $\Hom_{\bX}(X,Y(i(h(m-2)+1)\ow))=0$ holds for $i\gg0$.
Therefore the complete orbit category $(\vect\bX)/G$ is well-defined.\\
(1) There are equivalences $\vect\bX\simeq\CM^{\bL}\!S\simeq\CM^\bZ\!R$, where the first equivalence is standard \cite{GL1}, and the second is \ref{WPL as qgrZ}. Furthermore, the following diagram commutes.
\[
\xymatrix{
\vect\bX\ar[r]\ar[d]_{(\os_{m-3})}&\CM^\bZ\!R\ar[d]^{(1)}\\
\vect\bX\ar[r]&\CM^\bZ\!R
}
\]
Since $\mathfrak{R}$ has only finitely many indecomposable CM modules (see e.g.\ \cite[15.14]{Y}), there is an equivalence $(\CM^\bZ\!R)/\bZ\simeq\CM\mathfrak{R}$.
Therefore $(\vect\bX)/G\simeq(\CM^\bZ\!R)/\bZ\simeq\CM\mathfrak{R}$.\\
(2) This follows by the equivalences in \eqref{equiv last 1}, the definition of $\cE$, and \ref{specials determined thm}.
\end{proof}
As one final observation, recall that for a canonical algebra $\Lambda$, the \emph{preprojective algebra} of $\Lambda$ is defined by
\[
\Pi:=\bigoplus_{i\ge0}\Pi_i,\ \ \ \Pi_i:=\Hom_{\Db(\mod \Lambda)}(\Lambda,\tau^{-i}\Lambda),
\]
where $\tau$ is the Auslander-Reiten translation in the derived category $\Db(\mod \Lambda)$. Moreover, for a positive integer $t$, we denote the $t$-th Veronese subring of $\Pi$ by
\[
\Pi^{(t)}:=\bigoplus_{i\ge0}\Pi_{ti}.
\]
As notation we write $\Gamma_m$ for the reconstruction algebra of $R$, where $R$ is the $(m-3)$-Wahl Veronese subring associated with $(p_1,p_2,p_3)=(2,3,3)$, $(2,3,4)$ or $(2,3,5)$ and $m\ge3$, which correspond to one of the types $\mathbb{T}$, $\mathbb{O}$ or $\mathbb{I}$ in \ref{Veronese=quotient}.
The following is an analogue of \ref{deg 0 general prop}, but also describes the other graded pieces.
\begin{prop}\label{VeroneseGL2}
There is an isomorphism of $\bZ$-graded algebras
\[
\Pi^{(h(m-2)+1)}\cong \Gamma_m.
\]
\end{prop}
\begin{proof}
By \ref{h s lemma} we know that $(h(m-2)+1)\ow=-\os_{m-3}$.
Setting $M=\bigoplus_{\oy\in[0,\oc\,]}S(\oy)$, then $\Pi^{(h(m-2)+1)}_i$ for $i\ge0$ is given by
\begin{align*}
\Hom_{\Db(\mod\Lambda)}(\Lambda,\tau^{-(h(m-2)+1)i}\Lambda)&\cong\Hom_S^{\bL}(M,M(-i(h(m-2)+1)\ow))\\
&\cong\Hom_S^{\bL}(M,M(i\os_{m-3}))\\
&\cong(\Gamma_m)_i.
\end{align*}
Thus all the graded pieces match. It is easy to see that the isomorphisms are natural, and so give an isomorphism of graded rings.
\end{proof}
\begin{remark}
By \ref{VeroneseGL2}, it follows that in fact on the abelian level
\[
\qgr^{\bZ}\! \Gamma_m\simeq \qgr^{\bZ}\! \Pi^{(h(m-2)+1)}
\]
and so, combining \ref{WPL as qgrZ} and \ref{qgrR via qgrLambda},
\[
\coh \bX\simeq \qgr^{\bZ}\! \Pi^{(h(m-2)+1)}
\]
for any $m\geq 3$. This is a stronger version of results of \cite{GL1} and Minamoto \cite{Minamoto}, which combine to say that for the weighted projective lines of non-tubular type there are derived equivalences
\[
\Db(\coh \bX)\simeq \Db(\mod \Lambda)\simeq \Db(\qgr^{\bZ}\! \Pi).
\]
\end{remark}
| 67,154
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Here’s what we do and don’t know about rosacea.
- It is not contagious
- It is inflammatory
- It appears on the face
- It looks like eczema, acne, or an allergy
- Everyone’s rosacea severity is different
- There are treatments for rosacea
Rosacea has four subtypes
- Erythematotelangiectatic – redness with visible blood vessels
- Papulopustular – redness with visible blood vessels, and tiny acne-like pustules
- Phymatous – redness with visible blood vessels, and tiny acne-like pustules, and thickened skin on the nose (rhinophyma)
- Ocular – redness with visible blood vessels, and tiny acne-like pustules, and thickened skin on the nose (rhinophyma), but with dry eyes that tear and burn
The causes (that we know of) of rosacea
There is no one cause that we know of, because triggers seem to differ, as do solutions for each of you. Things that may trigger rosacea to initially develop could include:
- Mites (demodex mites, which live near hair follicles in mammals)
- Issues with blood vessels
- SIBO (small intestinal bacterial overgrowth)
- Leaky gut syndrome
- Helicobacter pylori
The mites
Rosacea sufferers have more of these mites than other people. (Read the study) This might make it worse or cause the problem, due to the mites being an irritant. Another idea is that the mites release a bacteria when they die, causing inflammation.
The blood vessels
The blood vessels in your face may become enlarged and inflamed, but we’re not sure why.
Intestinal inflammation or infection
It has been put forward that some rosacea could be caused by digestive problems like Helicobacter pylori or small intestinal bacterial overgrowth (SIBO). The role of these conditions in rosacea is under debate, with evidence proving changeable and difficult to pin down.
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\begin{document}
\renewcommand{\proofname}{Proof}
\renewcommand{\bibname}{References}
\maketitle
\begin{abstract}
We introduce the first bases of algebraic geometry over any commutative field $k$ inside the affine spaces $k^n$ themselves, rather than in an algebraically closed extension of $k$ or an equivalent setting. This concrete approach relies on the transposition in non-algebraically closed fields of McKenna's idea of (Galois-theoretic) normic forms, which are homogeneous polynomials with no non-trivial zeros, and builds upon an \og equiresidual" generalisation of Hilbert's Nullstellensatz and an associated radical in finitely generated $k$-algebras. It is natural to work out the usual algebraic constructions underlying affine algebraic geometry inside $k^n$ by using a new type of algebras over $k$ which correspond to \og canonical" localisations of $k$-algebras, associated to the set of polynomials over $k$ with no inner zero. The theory leads to a fruitful characterisation of the sections of the sheaf of regular functions over an affine algebraic set, in that it permits us to dualise the (equiresidual) affine algebraic varieties over $k$ using an analogue of reduced algebras of finite type and a maximal spectrum functor.
\end{abstract}
\section{Introduction and preliminaries}
Is it possible to develop a relevant algebraic geometry over any commutative field, i.e. without the hypothesis that the field is algebraically closed ? The usual answer is : yes, embed your favorite field $k$ into an algebraically closed field $K$ (sometimes with infinite transcendance degree, as in Weil's approach, see Chapter 10 of \cite{FA} for instance), and do the algebraic geometry in $K$ with parameters in $k$. Or, to be more fashionable, work in a suitable category of schemes over $k$, considered itself as a one-element scheme (see Chapter II of \cite{HAG} for the principle). A third and subtle possibility is to consider algebraic spaces over $k$, built from maximal spectra of finitely generated $k$-algebras (see Chapter 11 in \cite{MAG}). All these solutions have one thing in common : one comes down to classical algebraic geometry over algebraically closed fields over $k$, virtually considering rational points of \og geometric objects" in all finite or finitely generated extensions of $k$, and a form of another of Hilbert's Nullstensatz is implied.
Another solution is, in certain very specific cases, to develop whole analogues of complex algebraic geometry, by using some specific features of the field or a related family of fields often identified by a set of (first order) axioms. This is the case of the formidable example of real algebraic geometry (\cite{RAG}), where one abstracts the essential properties of $\RR$ which make it possible to develop a peculiar approach to algebraic geometry in it, with its unique and additional features, and then develop the theory in the category of real closed fields and related algebraic structures. This kind of situation is often strongly connected to first order logic and model-theoretic considerations. In particular, one knows in some core examples how to interpret the model-theoretic notion of quantifier elimination as some analogue of Chevalley's theorem on constructible sets (see Proposition 5.2.2 in \cite{RAG} for real algebraic geometry, and \cite{BEL2} for a $p$-adic analogue).
As we were considering the basics of a wide generalisation of this second approach (expanding the ideas underlying our preceding \cite{BER1} and \cite{BER2}), which will hopefully appear in its time, it striked us that our first question is a very legitimate one, and should be given a definite and simple answer, but in the same spirit as basic linear algebra and affine geometry are done over any field, or as basic algebraic geometry is done over any algebraically closed field. We thus wish to develop some relevant algebraic geometry over any commutative field $k$ in an intrinsic manner, and in particular without working explicitly or implicitly in algebraically closed fields containing $k$ or in a related axiomatisable family of fields. Interesting connexions between algebraic geometry and positive logic revealed how to do this by purely algebraic means, i.e. without model-theoretic methods. At least, it is possible for a start to generalise the theory to as far as algebraic varieties, as to encompass for instance all quasi-projective varieties, which we believe is a very good start. In this present work we want to expound the foundations of this approach, algebraic and affine, saving the theory of algebraic varieties for a further publication.\\
In section \ref{EQNUL}, we explain why a certain \emph{equiresidual} Nullstellensatz (Theorem \ref{NSREL}) holds in every commutative field. This rests on an analogue of a model-theoretic lemma of McKenna about the existence in all non-algebraically closed fields of homogeneous polynomials having only the trivial zero. Characterising the maximal ideals of finitely generated algebras over a field $k$ which have points rational in $k$, which we call \emph{special}, we define an analogue of the classical radical of an ideal - at least for finitely generated $k$-algebras - the \emph{equiresidual radical}. We also define the key algebraic construction which we will use, the \emph{canonical localisation} of an algebra over the base field, which applied to localisation at one element leads to an essential characterisation of the equiresidual radical (Theorem \ref{CARERAD}).
In section \ref{AFFSUB}, we first develop the abstract counterpart of canonical localisation, the notion of a $*$-algebra over a field, which is the \og right" category of algebras in which it is suitable to work out this inner algebraic geometry in general, in connection with \emph{special} algebras - a counterpart of reduced algebras as they appear in the classical affine algebro-geometric context; it is the occasion to introduce and characterise the \emph{special} ideals, which are equal to their equiresidual radical. Secondly, we establish the usual \og dictionary" between specific ideals and algebraic sets in affine spaces. Thirdly, we carefully study the algebras of sections of the sheaf of regular functions over an affine algebraic subvariety. Here lies our core result, Theorem \ref{MAG.3.6.a} : the affine sheaves of regular functions are sheaves of special $*$-algebras, and their algebras of sections are essentially the canonical localisations of the usual coordinate algebras.
In section \ref{SEQVAR}, we first introduce a natural category of locally ringed spaces over a base field which locally look like affine algebraic subvarieties, thus containing a subcategory of \emph{equiresidual affine algebraic varieties}, the abstract counterparts of affine subvarieties; we also give a corresponding abstract characterisation of the algebras of global sections of the structure sheaves of these, the \emph{affine $*$-algebras}, i.e. the special $*$-algebras of finite type as such. Secondly, we show that a natural maximal spectrum functor turns these algebras into affine algebraic equivarieties. Finally, building upon section \ref{AFFSUB} we prove that the global sections functor and the maximal spectrum functor are indeed a duality between both categories (Theorem \ref{DUALAFF}).
\subsection*{Preliminaries and conventions}
All rings and fields considered are implicitly unitary and commutative and we use some standard notation, terminology and folklore from commutative algebra and algebraic geometry, which we briefly review and complete. If $k$ is a field and $I$ is an ideal of a polynomial algebra $k[X_1,\ldots,X_n]$, the corresponding (affine) algebraic set of $k^n$ is noted $\ms Z(I)=\{P\in k^n : \forall f\in I, f(P)=0\}$. These algebraic sets of $k^n$ are the closed sets of a Noetherian topology called the \emph{Zariski topology}; recall that in general any nonempty open subset of an irreducible closed set is dense and irreducible (\cite{HAG}, Example 1.1.3). If $V\subs k^n$ is an algebraic set, the \emph{coordinate ring (or algebra)} of $V$ is the $k$-algebra $k[V]:=k[X_1,\ldots,X_n]/\ms I(V)$, where $\ms I(V)=\{f\in k[\ov X] :\forall P\in V,\ f(P)=0\}$; any element $f\in k[V]$ defines a \emph{function} $V\to k$, $P\mapsto F(P)$, for any $F\in k[\ov X]$ such that $F+\ms I(V)=f$. By definition of the induced Zariski topology on $V$, any basic open subset will be denoted by $D_V(f)=\{P\in V : f(P)\neq 0\}=V-\ms Z_V(f)$ for a certain $f\in k[V]$.
If $U\subs V$ is an open subset (for the induced Zariski topology on $V$), a function $f:U\to k$ is called \emph{regular at $P\in V$}, if there exists an open neighbourhood $U_P\subs U$ of $P$ in $U$ and elements $g,h\in k[V]$ such that for all $Q\in U_P$, $h(Q)\neq 0$ and $f(Q)=g(Q)/h(Q)$; $f$ is called \emph{regular (over $U$)} if $f$ is regular at every $P\in U$ (notice that $f$ is then continuous). The set of regular functions over $U$ is written $\O_V(U)$ and $\O_V$ is a sheaf of $k$-algebras, called the \emph{sheaf of regular functions on $V$}. An \emph{affine algebraic subvariety of $k^n$} is a pair $(V,\O_V)$, where $V\subs k^n$ is an algebraic set and $\O_V$ is its sheaf of regular functions. An element of the stalk $\O_{V,P}$ of $\O_V$ at $P\in V$ will be noted $[g,U]$, where $P\in U\subs V$ and $g\in \O_V(U)$. The following proposition - which we will refer to as the \og small lemma" - should be folkloric but we have never read it elsewhere (in usual textbooks on algebraic geometry, it is proved on algebraically closed fields as a consequence of Hilbert's Nullstellensatz, see \cite{HAG}, Theorem I.3.2 for instance !) :
\begin{prop}[\og Small lemma"]
\label{SMALEM}
For any affine algebraic subvariety $V\subs k^n$, for any $P\in V$, we have $\O_{V,P}\cong k[V]_{\m_P}$ for $\m_P=\{f\in k[V] : f(P)=0\}$. In particular, the structural morphism $k\to \ov{\O_{V,P}}$ is an isomorphism.
\end{prop}
\begin{proof}
Write $A=k[V]$ and $\m=\m_P$. Let $[f]\in \O_P=\O_{V,P}$ : there is a neighbourhood $U$ of $P$ in $V$ and $a,g\in A$ with $g(Q)\neq 0$ and $f(Q)=a(Q)/g(Q)$ for every $Q\in U$, whence $U\subs D_V(g)$ and we may assume that $U=D_V(g)$ with the same data. As $g\in A-\m$, define $\phi([f]):=a/g\in A_\m$ : if $[f]=[b/h]$ in $\O_P$, there exists a basic open neighbourhood $U'=D_V(l)\subs U$ of $P$ in $V$ on which $a/g\equiv b/h$; we have $D_V(l)\subs D_V(g)\cap D_V(h)=D_V(gh)$ and $l\in A -\m$, and the regular map defined by $(ah-bg)/gh$ on $D_V(l)=D_V(ghl)$ is zero, hence $ahl-bgl$ is also zero on $D_V(l)$, whereas for $Q\in V-D_V(l)$, we have $ghl(Q)=0$, and therefore $(ahl-bgl)(ghl)$ is zero on $V$, and hence in $A$. As $ghl\notin \m$, we get $ahl-bgl=0$ in $A_\m$, whereby $a/g=b/h$ in $A_\m$ and $\phi$ is well defined, and obviously a $k$-morphism. Finally, if $a/g\in A_\m$, the regular map defined by $a/g$ on $D_V(g)$ has $\phi([a/g,D_V(g)])=a/g$, and if $\phi([f])=0$ with $f$ defined as before on $D_V(g)$ by $a/g$ say, as $a/g=0$ in $A_\m$ there is $h\in A-\m$ with $ha=0$ in $A$, whence $[a/g]=[ah/gh|_{D_V(gh)}]=0$, and $\phi$ is an isomorphism. The isomorphism $\O_{V,P}\cong k[V]_{\m_P}$ induces a residual isomorphism $\ov{\O_{V,P}}\cong k[V]_{\m_P}/\m_P k[V]_{\m_P}$ over $k$, and by definition of $\m_P$ the structural morphism $k\to \ov{\O_{V,P}}$ is an isomorphism.
\end{proof}
\noindent If $W\subs k^m$ is another algebraic set, a \emph{regular morphism} from $V$ to $W$ is a map $f:V\to W$ such that there exist $f_1,\ldots,f_m\in k[V]$ with $f(P)=(f_1(P),\ldots,f_m(P))$ for all $P\in V$. Any regular morphism $f=(f_1,\ldots,f_m):V\to W$ induces in turn a $k$-algebra morphism $k[f]:k[W]\to k[V]$ in the usual way : to every $g=G+\ms I(W)\in k[W]$ we associate $G(F_1,\ldots,F_m)+\ms I(V)$, if $f_i=F_i+\ms I(V)$ for $i=1,\ldots,m$, and this defines a full and faithful functor $k[-]$ from the dual category of affine algebraic sets and regular morphisms into the category of reduced $k$-algebras of finite type. If $f:V\to W$ is a regular morphism of affine algebraic subvarieties, for every open subset $U\subs W$ and for every $s\in \O_W(U)$, we have $f\circ s\in \O_V(f^{-1}U)$, the map $f^\#_U:s\in\O_W(U)\mapsto f\circ s\in \O_V(f^{-1}U)$ is a morphism of $k$-algebras and the $f^\#_U$'s define a sheaf morphism $f^\#:\O_W\to f_*\O_V$. For every $P\in V$ we have a residual $k$-morphism $f^\#_P:\O_{W,f(P)}\to \O_{V,P}$ induced by the universal property of stalks considered as inductive limits, which is local by the small lemma \ref{SMALEM}, so $(f,f^\#):(W,\O_W)\to (V,\O_V)$ is a morphism of locally ringed spaces in $k$-algebras, and we have a functor $f\mapsto (f, f^\#)$ from the dual category of affine algebraic subvarieties to locally ringed spaces in $k$-algebras. If $A$ is any ring, we note $Spm(A)$ the maximal spectrum of $A$, i.e. the set of all maximal ideals of $A$, implicitly topologised as usual by taking as basic open sets the subsets of the form $D(f)=\{\m\in Spm(A) : f\notin \m\}$; this is the \emph{Zariski topology on $A$}.
We refer the reader to Chapter II of \cite{HAG}, for instance, about generalities on sheaves and locally ringed spaces. Not being an original algebraic geometer, we also apologise to the educated reader for any clumsiness in notation, conception, or reference, and for any presumption of demonstrating anything which is already well known to the specialist.
\section{The equiresidual Nullstellensatz and its associated radical}\label{EQNUL}
\subsection*{Normic forms and the Äquinullstellensatz}
In a context loaded with first order logic, McKenna ingeniously introduces the notion of a \emph{normic form} in a first order theory of fields (\cite{MK}, Lemma 4), which permits him to deal with the characterisation of analogues of the radical of an ideal. We adapt his definition as the following
\begin{defi}
If $k$ is a field, a \emph{normic form over $k$} is a homogeneous polynomial $P(X_1,\ldots,X_n)$ with coefficients in $k$, such that the only $\ov a\in k^n$ for which $P(\ov a)=0$ is $\ov 0$.
\end{defi}
\begin{rem}
This is \emph{a priori} only an analogue of McKenna'a notion, but both have a common generalisation thanks to basic positive logic (see \cite{PAG1}).
\end{rem}
\noindent In general, the only constant normic form over a field $k$ is $0$, and the normic forms in one variable are the nonzero monomials. Normic forms are useful - at least in \emph{non}-algebraically closed fields - in order to reduce the description of algebraic sets to sets of zeros of a unique polynomial, so in this respect they become interesting with at least two variables. Notice that if $k$ is algebraically closed and $P\in k[X_1,\ldots,X_n]$ with $n\geq 2$, $P$ always has a nontrivial zero, so $k$ does \emph{not} have such forms ! The miracle is that however, by elementary Galois theory they always exist over any other field :
\begin{prop}[Normic forms over non-algebraically closed fields]\label{EXNORM0}
If $k$ is a field, \emph{not} algebraically closed, then there exist normic forms of an arbitrary number of variables over $k$.
\end{prop}
\begin{proof}
We adapt the proof of McKenna (\cite{MK}, Lemma 4) to the present context. Composing polynomials and substituting zeros for certain variables, it suffices to show that there exists a normic form in two variables over $k$. As $k$ is not algebraically closed, there exists a proper algebraic extension $k\to k(\alpha)$ of $k$ and we distinguish two cases. First, if $k$ is separably closed, we have $char(k)=p>0$ and by Proposition V.6.1 in \cite{Lang}, there exists $m\in \NN$ such that $\alpha^{p^m}$ is separably algebraic over $k$, and thus $\alpha^{p^m}\in k$; we choose $m$ minimal with this property, we have $m>0$ and $\alpha^{p^{m-1}}\notin k$,
and we let $N(X,Y):=X^{p^m}-\alpha^{p^m}Y^{p^m}\in k[X,Y]$. If $p=2$, we have $N(X,Y)=(X^{2^{m-1}}-\alpha^{2^{m-1}}Y^{2^{m-1}})(X^{2^{m-1}}+\alpha^{2^{m-1}}Y^{2^{m-1}})$ and if $a,b\in k$ are such that $N(a,b)=0$ with $b\neq 0$, distinguishing cases we have $\alpha^{2^{m-1}}\in k$, which is impossible, so $b=0$, and also $a=0$. If $p\neq 2$, we have $N(X,Y)=(X-\alpha Y)^{p^m}$, and if $a,b\in k$ and $N(a,b)=0$ with $b\neq 0$, we have $a/b=\alpha\in k$, which contradicts the choice of $\alpha$, so $b=0$ and also $a=0$, and therefore $N(X,Y)$ is a normic form over $k$. Secondly, if $k$ is not separably closed, we may assume that $\alpha$ is separably algebraic over $k$ and any splitting field $k\to K$ for $\alpha$ is a finite separable algebraic extension by Theorem V.4.4 of \cite{Lang}, so a Galois extension, generated by a single element $\beta$ by Abel's Theorem (\cite{Lang}, Theorem V.4.6) : we have $K=k[\beta]=k(\beta)$ and the polynomial $N(X,Y):=\prod_{\sigma\in Gal(K/k)} (X-\beta^\sigma Y)$ is a member of $k[X,Y]$ by the fundamental theorem of Galois theory (\cite{Lang}, Theorem VI.1.1). Let again $a,b\in k$ with $N(a,b)=0$ : if $b\neq 0$, as $\prod_{\sigma\in Gal(K/k)} (a-\beta^\sigma b)=0$ there exists $\tau\in Gal(K/k)$ such that $\beta^\tau=a/b\in k$, which is impossible (all the conjugates of $\beta$ generate $K$ over $k$). We conclude that $b=0$, so $a=0$ also, therefore $N(X,Y)$ is a normic form over $k$.
\end{proof}
\noindent Combining this phenomenon with the exclusion, in finitely generated algebras over a field $k$, of ideals which contain certain functions with no zero rational over $k$, we may generalise Hilbert's Nullstellensatz as the following
\begin{thm}[\og Äquinullstellensatz"]\label{NSREL}
Let $k$ be any field, $A$ a finitely generated $k$-algebra, and $S$ the set of all $f\in A$ such that $\phi(f)\neq 0$ for all $k$-morphisms $\phi:A\to k$. Every ideal $I$ of $A$ disjoint from $S$ and maximal as such is a maximal ideal such that $A/I\cong k$ (and reciprocally).
\end{thm}
\begin{proof}
If $k$ is algebraically closed, then $S=k^\xx$ and the result is a consequence of Hilbert's Nullstellensatz, so we now suppose that $k$ is not algebraically closed. If $P\in k[\ov X]=k[X_1,\ldots,X_n]$ has a zero $[\ov f]=\ov f+I$ in $A/I$ for $\ov f\in A^n$, we have $P(\ov f)\in I$, and as $I\cap S=\emptyset$, there exists a $k$-morphism $\phi:A\to k$ such that $P(\phi(\ov f))=\phi(P(\ov f))=0$, and $P$ already has a zero in $k$. In particular, if $I=(P_1,\ldots,P_m)$ and $N(X_1,\ldots,X_m)$ is a normic form for $k$ by Proposition \ref{EXNORM0}, as $N(P_1,\ldots,P_m)$ has a zero in $A/I$, it has a zero in $k$ by what precedes, and as $N$ is a normic form, $I$ itself has a zero in $k$, corresponding by evaluation to a $k$-morphism $e:A/I\to k$. Now the composite $k$-morphism $\phi :A\to A/I\to k$ has $I\subs Ker(\phi)$, and if $P\in Ker(\phi)$, by definition we have $P\notin S$, so $Ker(\phi)\cap S=\emptyset$ : by maximality of $I$ with this last property, we have $I=Ker(\phi)$, so $e:A/I\to k$ is an isomorphism, and $I$ is maximal.
\end{proof}
\begin{rem}
i) Finiteness is needed in both cases, in the first for the application of Hilbert's Nullstellensatz, in the second for the application of a normic form to a finitely generated ideal.\\
ii) For any $k$-algebra $A$ and ideal $I$ of $A$, if $\phi:A/I\cong k$ is an isomorphism, $\phi$ is necessarily the inverse of the structural morphism $k\to A/I$, so the ideals of the statement are exactly those for which $k\cong A/I$.
\end{rem}
\begin{cor}\label{CORNSREL}
If $k$ is any field, $V\subs k^n$ is an affine algebraic subvariety, and $S=\{g\in k[V] :\forall P\in V,g(P)\neq 0\}$, an ideal $I$ of $k[V]$ has a zero in $V$ if and only if $I\cap S=\emptyset$.
\end{cor}
\begin{proof}
If $I$ has a zero in $V$, certainly we have $I\cap S=\emptyset$. Conversely, if $I\cap S=\emptyset$, by Noetherianity of $k[V]$ there exists an ideal $\m$ of $k[V]$, containing $I$, disjoint from $S$, and maximal with this property : by Theorem \ref{NSREL}, the structural morphism $k\to k[V]/\m$ is an isomorphism, which means that $I$ has a zero in $V$.
\end{proof}
\noindent As a first significant geometric consequence of the Äquinullstellensatz, we may characterise the global sections of the sheaf of regular functions on an irreducible affine algebraic subvariety, a result which we will generalise in section \ref{AFFSUB}.
\begin{prop}\label{CHSEC}
If $V\subs k^n$ is irreducible and $k\{V\}:=k[V]_S$, where $S=\{g\in k[V]:\forall P\in V,g(P)\neq 0\}$, then $\Gamma(V,\O_V)\cong k\{V\}$.
\end{prop}
\begin{proof}
Let $f\in \Gamma(V,\O_V)$ and for every $P\in V$, $U_P\subs V$ an open neighbourhood of $P$ in $V$ such that $f|_{U_P}\equiv u_P/v_P$ (i.e. such that $v_P(Q)\neq 0$ and $f(Q)=u_P(Q)/v_P(Q)$ for all $Q\in U_P$), with $u_P,v_P\in k[V]$. As $v_P\neq 0$ for all $P$, define a map $\phi:\Gamma(V,\O_V)\into k(V)$ by $f\mapsto u_P/v_P$ for any $P$; if $P,Q\in V$, as $V$ is irreducible $U_P$ is dense in $V$, so $O:=U_P\cap U_Q\neq \emptyset$ and for every $R\in O$ we have $f(R)=u_P(R)/v_P(R)=u_Q(R)/v_Q(R)$ so $u_Pv_Q|_O=u_Qv_P|_O$ and by density of $O\neq\emptyset$ in $V$, as the diagonal $\Delta_V$ is closed in $V\xx V$ we have $u_Pv_Q=u_Qv_P$ in $k[V]$, and therefore $u_P/v_P=u_Q/v_Q$ and $\phi$ is well defined, and obviously a $k$-morphism. If $\phi(f)=u_P/v_P=0$, we have $u_P=0\in k(V)$, so $f|_{U_P}\equiv 0$ and as $f$ is continuous and $U_P$ is dense, as $\Delta_V$ is closed again we have $f\equiv 0$, and $\phi$ is injective : denote by $A$ its isomorphic image in $k(V)$ and note that by definition, we have $k\{V\}\subs A$. Now let $I$ be the ideal of $k[V]$ generated by the $v_P$'s, $P\in V$ : if $I\cap S=\emptyset$, by the Äquinullstellensatz (\ref{NSREL}) $I$ has a rational point $Q\in V$, for which $v_Q(Q)=0$, which is impossible, so there exist $v\in I\cap S$, $r\in\NN$, $P_1,\ldots,P_r\in V$ and $\alpha_1,\ldots,\alpha_r\in k[V]$ with $v=\sum_{i=1}^r \alpha_i v_{P_i}$, from which we get, in $A$, $\phi(f)v=\sum_i \alpha_i \phi(f) v_{P_i}=\sum_i \alpha_i u_{P_i}$, and therefore $\phi(f)=(1/v)\sum_i \alpha_i u_{P_i}\in k\{V\}$, and we conclude that $k\{V\}=A$, i.e. $ \Gamma(V,\O_V)\cong k\{V\}$.
\end{proof}
\subsection*{Equiradicals and canonical localisation}
If $k[\ov X]=k[X_1,\ldots,X_n]$ is a polynomial algebra and $I$ is an ideal of $k[\ov X]$, the elements of $\ms Z(I)$ are in bijection with the $k$-morphisms $\phi:k[\ov X]\to k$ such that $I\subs Ker(\phi)$; we let $e_P:k[\ov X]\to k$ be the evaluation morphism at $P\in k^n$. In other words, if $S=\{f\in k[\ov X] | \forall \phi:k[\ov X]\to k,\phi(f)\neq 0\}$, for every point $P\in \ms Z(I)$ we have $Ker(e_P)\cap S=\emptyset$, and conversely every maximal ideal disjoint from $S$ and containing $I$ has the form $e_P$ for $P\in \ms Z(I)$ by Theorem \ref{NSREL}. It follows that $\ms I(\ms Z(I))$, which is the kernel of the product $k$-morphism $e_I:k[\ov X]\to k^{\ms Z(I)}$ of the morphisms $e_P$'s for $P\in \ms Z(I)$, is the intersection of all maximal ideals of $k[\ov X]$ containing $I$ and disjoint from $S$. Abstracting this notion we adopt the following
\begin{defi}\label{EQUIRAD}
If $A$ is a $k$-algebra, say that a maximal ideal $\m$ of $A$ is \emph{special} if the structural morphism $k\to A/\m$ is an isomorphism. If $I$ is any ideal of $A$, the \emph{equiresidual radical of $I$}, or \emph{equiradical of $I$}, noted $\erad I$, is the intersection of all special maximal ideals of $A$ containing $I$.
\end{defi}
\begin{rem}\label{EQUIREM}
i) If $A$ is of finite type and $S=\{f\in A | \forall \phi:A\to k,\phi(f)\neq 0\}$ as before, then by the Äquinullstellensatz (\ref{NSREL}) a maximal ideal $\m$ of $A$ is special if and only if $\m\cap S=\emptyset$.\\
ii) If $A$ is the coordinate algebra $k[V]$ of some affine algebraic subvariety $V\subs k^n$, then the equiradical of an ideal $I$ of $A$ is nothing else than $\ms I(\ms Z_V(I))=\{f\in A :\forall P\in \ms Z_V(I),f(P)=0\}$. All this could seem trivial, were it not for the existence of normic forms which make it possible to \og encode" this information in the multiplicative set $S$ in case $k$ is not algebraically closed.\\
iii) Another solution is to save the expression \og special maximal ideal" for a maximal ideal $\m$ such that $A/\m$ preserves the algebraic signature (Definition \ref{ALGSIG}), as in \cite{PAG1}. Both notions coincide for finitely generated $k$-algebras, so we keep it this way in order to connect with the general concept of a \emph{special algebra} (Definition \ref{SPEALG}).
\end{rem}
\noindent If $A$ is a $k$-algebra, the set $S$ as defined above is multiplicative; in case $A$ is of finite type, by what precedes we may identify the special maximal ideals of $A$ with the maximal ideals of $A_S$ by localisation. This leads to a transposition of the usual algebraic constructions surrounding classical algebraic geometry into this kind of localised algebras, which we begin to study here using a more convenient description of $A_S$, leading to a profitable characterisation of the equiradical, thanks to the following notions which are inspired by Theorem 2 of \cite{MK} and Theorem 2.1 of \cite{B-H}.
\begin{defi}\label{ALGSIG}
i) The \emph{algebraic signature of $k$} is the set $\ms D$ of all polynomials in finitely many variables over $k$ which have no zero rational in $k$.\\
ii) If $A$ is a $k$-algebra, we note $M_A$ the multiplicative subset of all $D(\ov a)$ for $D\in \ms D$ and $\ov a\in A$, and we call $A_M:=A_{M_A}$ the \emph{canonical localisation of $A$}.
\end{defi}
\begin{rem}
i) The algebraic signature is an analogue of McKenna's \og determining sets" (\cite{MK}, Theorem 2). As with normic forms, both notions have a common natural generalisation using positive logic (see \cite{PAG1} again).\\
ii) If $k$ is algebraically closed, then $\ms D=k^\xx$, so for every $k$-algebra $A$, we have $M_A\cong k^\xx$ and $A\cong A_M$.
\end{rem}
\begin{lem}\label{CARSPE}
If $A$ is a finitely generated $k$-algebra and $J$ is an ideal of $A$, then $J\cap S=\emptyset\Iff J\cap M_A=\emptyset$.
\end{lem}
\begin{proof}
It suffices to prove it for $A=k[\ov x]=k[X_1,\ldots,X_n]/I$. If $f\in J\cap S$, write $f=F+I$ with $F\in k[\ov X]$, and let $P_i:i=1,\ldots,m$ be finitely many generators of $I$. Suppose $k$ is algebraically closed, by definition of $S$ the ideal $(F,I)$ of $k[\ov X]$ has no zero in $k$; by Hilbert's Nullstellensatz we have $1\in \sqrt{(F,I)}$, in other words there are polynomials $G,H_i\in k[\ov X]$ such that $1=GF+\sum_i H_iP_i$, whence $1=gf$ in $A$, for $g=G+I$; it follows that $1\in J$, so $J=A$ and $J\cap M_A\neq\emptyset$. Suppose $k$ is not algebraically closed, and $N(Y,Z_1,\ldots,Z_m)$ is an appropriate normic form over $k$ by Proposition \ref{EXNORM0} : by definition of $S$, $F$ and the $P_i$'s have no common zero in $k$, so the polynomial $D=N(F,P_i:i)$ has no zero in $k$ and is therefore a member of $\ms D$. It follows that $g:=N(f,\ov 0)=N(F,P_i:i)+I=D(\ov x+I)$ is both a member of $J$ (as a $k$-linear combination of powers of $f$) and a member of $M_A$. Conversely, suppose $f\in J\cap M_A$, then $f=D(\ov g)$ for some $D\in \ms D$ and $\ov g=\ov G+I$; if $\phi:A\to k$ is a k-morphism, we have $\phi(f)=\phi(D(\ov g))=D(\phi(\ov g))\neq 0$ by definition of $\ms D$, so $f\in J\cap S$, which is not empty, and the lemma is proved.
\end{proof}
\begin{rem}
Keeping in mind the first point of Remark \ref{EQUIREM}, we now see that the special maximal ideals of a finitely generated $k$-algebra $A$ are the maximal ideals which are disjoint from $M_A$. Beware that this is not true in general $k$-algebras (see Example \ref{CORFUNEX}).
\end{rem}
\begin{prop}
For every finitely generated $k$-algebra $A$, we have $A_M\cong A_S$.
\end{prop}
\begin{proof}
As $M_A\subs S$, it suffices to show by the universal properties of $A_M$ and $A_S$ that every member of $S$ becomes invertible in $A_M$. We have $A_M^\xx=\bigcap\{\m^c : \m\in Spm(A_M)\}$, and $\m\in Spm(A_M)\Iff \m=\n A_M$ for $\n$ disjoint from $M_A$ and maximal as such $\Iff \m=\n A_M$ for $\n$ disjoint from $S$ and maximal as such (by Lemma \ref{CARSPE}) $\Iff \m=\n A_M$ for $\n$ maximal and special by Theorem \ref{NSREL}. Now let $s\in S$ : $s$ belongs to no special maximal ideal of $A$, so by what precedes $s$ is invertible in $A_M^\xx$ and the proposition is proved.
\end{proof}
\begin{rem}
A direct proof in the non-algebraically closed case along \ref{CARSPE} is interesting : if the members of $M_A$ are invertible, an element of $S$ has the form $F+I$ with $(F,I)$ having no zero in $k$, so $N(f,\ov 0)$ is in $M_A$, so is invertible; now $N(f,\ov 0)$ is precisely the monomial where only the variable corresponding to $f$ occurs, with a power $\geq 1$, so that inverting $N(f,\ov 0)$ entails inverting $f$.
\end{rem}
\noindent The following lemma is a generalisation of the existence of \og rational points" (i.e. morphisms to the base field) for any non-trivial finitely generated algebra over an algebraically closed field.
\begin{lem}\label{CARPRE}
If $A$ is a finitely generated $k$-algebra, then $A_M\neq 0$ if and only if there exists a $k$-morphism $A_M\to k$.
\end{lem}
\begin{proof}
It suffices to prove the direct sense. If $A_M\neq 0$, there exists by Noetherianity a maximal ideal $\m$ of $A_M$, and we let $\p:=\m\cap A$, an ideal of $A$ disjoint from $M_A$, and maximal as such. By Lemma \ref{CARSPE}, $\p$ is disjoint from $S$, and maximal as such, so by the Äquinullstellensatz (\ref{NSREL}) $\p$ is maximal and $A/\p\cong k$. As $A_M/\m\cong (A/\p)_M\cong k$ as $k$-algebras, $\m$ is the kernel of a morphism $A_M\to k$ and the proof is complete.
\end{proof}
\noindent In order to characterise the equiradical in finitely generated $k$-algebras $A$, we are going to use localisation at one element. If $a\in A$, we let $\Sigma_a$ be the multiplicative subset generated by all elements of the form $a^mD^\#(\ov b,a^n)$, for $D(\ov X)\in \ms D$ of degree $d$ say, $D^\#(\ov X,Y)=Y^dD(\ov X/Y)$ the homogenisation of $D$, $m,n\in\NN$ and $\ov b$ an appropriate tuple from $A$. We note $A_{\<a\>}$ the localisation $\Sigma_a^{-1}A$.
\begin{lem}\label{CARUNLOC}
If $A$ is a finitely generated $k$-algebra and $a\in A$, the map $c/a^mD^\#(\ov b,a^n)\in A_{\<a\>}\mapsto (c/a^m)/a^{nd} D(\ov b/a^n)\in (A_a)_M$ is a $k$-isomorphism $A_{\<a\>}\cong (A_a)_M$.
\end{lem}
\begin{proof}
Let $l_a:A\to A_a$ be the localisation at $a$, $l_M:A_a\to (A_a)_M$ the canonical localisation and $f_a:A\to A_{\<a\>}$ the localisation by $\Sigma_a$. As $a\in (A_{\<a\>})^\xx$, there exists a unique morphism $\phi_a:A_a\to A_{\<a\>}$ such that $\phi_a\circ l_a=f_a$. For $D(\ov x)\in \ms D$ and $\ov b/a^m$ a corresponding tuple in $A_a$, we have $D(\ov b/a^m)=(1/a^{md})D^\#(\ov b,a^m)$, an element of $A_a$ which becomes invertible in $A_{\<a\>}$. By the universal property of $l_M$ (as a morphism of $A$-algebras), there exists a unique $A$-morphism $\phi:(A_a)_M\to A_{\<a\>}$ such that $\phi\circ l_M=\phi_a$, and by the universal property of $l_a$ this is the unique such that $\phi\circ l_M\circ l_a=f_a$. The other way round, any non-negative power $a^m$ of $a$ is invertible in $(A_a)_M$ and in $A_a$ we have $D^\#(\ov b,a^m)=a^{md}D(\ov b/a^m)$, which also becomes invertible in $(A_a)_M$. By the universal property of $f_a$, there exists a unique $\psi:A_{\<a\>}\to (A_a)_M$ such that $\psi\circ f_a=l_M\circ l_a$. We have $\psi\phi l_Ml_a=\psi f_a=l_Ml_a$ and by the universal properties of localisation this entails $\psi\phi=1$; likewise, we have $\phi\psi f_a=f_a$ and for the same reason we have $\phi\psi=1$, so that $\phi$ and $\psi$ are reciprocal isomorphisms. Now by definition, we have $\phi((c/a^m)/D(\ov b/a^n))=ca^{nd}/a^mD^\#(\ov b,a^n)$ and $\psi(c/a^mD^\#(\ov b,a^n))=(c/a^m)/a^{nd}D(\ov b/a^n)$. The maps are represented on the following diagram :
$$\begin{CD} A @>l_a>> A_a @>l_M>> (A_a)_M @= (A_a)_M\\
@Vf_aVV @V\phi_a VV @V\phi VV @AA\psi A\\
A_{\<a\>} @= A_{\<a\>} @= A_{\<a\>} @= A_{\<a\>}. \end{CD}$$
\end{proof}
\begin{thm}\label{CARERAD}
For a finitely generated $k$-algebra $A$ and an ideal $I$ of $A$, we have
$\erad I=\{a\in A : I\cap \Sigma_a\neq\emptyset\}$.
\end{thm}
\begin{proof}
Suppose $a\notin \erad I$ : by definition there exists a special maximal ideal $\m$ of $A$ containing $I$ and such that $a\notin \m$. For all $m,n\in\NN$, we have $a^m,a^n\notin\m$ and as $A/\m\cong k$, for all $D\in \ms D$ with degree $d$ and appropriate $\ov b\in A$ we have $a^mD^\#(\ov b,a^n)\notin \m$ (otherwise $D([\ov b]/[\ov a^n])=(1/[a^{nd}])D^\#([\ov b],[\ov a^n])=0$ in $A/\m$, contradicting the choice of $D$ and $\m$), so $I\cap \Sigma_a=\emptyset$ by primality of $\m$. Conversely, if $I\cap \Sigma_a=\emptyset$, then $A_{\<a\>}/\Sigma_a^{-1} I\neq 0$ and as $(A/I)_{\<a+I\>}\cong\Sigma_a^{-1}(A/I)\cong A_{\<a\>}/\Sigma_a^{-1} I$, there exists a $k$-morphism $(A/I)_{\<a+I\>}\to k$.
Indeed, we have $(A/I)_{\<a+I\>}\cong ((A/I)_{a+I})_M$ by Lemma \ref{CARUNLOC}, and as $(A/I)_{a+I}$ is finitely generated over $k$, we may apply Lemma \ref{CARPRE}. Let then $\m$ be the kernel of the composite morphism $A\to A/I\to (A/I)_{\<a+I\>}\to k$ : we have $a\notin \m$, and as $\m$ is special we get $a\notin \erad I$.
\end{proof}
\noindent This theorem is the key ingredient to the characterisation of the algebras of sections of regular functions over an open subset of an affine algebraic subvariety, which is the core result of the next section and of the article.
\section{Affine algebraic subvarieties}\label{AFFSUB}
\subsection*{$*$-Algebras and special algebras}\label{SPECALG}
The following definition captures the intrinsic algebraic properties of canonical localisations.
\begin{defi}
Say that a $k$-algebra $A$ is a \emph{$*$-algebra (over $k$)} if every element of $M_A$ is invertible in $A$.
\end{defi}
\begin{ex}\label{CORFUNEX}
For every irreducible affine algebraic subvariety $V\subs k^n$, $k(V)$ is a $*$-algebra : if $D\in\ms D$ and $\ov f/g\in k(V)$, we have $D(\ov f/g)=D^\#(\ov f/g,1)=(1/g^d)D^\#(\ov f,g)$ and as $g\neq 0$, we have $D^\#(\ov f,g)\neq 0$ by Lemma \ref{CARSPESIG}, so $D(\ov f/g)\in k(V)^\xx$.
\end{ex}
\begin{lem}\label{CARSTAR}
If $A$ is a $k$-algebra and $l_M:A\to A_M$ its canonical localisation, then $A$ is a $*$-algebra if and only if $l_M$ is an isomorphism. In particular, $A_M$ is a $*$-algebra for every $k$-algebra $A$.
\end{lem}
\begin{proof}
If $A$ is a $*$-algebra, then for every $k$-morphism $f:A\to B$ with $f(M_A)\subs B^\xx$, there exists a unique $g:A\to B$ such that $g\circ 1_A=f$, so $1_A$ has the universal property of $l_M$, which is therefore an isomorphism. Conversely, if we assume that $l_M:A\to A_M$ is an isomorphism, it suffices to show that $A_M$ in general is a $*$-algebra. Let thus $D(\ov x)\in \ms D$ and $\ov a/m\in A_M$ an appropriate tuple : $m$ has the form $D_1(\ov a_1)$ for $D_1\in \ms D$ and $D(\ov a/m)=D^\#(\ov a/m,1)=(1/m^d)D^\#(\ov a,m)$. Let $D_2(\ov x,\ov x_1)=D^\#(\ov x,D_1(\ov x_1))$ : if $\ov b,\ov b_1\in k$, we have $D_1(\ov b_1)\neq 0$, hence $D^\#(\ov b,D_1(\ov b_1))\neq 0$ (otherwise $D(\ov b /D_1(\ov b_1))=(1/D_1(\ov b_1)^d)D^\#(\ov b,D_1(\ov b_1))=0$), so $D_2\in \ms D$, and therefore $m^dD(\ov a/m)=D_2(\ov a,D_1(\ov a_1))\in A_M^\xx$, whence $D(\ov a/m)\in A_M^\xx$, so $A_M$ is a $*$-algebra.
\end{proof}
\begin{rem}
i) The key ingredient of the proof is borrowed from \cite{B-H}, Theorem 2.1.\\
ii) By the properties of localisation, to every morphism of $k$-algebras $\phi:A\to B$, we may associate a morphism of $*$-algebras $\phi_M:A_M\to B_M$ in an obvious way. Canonical localisation is thus a functor from +$k$-algebras to $*$-algebras, left adjoint to the forgetful functor. This last category has many interesting properties, being in particular locally finitely presentable (see \cite{LPAC} for instance). We will not go into the category-theoretic detail here, but we will use a notion of $*$-algebra of finite type (as such) in section \ref{SEQVAR}.
\end{rem}
\noindent The following very simple definition, inspired by the first order theory of quasivarieties (the curious reader might want to have a glance at sections 9.1 and 9.2 of \cite{HMT}), generalises reduced algebras of finite type over (algebraically closed) fields.
\begin{defi}\label{SPEALG}
Say that a $k$-algebra $A$ is \emph{special} if $A$ embeds as a $k$-algebra into a power of $k$. Say that an ideal $I$ of a $k$-algebra $A$ is \emph{special} if $I=\erad I$.
\end{defi}
\noindent It is obvious that a maximal ideal $\m$ of $A$ is special in the sense of the present definition if and only if it is in the sense of Definition \ref{EQUIRAD}.
\begin{lem}\label{CARSPEQUO}
If $A$ is a $k$-algebra and $I$ an ideal of $A$, then $A/I$ is special if and only if $I$ is special.
\end{lem}
\begin{proof}
Suppose $A/I$ is special : there exists a set $S$ and an embedding $\phi:A/I\into k^S$ of $k$-algebras. If $a\in A-I$, we have $\phi(a+I)\neq 0$, so there exists $s\in S$ such that $p_s\circ \phi(a)\neq 0$, where $p_s: k^S\to k$ is the $s$-th projection. It follows that $a\notin \m:=Ker(p_s\circ \phi\circ \pi_I)$, for $\pi_I:A\onto A/I$ the canonical projection; as $\m$ is special, we have $a\notin \erad I$, so $I=\erad I$. Conversely, if $I=\erad I$, then by definition of $\erad I$ the quotient $A/I=A/\erad I$ embeds into $k^S$, where $S$ is the set of special maximal ideals containing $I$, so $A/I$ is special.
\end{proof}
\begin{lem}\label{CARSPESIG}
An ideal $I$ of a $k$-algebra $A$ is special (i.e. $I=\erad I$) if and only if for all $D(\ov x)\in \ms D$, $\ov a,b\in A$ and $m,n\in\NN$ such that $b^mD^\#(\ov a,b^n)\in I$, we have $b\in I$. In particular, an algebra $A$ is special if and only if for all $D(\ov x)\in \ms D$ and $\ov a,b\in A$ such that $b^mD^\#(\ov a,b^n)=0$, we have $b=0$.
\end{lem}
\begin{proof}
Suppose $I$ is special : by Lemma \ref{CARSPEQUO}, $A/I$ is special, so there exists an embedding $\phi:A/I\into k^S$ for a set $S$. Let $D(\ov x)\in \ms D$, $\ov a,b\in A$ and $m,n\in\NN$ be such that $b^mD^\#(\ov a,b^n)\in I$ : write $\ov a_I=\ov a+I$, $b_I=b+I$, for each $s\in S$ we have $\phi b_I(s)^mD^\#(\phi\ov a_I(s),\phi b_I(s)^n)=0$ in $k$. If $\phi b_I(s)\neq 0$, then $D(\phi\ov a_I(s)/\phi b_I(s)^n)=\phi b_I(s)^{-nd}D^\#(\phi \ov a_I(s),\phi b_I(s)^n)=0$, which contradicts the definition of $\ms D$, so $\phi b_I(s)=0$ for all $s$, whence $\phi(b_I)=0$ and therefore $b_I=0$, i.e. $b\in I$. Conversely, suppose the property holds, and let $b\notin I$. If $D(\ov x)\in \ms D$, $\ov a\in A$ and $n\in\NN$, suppose that $D([\ov a]/[b^n])=0$ in $B:=(A/I)_{[b]}$ : we get $D^\#([\ov a],[b^n])=0$, thus there exists $m\in\NN$ such that $b^m D^\#(\ov a,b^n)=0$ in $A/I$, whence $b\in I$, which contradicts our assumption. We get $D([\ov a]/[b^n])\neq 0$, so $0\notin M_B$, whence $B_M\neq 0$ and by Lemma \ref{CARPRE} there exists a $k$-morphism $B_M\to k$ : the kernel $\m$ of the composite morphism $A\to (A/I)_{[b]}\to B_M\to k$ is special and contains $I$ but not $b$, so $b\notin \erad I$. We conclude that $I=\erad I$, i.e. that $I$ is special.
\end{proof}
\begin{rem}\label{HOMSIGN}
Special ideals may as well be characterised by the \emph{homogeneous signature} of $k$, which is the set $\ms H$ of all homogeneous polynomials $P$ over $k$ with no non-trivial zero rational in $k$ : an ideal $I$ is special if and only if for every $P(\ov x,y)\in \ms H$ and $\ov a,b\in A$ such that $P(\ov a,b)\in I$, we have $b\in I$. We do not need this here but we will expand on the subject in \cite{PAG1}.
\end{rem}
\noindent The total ring of fractions of a coordinate ring generalises the function field of an irreducible affine subvariety; we need to check that the construction preserves the fact of being a special algebra.
\begin{prop}\label{THFRAC}
If $A$ is a special $k$-algebra, then the total ring $\Phi A$ of fractions of $A$ is a special $*$-algebra, as well as the canonical localisation $A_M$.
\end{prop}
\begin{proof}
Let $D(\ov x)\in \ms D$ and $\ov a/s,b/s\in \Phi A$ with the same denominator : if $D^\#(\ov a/s,b/s)=0$, we have $D^\#(\ov a,b)=s^d D^\#(\ov a/s,b/s)=0$ in $A$ (for $d=deg(D)$), whence $b=0$ by Lemma \ref{CARSPESIG}, so $b/s=0$, and by the same lemma $\Phi A$ is special. Furthermore, let $D(\ov x)\in \ms D$ and $\ov a/s$ an appropriate tuple from $\Phi A$ : if $b\in A$ and $D^\#(\ov a,s)b=0$ in $A$, either $D^\#$ is constant (and nonzero) and thus $b=0$, or $deg(D^\#)=d>0$ and $0=D^\#(\ov a,s)b^d=D^\#(b\ov a,bs)$ and by Lemma \ref{CARSPESIG} again, we get $bs=0$, whence $b=0$ because $s$ is simplifiable; it follows that $D^\#(\ov a,s)$ is simplifiable, so $D(\ov a/s)=(1/s^d)D^\#(\ov a,s)\in (\Phi A)^\xx$, which is therefore a $*$-algebra. Now let $m\in M_A$ and $f\in A$ such that $fm=0$; as $A$ is special, by definition there exists an embedding $\phi:A\into k^S$ for a set $S$, and for each $s\in S$ we have $\phi(f)(s).\phi(m)(s)=0$ in $k$; write $m=D(\ov a)$ : as $\phi(m)(s)=D(\phi(\ov a)(s))$, we get $\phi(f)=0$ by definition of $\ms D$, so $f=0$ and $m$ is simplifiable. It follows that $A_M$ embeds into $\Phi A$ over $A$, and is special as a subalgebra of a special algebra.
\end{proof}
\subsection*{Points and subvarieties in affine spaces}
In classical algebraic geometry (i.e. over an algebraically closed field $k$), we have a well known correspondance between algebraic subsets of $k^n$ and radical ideals of $k[X_1,\ldots,X_n]$ (\cite{HAG}, Corollary I.1.4). This is true in general if we replace radical ideals by special ideals. We begin with the case of points (recall that if $(a_1,\ldots,a_n)\in k^n$ and $k[x_1,\ldots,x_n]=k[X_1,\ldots,X_n]/(X_1-a_1,\ldots,X_n-a_n)$, in $k[\ov x]$ we have $x_i=a_i$ for each $i$, so the structural morphism $k\to k[\ov x]$ is an isomorphism and $(X_1-a_1,\ldots,X_n-a_n)$ is a maximal ideal).
\begin{lem}\label{BIJPNT}
For every $n\in\NN$, the map $P\in k^n\mapsto Ker(e_P)$ is a bijection between the points of the affine $n$-space and the special maximal ideals of $k[X_1,\ldots,X_n]$, which are therefore of the form $(X_1-a_1,\ldots,X_n-a_n)$ for $a_1,\ldots,a_n\in k$ (and the reciprocal bijection is given by $\m\mapsto \ms Z(\m)$).
\end{lem}
\begin{proof}
Write $A=k[X_1,\ldots,X_n]$. If $P=(a_1,\ldots,a_n)$, we have $(X_1-a_1,\ldots,X_n-a_n)= Ker(e_P)$ by what precedes; if $Q=(b_1,\ldots,b_n)$ and $Q\neq P$, it follows that $Ker(e_P)\neq Ker(e_Q)$, and the map is injective. As for surjectivity, if $\m$ is a special maximal ideal of $A$, we have $\m=\erad \m$ by definition, so by Theorem \ref{NSREL} we have $\ms I(\ms Z(\m))=\erad\m=\m\neq A$, whence $\ms Z(\m)\neq\emptyset$ and there exists a zero $P$ of $\m$ in $k^n$, so that $Ker(e_P)\subs \m$; by maximality of $Ker(e_P)$, we have $Ker(e_P)=\m$ and the map is surjective.
\end{proof}
\noindent Points are particular cases of irreducible subvarieties, which have in general the same useful characterisation as in algebraically closed fields. Recall that if $V$ is an affine subvariety and $f\in k[V]$, we let $\ms Z_V(f)=\{P\in V : f(P)=0\}$.
\begin{lem}\label{CARIRRAFF}
If $V\subs k^n$ is an affine subvariety, then $V$ is irreducible if and only if $\Gamma(V,\O_V)$ is an integral domain, if and only if $k[V]$ is an integral domain.
\end{lem}
\begin{proof}
Suppose $V$ is irreducible and $f,g\in J(V):=\Gamma(V,\O_V)$ are such that $fg=0$ : for all $P\in V$, we have $f(P)g(P)=0$, so $V=\ms Z_V(f)\cup\ms Z_V(g)$; as $V$ is irreducible and $\ms Z_V(f),\ms Z_V(g)$ are closed, we have $V=\ms Z_V(f)$ or $V=\ms Z_V(g)$, i.e. $f=0$ or $g=0$, and $J(V)$ is an integral domain, as well as $k[V]$, which embeds into $J(V)$. Next, suppose $k[V]$ is an interal domain, and let $V=V_1\cup V_2$, with $V_1=\ms Z_V(I_1)$ and $V_2=\ms Z_V(I_2)$ for $I_1,I_2$ ideals of $k[V]$, and distinguish two cases : if $I_1=(0)$, then $V=V_1$, whereas if $I_1\neq (0)$, there exists $f\in I_1$, $f\neq 0$; for every $P\in V$ and $g\in I_2$ we now have $fg(P)=0$ (because $V=V_1\cup V_2$), so $fg=0$ and as $k[V]$ is integral, we have $g=0$, and therefore $I_2=(0)$ and $V=V_2$. We conclude that $V$ is irreducible.
\end{proof}
\noindent Let $I$ be an ideal of $k[X_1,\ldots,X_n]$ : we have $\ms I(\ms Z(I))=\erad I$ by the Äquinullstellensatz (\ref{NSREL}), so $\ms Z(I)=\ms Z(\erad I)$, and thus every algebraic set is the zero set of a special ideal. The correspondance is thus given as the following
\begin{prop}
The map $I\mapsto \ms Z(I)$ induces an order-reversing bijection between special ideals of $k[\ov X]$ and algebraic subsets of $k^n$, which restricts to a bijection between prime and special ideals and irreducible algebraic sets, which restricts to a bijection between maximal and special ideals and points of $k^n$.
\end{prop}
\begin{proof}
Suppose $I,J$ are special and $\ms Z(I)=\ms Z(J)$ : we have $\ms I(\ms Z(I))=\ms I(\ms Z(J))$, so $I=\erad I=\erad J=J$ by what precedes, so the map is injective on special ideals. If $V=\ms Z(I)\subs k^n$ is an algebraic subset, we have seen that $V=\ms Z(\erad I)$ so the map is surjective on special ideals, it is a bijection. By Lemma \ref{CARIRRAFF}, a special ideal $I$ is prime if and only if $\ms Z(I)$ is irreducible, which establishes the second part of the statement. Finally, a special ideal $I$ is maximal if and only if it is a special maximal ideal, if and only if $\ms Z(\m)$ is a point by Lemma \ref{BIJPNT}.
\end{proof}
\begin{rem}
The picture may be completed as usual by a description of the topological closure $\ov S=\ms Z(\ms I(S))$ of any subset $S\subs k^n$, and by the relativisation of the correspondance to any affine subvariety.
\end{rem}
\subsection*{Sheaves of regular functions}
If $V\subs k^n$ is an affine algebraic subvariety and $h,h'\in k[V]$, we have $$(*)\ D_V(h)\subs D_V(h') \Iff \ms Z_V(h)\supseteq \ms Z_V(h') \Iff \erad{(h)}\subs \erad{(h')} \Iff h\in \erad{(h')} \Iff \exists \alpha\in \Sigma_h\cap (h')$$ (by definition of $\erad{}$ and Theorem \ref{CARERAD}). Now let $g,h\in k[V]$ and $\alpha\in \Sigma_h$ : if $P\in D_V(h)$ we have $h(P)\neq 0$ and one easily checks that $\alpha(P)\neq 0$, so $h(P)\alpha(P)\neq 0$, and $g/h\alpha$ defines a regular function on $D_V(h)$, i.e. an element of $\O_V(D_V(h))$.
\begin{lem}\label{MAG.3.5}
The map $P\in D_V(h)\mapsto g(P)/h\alpha(P)$ is zero on $D_V(h)$ if and only if $gh=0$ in $k[V]$.
\end{lem}
\begin{proof}
If $g/h\alpha\equiv 0$ on $D_V(h)$, then as $h=0$ on $\ms Z_V(h)=V-D_V(h)$, we have $gh\equiv 0$ on $V$, i.e. $gh=0$ in $k[V]$. Conversely, if $gh=0$, then for every $P\in V$ we have $g(P)h(P)=0$, so $g(P)=0$ if $P\in D_V(h)$, as well as $g(P)/h(P)\alpha(P)$.
\end{proof}
\noindent Although the next lemma should be considered as folklore, we include it for the sake of completeness.
\begin{lem}\label{ISOSUB}
For any affine algebraic subvariety $V\subs k^n$, any basic open subset of $V$ is isomorphic, as a locally ringed space in $k$-algebras, to an affine algebraic subvariety of $k^{n+1}$.
\end{lem}
\begin{proof}
Write $V=\ms Z(I)$ for $I$ an ideal of $k[\ov X]=k[X_1,\ldots,X_n]$. A basic open subset of $V$ has the form $D_V(h)=\{P\in V : h(P)\neq 0\}$ with $h\in k[V]$, and the restriction of $\O_V$ to $D_V(h)$ is $\O_V$ itself; if $H\in k[\ov X]$ with $h=H+\ms I(V)$, for $W=\ms Z(I,YH-1)\subs k^{n+1}$ and the sheaf $\O_W$ of regular functions on $W$, one easily checks that the projection map $\phi:W\to D_V(h)$ is a homeomorphism. Now if $U\subs D_V(h)$ is open and $f\in \O_V(U)$, we let $g:(\ov a,b)\in \phi^{-1}U\mapsto f(\ov a)\in k$; as $f$ is regular, for each $\ov a\in U$ there is an open $U_{\ov a}\subs U$ and $L,M\in k[\ov X]$ such that $\ov a\in U_{\ov a}$ and for all $(\ov c,d)\in \phi^{-1}U_{\ov a}$, $g(\ov c,d)=L(\ov c)/M(\ov c)$, which shows that $g\in \O_W(\phi^{-1}U)$, and if we let $\phi^\#_U(f):=g$ we have defined a morphism of $k$-algebras and a natural transformation $\phi^\#:\O_V|_{D_V(h)}\to \phi_*\O_W$. The other way round, if $g\in \phi_*\O_W(U)=\O_W(\phi^{-1}U)$, we let $f:\ov a\in U\mapsto g(\ov a,1/H(\ov a))$ : for each $(\ov a,b)\in \phi^{-1} U$ there is an open $U_{\ov a,b}\subs \phi^{-1} U$ and $L,M\in k[\ov X,Y]$ such that $(\ov a,b)\in U_{\ov a,b}$ and for all $\ov c\in \phi(U_{\ov a,b})$ we have $f(\ov c)=L(\ov c,1/H(\ov c))/M(\ov c,1/H(\ov c))=(H(\ov c))^{deg(L)-deg(M)}L^\#(\ov c H(\ov c),1,H(\ov c))/M^\#(\ov c H(\ov c),1,H(\ov c))$, so $f$ is regular and if we let $\psi^\#_U(g):=f$ we have defined a morphism of $k$-algebras and a natural transformation $\psi^\#:\phi_*\O_W\to \O_V|_{D_V(h)}$, and one checks that $\phi^\#$ are $\psi^\#$ mutually inverse isomorphisms.
\end{proof}
\begin{lem}\label{COMPSUB}
Every affine algebraic subvariety is compact for the Zariski topology.
\end{lem}
\begin{proof}
Let $V\subs k^n$ be such a subvariety, and suppose that $V=\bigcup_I D_V(f_i)$, a cover by basic open subsets. We have $\emptyset=\bigcap_I \ms Z_V(f_i)=\ms Z_V(\sum_I k[V] f_i)$, so $k[V]=\ms I(\emptyset)=\ms I(\ms Z_V(\sum_I k[V] f_i))=\erad{\sum_I k[V]f_i}$ by Theorem \ref{NSREL}, and thus by Theorem \ref{CARERAD} there exists $m\in \Sigma_1=M_{k[V]}$, a finite subset $I_0$ of $I$ and $(a_i:i\in I_0)$ in $k[V]$ such that $m=\sum_{I_0} a_i f_i$, and therefore $\emptyset=\ms Z_V(m)=\ms Z_V(\sum_{I_0} k[V]f_i)=\bigcap_{I_0} \ms Z_V(f_i)$, whence $V=\bigcup_{I_0} D_V(f_i)$.
\end{proof}
\noindent So far we have defined a natural map $k[V]_{<h>}\to \Gamma(D_V(h),\O_V)$, $g/\alpha\mapsto [P\mapsto g(P)/\alpha(P)]$, which is an injective $k$-morphism : if the member on the right is zero, then the regular map defined by $gh/h\alpha$ is zero on $D_V(h)$, so $gh^2=0$ in $k[V]$ by Lemma \ref{MAG.3.5}, and $g/\alpha=gh^2/h^2\alpha=0$ in $k[V]_{<h>}$. The following theorem is our core result, bringing together the preceding algebraic theory and the affine geometric theory, and is inspired by \cite{MAG}, Proposition 3.6(a).
\begin{thm}\label{MAG.3.6.a}
The morphism $k[V]_{<h>}\to \Gamma(D_V(h),\O_V)$ is an isomorphism. In particular, the sheaf $\O_V$ is a sheaf of special $*$-algebras over $V$.
\end{thm}
\begin{proof}
As for the first assertion, it only remains to prove that the morphism is surjective. Let $f\in \Gamma(D_V(h),\O_V)$ : there exists an open cover $D_V(h)=\bigcup_i U_i$, as well as $g_i,h_i\in k[V]$ for each $i$, such that for every $i$ and $P\in U_i$, $h_i(P)\neq 0$ and $f(P)=g_i(P)/h_i(P)$. Replacing the $U_i$'s by basic open subsets, we may suppose that $U_i=D_V(a_i)$ for all $i$, with $a_i\in k[V$] : we have $D_V(a_i)\subs D_V(h_i)$ and by $(*)$, for every $i$ there exists $\alpha_i\in \Sigma_{a_i}$ and $g_i'\in k[V]$ with $\alpha_i=g_i'h_i$; on $D_V(a_i)$, $f$ is represented by $g_ig_i'/h_ig_i'=g_ig_i'a_i/a_i\alpha_i$ : replacing $g_i$ by $g_ig_i'a_i$, and $h_i$ by $a_i\alpha_i$, as $D_V(a_i)=D_V(a_i\alpha_i)$ we may suppose that $U_i=D_V(h_i)$ for all $i$; we have $D_V(h)=\bigcup_i D_V(h_i)$, and $f$ is represented on $D_V(h_i)$ by $g_i/h_i$. By Lemmas \ref{ISOSUB} and \ref{COMPSUB}, $D_V(h)$ is compact so we may suppose that this cover is finite, and as the functions represented by $g_i/h_i$ and $g_j/h_j$ on $D_V(h_i)\cap D_V(h_j)=D_V(h_ih_j)$ are equal, we have $(g_ih_j-g_jh_i)/h_ih_j\equiv 0$ on $D_V(h_ih_j)$, whence by Lemma \ref{MAG.3.5} $h_ih_j(g_ih_j-g_jh_i)=0$ in $k[V]$, i.e. $h_ih_j^2g_i=h_i^2h_jg_j$. Now we have $D_V(h)=\bigcup_{i=1}^m D_V(h_i)=\bigcup_{i=1}^m D_V(h_i^2)$, so $\ms Z_V((h))=\ms Z_V((h_1^2,\ldots,h_m^2))$, whence $h\in \ms I(\ms Z_V((h_1^2,\ldots,h_m^2))=\erad{(h_1^2,\ldots,h_m^2)}$ by the Äquinullstellensatz (\ref{NSREL}) again, and by Theorem \ref{CARERAD} there exists $\alpha\in \Sigma_h$ and $a_i\in k[V]$ such that $\alpha=\sum_{i=1}^m a_i h_i^2$ : we want to show that $f$ is represented on $D_V(h)$ by $(\sum_i a_ig_ih_i) / \alpha$. Let $P\in D_V(h)$ : for each $j$ such that $P\in D_V(h_j)$, we have $h_j^2\sum_{i=1}^m a_ig_ih_i=\sum_{i=1}^m a_i g_j h_j h_i^2=g_j h_j \alpha$ by what precedes. Now $f$ is represented on $D_V(h_j)$ by $g_j/h_j$, so $fh_j\equiv g_j$ on $D_V(h_j)$, on which therefore we have $fh_j^2\alpha\equiv g_jh_j\alpha\equiv h_j^2\sum_{i=1}^m a_ig_ih_i$ as maps. As $h_j(P)^2\neq 0$ for each $P\in D_V(h_j)$, on $D_V(h_j)$ we have $f\alpha\equiv\sum_i a_ig_ih_i$ as maps, so that $f$ is represented on $D_V(h_j)$ by $(\sum_i a_i g_i h_i)/ \alpha$, and as this is true for every $j$, this is true on $D_V(h)$, so finally the morphism $k[V]_{<h>}\to \Gamma(D_V(h),\O_V)$ is surjective, it is an isomorphism. As for the second assertion, for every open subset $U\subs V$, we have $U=\bigcup\{ D_V(f) : D_V(f)\subs U, f\in k[V]\}$, and therefore $\O_V(U)=\proj_{D(f)\subs U} \O_V(D(f))$. Now by what precedes each $\ms O_V(D(f))$ is a $*$-algebra by Lemmas \ref{CARUNLOC} and \ref{CARSTAR}, and a special algebra as well, because $D_V(f)$ is isomorphic to an affine algebraic subvariety by Lemma \ref{ISOSUB}; as $*$-algebras and special algebras are clearly closed under projective limits, $\ms O_V(U)$ is a special $*$-algebra.
\end{proof}
\begin{cor}\label{CARREGMAP3}
For every affine algebraic subvariety $V\subs k^n$, the $k$-algebra $\Gamma(V,\O_V)$ of everywhere regular functions on $V$ is isomorphic to $k[V]_M$, by the natural $k$-morphism $k[V]_M\to \Gamma(V,\O_V)$.
\end{cor}
\begin{proof}
By Theorem \ref{MAG.3.6.a}, we have $\Gamma(V,\O_V)=\Gamma(D_V(1),\O_V))\cong k[V]_{\<1\>}=k[V]_M$.
\end{proof}
\begin{rem}
i) This is the generalisation over an arbitrary field of the characterisation of the algebra of global sections of the sheaf of regular functions on an affine algebraic subvariety (if $k$ is algebraically closed, this algebra is essentially $k[V]$).\\
ii) Of course, in general $k[V]_M$ is bigger than $k[V]$. For instance, if $k$ is a subfield of $\RR$, the rational function $x\in k\mapsto 1/x^2+1$ is regular over $k$, but the rational fraction $1/X^2+1$ is not in $k[X]$, the coordinate algebra of $k$, so $(1/X^2+1)\in k[X]_M-k[X]$.
\end{rem}
\section{Equiresidual affine algebraic varieties over a field}\label{SEQVAR}
\subsection*{Equiresidual varieties and affine $*$-algebras}
Let $k$ by any field. The co-restriction to special $k$-algebras of the coordinate algebra functor $k[-]$ is in fact a duality :
\begin{prop}\label{AFFDUAL}
The functor $k[-]$ is an duality between the categories of affine algebraic subvarieties of $k$ and finitely generated special $k$-algebras.
\end{prop}
\begin{proof}
We focus on essential surjectivity, so let $A$ be a special $k$-algebra of finite type : $A$ is isomorphic to an algebra of the form $k[\ov X]/I$, where $\ov X=(X_1,\ldots,X_n)$. Consider the affine algebraic subvariety $V:=\ms Z(I)\subs k^n$ : we have $k[V]:=k[\ov X]/\ms I(V)=k[\ov X]/\erad I$ (by Theorem \ref{NSREL}) $=k[\ov X]/I$ by Lemma \ref{CARSPEQUO} because $A$ is special; in short, $A$ is isomorphic to $k[V]$, and $k[-]$ is an equivalence.
\end{proof}
\noindent As in the classical context, we may want to work with a category of locally ringed spaces in $k$-algebras which are locally isomorphic to affine algebraic subvarieties (among which we will find the \og equiresidual" version of algebraic equivarieties, see \cite{EQAG2}). This category will comprise the spaces which are essentially affine subvarieties, like basic open subsets of these. We start from a broad definition which we will use in subsequent work.
\begin{defi}
i) An \emph{equiresidual variety over $k$}, or \emph{equivariety over $k$} for short, is a locally ringed space in $k$-algebras $(V,\O_V)$, such that for every $P\in V$, there exists an open neighbourhood $U$ of $P$ in $V$ for which $(U,\O_V|_U)$ is isomorphic to an affine algebraic subvariety.\\
ii) An \emph{(equiresidual) affine algebraic variety over $k$} or \emph{affine (algebraic) equivariety} for short, is an equivariety over $k$ which is isomorphic to an affine algebraic subvariety of $k$.\\
We note $EVar^a_k$ the category of equiresidual affine algebraic varieties over $k$, with arrows the morphisms of locally ringed spaces in $k$-algebras. As we have seen in the introduction, any regular morphism of affine subvarieties \og is" naturally a morphism of equivarieties.
\end{defi}
\begin{rem}\label{EQVAR}
i) If $V$ is an equivariety over $k$, for $f\in\Gamma(V,\O_V)$ write $[f]_P\in \O_{V,P}$ its germ at $P$. If $U$ is an open neighbourhood of $P$ such that $(U,\O_V|_U)$ is isomorphic to an affine algebraic subvariety, as $\O_{V,P}\cong \O_V|_{U,P}$, by the small lemma \ref{SMALEM} the structural morphism $k\to \ov{\O_{V,P}}$ (the residual field of $\O_{V,P}$) is an isomorphism.\\
ii) In this situation, if $U\subs V$ is an open subset and $v\in \O_V(U)$ and for each $P\in U$, $\ov{[v]_P}\neq 0$ in $\ov{\O_{V,P}}$, we have $v\in \O_V(U)^\xx$. Indeed, it suffices to prove this for $V$ an affine subvariety, in which case it is obvious.
\end{rem}
\noindent Let $(\phi,\phi^\#):V\to W$ be a morphism of affine algebraic equivarieties over $k$, with $\phi^\#:\O_W\to \phi_* \O_V$. We have the $k$-algebra morphism $\phi^\#_W:\O_W(W)\to \phi_*\O_V(W)=\O_V(\phi^{-1} W)=\O_V(V)$, and we let $J(\phi)=\phi^\#_W$. This obviously defines a functor from the dual of $EVar^a_k$ to the category of $k$-algebras. By Proposition \ref{AFFDUAL}, the category $EVar_k^a$ and the category of special $k$-algebras of finite type are dual, but if $(V,\O_V)$ is an affine algebraic subvariety of $k^n$, the algebra of sections $J(V)=\Gamma(V,\O_V)$ is, as we have seen in Corollary \ref{CARREGMAP3}, naturally isomorphic to the $*$-algebra $k[V]_M$, which is not in general isomorphic to $k[V]$. The duality of Proposition \ref{AFFDUAL} cannot therefore be extended to $EVar^a_k$ by taking the only natural functor we have in mind, which is $J$, the global section functors, as it is the case in the classical setting. This suggests searching for another duality, by introducing the following type of $k$-algebras :
\begin{defi}
Say that a $*$-algebra $A$ over $k$ is :\\
i) \emph{affine}, if it is isomorphic to the algebra of global sections of regular functions on an affine algebraic subvariety of $k$\\
ii) \emph{of finite $*$-type} if there exists a surjective $k$-morphism of the form $k[X_1,\ldots,X_n]_M\onto A$ (or equivalently if $A$ is isomorphic to the canonical localisation of a $k$-algebra of finite type). By Corollary \ref{CARREGMAP3}, an affine $*$-algebra is of finite $*$-type.
\end{defi}
\begin{rem}
We have an analogue of the strong form of Hilbert's Nullstellensatz : if $k\to K$ is a field extension with $K$ also a $*$-algebra of finite $*$-type, then the extension is an isomorphism. Indeed, we have a $k$-isomorphism $k[\ov X]_M/\m_M\cong K$ with $\m$ a special maximal ideal of $k[\ov X]$ by \ref{NSREL} and \ref{CARSPE}, whence the sequence of $k$-isomorphisms $k\cong k[\ov X]/\m\cong (k[\ov X]/\m)_M \cong k[\ov X]_M/\m_M\cong K$.
\end{rem}
\noindent Now the functor $J$ has values in the category $*Aff_k$ of affine $*$-algebras. The obvious properties of such algebras almost readily suggest their following characterisation, which will be used to build the duality :
\begin{prop}\label{AFFSTAR}
If $A$ is a $k$-algebra, the following are equivalent :\\
i) $A$ is an affine $*$-algebra over $k$\\
ii) $A$ is a special $*$-algebra of finite $*$-type over $k$.
\end{prop}
\begin{proof}
(i)$\Imp$(ii) Let $V\subs k^n$ be an affine algebraic subvariety of $k^n$ say, such that $A\cong \Gamma(V,\O_V)$ : we have $\Gamma(V,\O_V)\cong k[V]_M$ by Corollary \ref{CARREGMAP3} and as $k[V]$ is special, so is
$k[V]_M$ by Proposition \ref{THFRAC}. Now the surjective morphism $k[\ov X]\onto k[V]$ gives by canonical localisation a surjective morphism $k[\ov X]_M\onto k[V]_M$, whence (ii).\\
(ii)$\Imp$(i) Let $\phi:k[X_1,\ldots,X_n]_M\onto A$ be a surjective $k$-morphism, and $B=\phi(k[X_1,\ldots,X_n])$ : we have $A\cong B_M$ and as $A$ is special, so is $B$ as a subalgebra; as a special $k$-algebra of finite type, $B$ is isomorphic to a coordinate algebra of an affine algebraic subvariety by Proposition \ref{AFFDUAL}, for instance the subvariety $V=\ms Z(I)$, for $I=Ker(\phi|_{k[\ov X]})$. In particular, we have $A\cong B_M\cong \Gamma(V,\O_V)$ by Corollary \ref{CARREGMAP3} again, therefore $A$ is an affine $*$-algebra over $k$.
\end{proof}
\subsection*{Maximal spectra of affine $*$-algebras}
\noindent The functor $J$ is a duality if and only if it has a right adjoint $K:*Aff_k\to (EVar^a_k)^o$ such that $Id\cong J\circ K$ and $Id\cong K\circ J$, in which case we have an \emph{adjoint equivalence} (\cite{CWM}, Theorem IV.4.1); this adjoint is provided by the \emph{maximal spectrum} of affine $*$-algebras.
In general, let $A$ be any $k$-algebra : following the same idea as for algebraic spaces (\cite{MAG}, Chapter 11), we may define a natural sheaf $\O_X$ on $X=Spm(A)$ as follows : for every open $U\subs X$ for the Zariski topology, we let $\O_X(U)$ be the set of all maps $s:U\to \coprod_{\m\in U} A/\m$, such that for all $\m\in U$, $s(\m)\in A/\m$ and there exists an open neighbourhood $U_\m\subs U$ of $\m$ and $u_\m,v_\m\in A$ for which $s|_{U_\m}\equiv [u_\m]/[v_\m]$ (by which we mean that for every $\n\in U_\m$ we have $v_\m\notin \n$ and $g(\n)=[u_\m]/[v_\m]$ in $A/\n$). Now let $(V,\O_V)$ be a (general) equivariety over $k$ and $A=\Gamma(V,\O_V)$ (a $*$-algebra by Theorem \ref{MAG.3.6.a}), and let $P\in V$ : as the structural morphism $k\to \ov{\O_{V,P}}$ is an isomorphism (see Remark \ref{EQVAR}), the natural morphism $\Gamma(V,\O_V)\to \O_{V,P}\onto \ov{\O_{V,P}}$ is surjective, with kernel $\m_P=\{g\in A : \ov{[g]_P}=0\}$ by definition and we have a $k$-isomorphism $i_P:A/\m_P\cong \ov{\O_{V,P}}$, $[g]\mapsto \ov{[g]_P}$. It is easy to see that the map $\phi_V:P\in V\mapsto \m_P\in X:=Spm(A)$ is continuous, and we may define a sheaf morphism $(\phi_V)^\#:\O_X\to (\phi_V)_*\O_V$ as follows. If $U\subs X$ is open and $s\in\O_X(U)$, for each $\m\in X$ let $U_\m$ and $u_\m,v_\m\in A=\Gamma(V,\O_V)$ as before; if $P\in V$, write $O_P=\phi_V^{-1} U_{\m_P}$ :
for each $Q\in O_P$, we have $\m_Q\in U_{\m_P}$ and $[b_{\m_P}]\neq 0$ in $A/\m_Q$, so $i_Q([b_{\m_P}])=\ov{[b_{\m_P}]_Q}\neq 0$ in $\ov{\O_{V,Q}}$, whence $b_{\m_P}|_{O_P}\in \O_V(O_P)^\xx$ by Remark \ref{EQVAR} and we define an element of $\O_V(O_P)$ by $t_P:=a_{\m_P}|_{O_P}/b_{\m_P}|_{O_P}$.
\begin{lem}\label{EQSEC}
If $s,t\in \O_V(U)$ are such that for each $P\in U$, $\ov{[s]_P}=\ov{[t]_P}$ in $\ov{\O_{V,P}}$, then $s=t$.
\end{lem}
\begin{proof}
By definition of an equivariety and the local character of equality for sections of a sheaf, it suffices to prove this for $V$ an affine algebraic subvariety of $k^n$, in which case it is obviously true because $s$ and $t$ are functions with values in $k$.
\end{proof}
\noindent Let $P,Q\in V$ : if $R\in O_P\cap O_Q$, we have $\m_R\in U_{\m_P}\cap U_{\m_Q}$, so that $s(\m_R)=[u_{\m_P}]/[v_{\m_P}]=[u_{\m_Q}]/[v_{\m_Q}]$ in $A/\m_R$, and thus $u_{\m_P}v_{\m_Q}-u_{\m_Q}v_{\m_P}\in\m_R$, which means that $\ov{[u_{\m_P}]_R}/\ov{[v_{\m_P}]_R}=\ov{[u_{\m_Q}]_R}/\ov{[v_{\m_Q}]_R}$ in $\ov{\O_{V,R}}$, and as this is true for each $R\in O_P\cap O_Q$, by Lemma \ref{EQSEC} we have $t_P|_{O_P\cap O_Q}=t_Q|_{O_P\cap O_Q}$, so the $t_P$'s define a unique section $t\in \O_V(U)$ and we let $(\phi_V)^\#_U(s):=t$. We have defined a map $(\phi_V)^\#_U:\O_X(U)\to \O_V(\phi_V^{-1}U)=(\phi_V)_*\O_V(U)$, and one checks that it is a morphism of $k$-algebras and defines a natural transformation $(\phi_V)^\#:\O_X\to (\phi_V)_*\O_V$, so we have a morphism of locally ringed spaces of $k$-algebras $(\phi_V,(\phi_V)^\#):(V,\O_V)\to (X,\O_X)$. Indeed, with the same notations
the induced morphism on the stalk at $P$ is $(\phi_V)^\#_P:\O_{X,\m_P}\to \O_{V,P}$, $[s,U]\mapsto [a_{\m_P}|_{O_P}/b_{\m_P}|_{O_P}]$,
and we may assume that $U=U_P$; as $k\to \ov{\O_{X,\m_P}}$ is an isomorphism, $(\phi_V)^\#_P$ is local. We now turn to maximal spectra of $*$-algebras of finite $*$-type.
\begin{prop}\label{BIJMAX}
For every finitely generated $k$-algebra $A=k[X_1,\ldots,X_n]/I$, the map $P\in \ms Z(I)\mapsto \m_P=\{f/g\in A_M : f(P)=0\}\in Spm(A_M)$ is a homeomorphism.
\end{prop}
\begin{proof}
Write $V=\ms Z(I)$, $P=(a_1,\ldots,a_n)\in V$ and $e'_P:A\to k$ be the evaluation at $P$, $f\mapsto f(P)$, factoring out the evaluation $e_P:k[X_1,\ldots,X_n]\to k$. By Lemma \ref{BIJPNT}, the map $\phi:P\in V\mapsto Ker(e'_P)$ is obviously bijective. As the canonical localisation $l_M:A\to A_M$ exchanges the special maximal ideals of $A$ and the maximal ideals of $A_M$ by Lemma \ref{CARSPE}, the map $\phi_M:V\to Spm(A_M)$, $P\mapsto Ker(e'_P)_M=\{f/g \in A_M : f(P)=0\}$, is also a bijection. Let $D(f/m)=\{\m_M\in Spm(A)_M : f/m\notin\m_M\}$ be a basic open subset of $Spm(A_M)$ : we have $\phi_M^{-1}(D(f/m))=\{P\in V : f\notin Ker(e'_P)\}=\{P\in V: f(P)\neq 0\}=D_V(f)\subs V$, an open subset, so $\phi_M$ is continuous; and if $D_V(f)=\{P\in V :f(P)\neq 0\}$ is a basic open subset of $V$, we have $\phi_M(D(f))=\{\m_M \in Spm(A_M) : f(\phi_M^{-1}\m_M)\neq 0\}=\{\m_M\in Spm(A_M) : f\notin \m\}=D(f/1)$, a basic open subset of $\subs Spm(A_M)$, so $\phi_M^{-1}$ is continuous, and $\phi_M$ is a homeomorphism.
\end{proof}
\noindent As Proposition \ref{BIJMAX} suggests, the maximal spectrum turns affine $*$-algebras into affine algebraic equivarieties :
\begin{prop}\label{SPMAFF}
If $A$ is an affine $*$-algebra and $X=Spm(A)$, then for the sheaf $\O_X$ of regular functions on $X$ as defined above, $(X,\O_X)$ is an equiresidual affine algebraic variety.
\end{prop}
\begin{proof}
By definition, if $A$ is an affine $*$-algebra it is isomorphic to $\Gamma(V,\O_V)$ for $V\subs k^n$ an affine algebraic subvariety, so we may assume that $A=\Gamma(V,\O_V)$. By Corollary \ref{CARREGMAP3} and Proposition \ref{BIJMAX}, the above map $\phi_V:P\in V\mapsto \m_P=\{g\in \Gamma(V,\O_V) : g(P)=0\}\in Spm(A)$ is a homeomorphism, and we want to show that $\phi_V^\#$ is an isomorphism. If $U\subs X$ is open and $s\in \O_X(U)$ is represented on the open neighbourhood $U_{\m_P}\subs U$ of $\m_P$ by $[u_{\m_P}]/[v_{\m_P}]$ as before for each $P\in \phi_V^{-1} U$, suppose $(\phi_V^\#)_U(s)\equiv 0$ : for every $P\in U$ we have $u_{\m_P}(P)/v_{\m_P}(P)=0$, so $u_{\m_P}(P)=0$ and $u_{\m_P}\in \m_P$, and therefore $s\equiv 0$ by its local characterisation on each $U_{\m_P}$, so $(\phi_V^\#)_U$ is injective. As for surjectivity, let $g\in \O_V(\phi^{-1}U)$ : for each $P\in \phi^{-1}(U)$ there exists an open $O_P\subs \phi^{-1}(U)$ and $u_{\m_P},v_{\m_P}\in k[V]$, such that $P\in O_P$ and $g|_{O_P}\equiv u_{\m_P}/v_{\m_P}$ (in the sense of regular functions). Applying $\phi$, $U_{\m_P}:=\phi(O_P)\subs U$ is an open neighbourhood of $\m_P$, $u_{\m_P}$ and $v_{\m_P}$ define regular functions on $V$ (we keep the same notation), and $v_{\m_P}(Q)\neq 0$ for all $Q\in O_P$, i.e. $v_{\m_P}\notin \m_Q$. Now let $s:\m\in U\mapsto [u_{\m_P}]/[v_{\m_P}]\in A/\m$, for the unique $P\in V$ such that $\m=\m_P$; as before, if $Q\in O_P$, as $g|_{O_P}\equiv u_{\m_P}/v_{\m_P}$ and $g|_{O_Q}\equiv u_{\m_Q}/v_{\m_Q}$ we have $u_{\m_P}(Q)/v_{\m_P}(Q)=u_{\m_Q}(Q)/v_{\m_Q}(Q)\in k$. This means that $u_{\m_P}v_{\m_Q}-u_{\m_Q}v_{\m_P}\in \m_Q$, so $[u_{\m_P}]/[v_{\m_P}]=[u_{\m_Q}]/[v_{\m_Q}]$ in $A/\m_Q$, therefore $s$ has a constant description on $U_{\m_P}$ for every $P\in U$, i.e. $s\in \O_X(U)$; now obviously we have $(\phi_V^\#)_U(s)=g$, so $(\phi_V^\#)_U$ is surjective, $\phi_V^\#$ is an isomorphism, and $(X,\O_X)$ is an affine equivariety.
\end{proof}
\noindent As for functoriality, if $f:A\to B$ is a $k$-morphism of $*$-algebras of finite $*$-type, we have a natural continuous map $\phi=f^a:Y:=Spm(B)\to X:=Spm(A)$, which sends a maximal ideal $\n$ of $B$ to $\m:=f^{-1}(\n)$ : $\m$ is a maximal ideal, which appears in the sequence of embeddings $k\into A/\m \into B/\n\cong k$ over $k$; write $f/\n:A/\m\cong B/\n$ the induced equiresidual isomorphism, we may now define a morphism of sheaves $(f^a)^\#:\O_X\to \phi_*\O_Y$ as follows : if $U\subs X$ is open, we let $(f^a)^\#_U(g\in \O_X(U)):=[\n\in (f^a)^{-1}(U)\mapsto (f/\n)(g((f^a)(\n))]\in \O_Y((f^a)^{-1}U)=(f^a)_*\O_Y(U)$. One checks that this is well defined and that for $\n\in (f^a)^{-1} U$, $(f^a)^\#_U(g)$ is represented on $(f^a)^{-1}U_{f^a(\n)}$ by $[f(u_{f^a(\n)})]/[f(v_{f^a(\n)})]$ if $g$ is represented on $U_{f^a(\n)}$ by $[u_{f^a(\n)}]/[v_{f^a(\n)}]$. In particular,by Proposition \ref{SPMAFF} we have a functor $K:f\mapsto (f^a,(f^a)^\#)$ from the category $*Aff_k$ of affine $*$-algebras over $k$, into the dual category $(EVar^a_k)^o$ of affine algebraic equivarieties over $k$.
\subsection*{The affine adjoint duality}
\noindent The last matter of business is to show that $J$ and $K$ are mutual quasi-inverses, i.e. to define natural isomorphisms $Id\cong J\circ K$ and $Id\cong K\circ J$. The proof of the following proposition is tedious but straightforward.
\begin{prop}\label{FUNCPHI}
The morphisms $(\phi_V,(\phi_V)^\#):(V,\O_V)\to (KJ(V),\O_{KJ(V)})$, for $V\in EVar^a_k$, define a natural transformation $\phi$ from $Id$ to $KJ$.
\end{prop}
\noindent Now if $V$ is any affine algebraic equivariety, there exists an isomorphism between $V$ and an affine algebraic subvariety $W$ of a $k^n$ say; we have seen in the proof of Proposition \ref{SPMAFF} that $\phi_W$ is a homeomorphism, so $\phi_V$ is also a homeomorphism by Proposition \ref{FUNCPHI}. We are now able to prove the
\begin{prop}\label{DUAL2}
The pair $(\phi_V,(\phi_V)^\#):(V,\O_V)\to(X,\O_X)$ is an isomorphism, and thus it defines a natural isomorphism $\phi:Id\cong K\circ J$ of endofunctors of $EVar^a_k$.
\end{prop}
\begin{proof}
Suppose that $U\subs X$ is open, $s\in \O_X(U)$, and $t=(\phi_V)^\#_U(s)\equiv 0$. If $\m\in U$, there exists a unique $P\in \phi_V^{-1}U$ such that $\m=\m_P=\phi(P)$, and by definition we have $t|_{U_P}=(u_P|_{\phi_V^{-1}U_P})/(v_P|_{\phi_V^{-1} U_P})$ for a representation $u_P/v_P$ of $s$ on $U_P$ say. As $t=0$ by hypothesis, we have $t|_{\phi_V^{-1} U_P}=0$, so $u_P|_{\phi_V^{-1} U_P}=0$, and in the residual field $\ov{\O_{V,P}}$ we get $\ov{[u_P|_{\phi_V^{-1} U_P}]}=0$, whence $[u_P]=i_P^{-1}(\ov{[u_P|_{\phi_V^{-1} U_P}]})=0$ in $J(V)/\m_P$ and finally $s(p)=[u_P]/[v_P]=0$. It follows that $s=0$, so $(\phi_V)^\#_U$ is injective and $(\phi_V)^\#$ is a monomorphism. As for surjectivity, we first suppose that $V$ is an affine algebraic subvariety of $k^n$ say and let $t\in \O_V(\phi_V^{-1}U)$ : we have $t:\phi_V^{-1} U\to k$ and for each $P\in O:=\phi_V^{-1}U$ there exists $O_P\subs O$ and $a_P,b_P\in k[V]$ such that $P\in O_P$ and $t|_{O_P}=\wt{a_P}|_{O_P}/\wt{b_P}|_{O_P}$, where $\wt {a_P}\in J(V)$ is the global section defined by $a_P$. Let $s:U\to \coprod_{\m\in U} J(V)/\m$, $\m=\m_P\mapsto [\wt{a_P}]/[\wt{b_P}]$; if $\n\in U_P=\phi_V(O_P)$, we have $\n=\m_Q$ for $Q\in O_P$, so $t(Q)=a_P(Q)/b_P(Q)$ but also $t(Q)=a_Q(Q)/b_Q(Q)$, whence $a_Pb_Q-a_Qb_P\in Ker(e_Q:k[V]\to k)$, and therefore $\wt{a_P}\wt{b_Q}-\wt{a_Q}\wt{b_P}\in \m_Q\subs J(V)$, so $s(\n)=s(\m_Q)=[\wt{a_Q}]/[\wt{b_Q}]=[\wt{a_P}]/[\wt{b_P}]$, and finally $s\in \O_X(U)$, as it has a locally constant description. Let now $P\in O$ : if we note $u:=(\phi_V)^\#_U(s)$, by definition we have $u|_{O_P}=\wt{a_P}|_{O_P}/\wt{b_P}|_{O_P}$, which is exactly $t|_{O_P}$; by characterisation of a sheaf, we have $t=u$, so $(\phi_V)^\#_U$ is surjective, and hence $(\phi_V,(\phi_V)^\#)$ is an isomorphism. In the general case, if $V$ is an affine equivariety there exists an affine subvariety $W$ and an isomorphism $\psi=(\psi,\psi^\#):W\to V$, and applying what precedes and the functoriality of Proposition \ref{FUNCPHI}, the following diagram
$$\begin{CD}
V @>\phi_V >> KJ(V)\\
@A\psi AA @AAKJ(\psi) A\\
W @>>\phi_W > KJ(W)\end{CD}$$
commutes and $\phi_V=KJ(\psi)\circ \phi_W\circ \psi^{-1}$ is an isomorphism, so $\phi:Id\cong K\circ J$ and the proof is complete.
\end{proof}
\noindent The second transformation is easier to describe. Let $A$ be an affine $*$-algebra and $f_A:A\to JK(A)$ be defined by $f_A(a\in A):=[\m\in KA=Spm(A)\mapsto [a]\in A/\m]$; it is obvious that $f_A$ is a morphism of $k$-algebras.
\begin{prop}
The $k$-algebra morphisms $f_A:A\to JK(A)=\Gamma(Spm(A),\O_{Spm(A)})$, for $A\in *Aff^a_k$, define a natural transformation $f:Id\to J\circ K$.
\end{prop}
\begin{proof}
Let $\phi:A\to B$ be a morphism of affine $*$-algebras over $k$ and $(X,\O_X):=K(A)$, $(Y,\O_Y):=K(B)$ : the morphism $K\phi:Y\to X$ is defined by $K\phi:\n\mapsto \phi^{-1}\n$ and the sheaf morphism $(K\phi)^\#:\O_X\to (K\phi)_*\O_Y$, for every open $U\subs X$, by $(K\phi)^\#_U(s\in \O_X(U)):\n\in (K\phi)^{-1}U\mapsto (\phi/\n)\circ s(\phi^{-1}\n)$, for $\phi/\n:A/\phi^{-1}\n\to B/\n$ the quotient morphism; the morphism $JK(\phi):JK(A)=\Gamma(X,\O_X)\to JK(B)=\Gamma(Y,\O_Y)$ is then simply $(K\phi)^\#_X : s\mapsto [\n\in Y\mapsto \phi/\n\circ s (\phi^{-1}\n)]$. It follows that for each $a\in A$, we have $JK\phi\circ f_A(a)=JK\phi([\m\in X\mapsto [a]\in A/\m])=[\n\in Y\mapsto \phi/\n([a]\in A/\phi^{-1}\n)]$. By definition of $\phi/\n$, it is the section $\n\in Y\mapsto [\phi(a)]\in B/\n$, which is exactly $f_B\circ \phi(a)$, so $f$ is a natural transformation.
\end{proof}
\begin{lem}\label{AFFJAC}
If $A$ is an affine $*$-algebra, then the Jacobson radical of $A$ is $(0)$.
\end{lem}
\begin{proof}
It suffices to prove it for $A=k[V]_M\cong\Gamma(V,\O_V)$, for $V\subs k^n$ an affine algebraic subvariety. Suppose that $f/g$ is in the Jacobson radical of $k[V]_M$ : in particular, for every $P\in V$ we have $f/g\in \m_P k[V]_M$, where $\m_P$ is the maximal ideal of $P$ in $k[V]$. In other words, for every $P\in V$ we have $f(P)=0$, and thus $f=0$ in $k[V]$. In particular, $f/g=0$ and the Jacobson radical of $k[V]_M$ is $(0)$.
\end{proof}
\begin{prop}\label{DUAL1}
For every affine $*$-algebra $A$, the $k$-algebra morphism $f_A:A\to KJ(A)$ is an isomorphism. In particular, the maps $f_A$ define a natural isomorphism $Id\cong J\circ K$ of endofunctors of $*Aff_k$.
\end{prop}
\begin{proof}
First, suppose that $A=\Gamma(V,\O_V)$ for an affine algebraic equivariety $(V,\O_V)$, $X=Spm(A)$ and $f_A:A\to JK(A)=\Gamma(X,\O_X)$. Let $a\in A$ be such that $f_A(a)=0$ : for each $P\in V$, we have $[a]=0$ in $A/\m_P$, so $a\in \m_P$; it follows that $a$ is in the Jacobson radical of $A$, which is zero by Lemma \ref{AFFJAC}, so $a=0$, and $f_A$ is injective. As for surjectivity, let $\phi_V:V\to X$ be the homeomorphism $P\mapsto \m_P$, $s\in\Gamma(X,\O_X)$ and $t=(\phi_V)^\#_X(s)$ : by definition, for every $P\in V$ there exists an open $O_P\subs V$ and $a_P,b_P \in A$ such that $t|_{O_P}\equiv a_P|_{O_P}/b_P|_{O_P}$ and $s|_{\phi(O_P)}\equiv [a_P]/[b_P]$. We have $f_A(t):\m\in X\mapsto [t]\in A/\m$, so let $\m\in X$ : as $\phi_V$ is a homeomorphism, write $\m=\m_P$ for a unique $P\in V$, and via the isomorphism $i_P:J(V)/\m_P\cong \ov{\O_{V,P}}$ we may write $i_P([t])=\ov{[t]_P}=\ov{[t|_{O_P}]_P}=\ov{[a_P|_{O_P}/b_P|_{O_P}]_P}=\ov{[a_P]_P}/\ov{[b_P]_P}=i_P(s(\m_P))$, whence $[t]=s(\m_P)$ in $A/\m_P=A/\m$, and as this is true for every $\m\in X$, finally we have $f_A(t)=s$, and $f_A$ is surjective : it is an isomorphism. In the general case, any affine $*$-algebra is by definition isomorphic to an algebra of the form $\Gamma(V,\O_V)$, so as in the end of the proof of Proposition \ref{DUAL2}, $f_A$ is an isomorphism as well.
\end{proof}
\noindent Assembling Propositions \ref{DUAL2} and \ref{DUAL1} we get the duality theorem :
\begin{thm}\label{DUALAFF}
The global sections functor $J:(EVar^a_k)^o\to *Aff_k$ is a duality between the categories $EVar^a_k$ and $*Aff_k$, with adjoint the maximal spectrum functor $K$.
\end{thm}
\section*{Conclusions}
So far we have a built a robust theory of (equiresidual) affine algebraic varieties over any field and a promising extension of the usual commutative algebra underlying affine algebraic geometry over algebraically closed fields. We have laid the groundwork for a theory of algebraic equivarieties, which we will develop in a forthcoming publication (\cite{EQAG2}), and in which we will expound the important particular case of quasi-projective equivarieties.
From this point on, several directions may be pursued. First it is desirable and possible to pursue this subject further and investigate some usual constructions and theorems from classical algebraic geometry in the present setting. For a start, we hope to tackle shortly the study of simple points and tangent spaces, of étale morphisms of algebraic equivarieties, and of differential forms, thanks to the formalism of canonical localisations and $*$-algebras. Further explorations may concern normal varieties and other subjects pertaining to the classical theory, which will have to be reinterpreted in the equiresidual approach.
Another obvious series of questions lies in the potential application of the present general theory to the study of the \og inner" algebraic geometry of any particular field, using the tools and concepts presented here. A good start would be to sketch some general features of algebraic geometry over the field $\QQ$ of rational numbers without working in its algebraic closure.
In this perspective, normic forms have played a fundamental role in the present work, but were only used as a "tool" for the Äquinullstellensatz and the Equiradical, whereas in general homogeneous polynomials with only the trivial zero may serve to characterise the special ideals (Remark \ref{HOMSIGN}). We plan to explore deeper this topic, hopefully connecting through Galois theory the present approach to algebraic geometry in an algebraic closure or a separable closure of the ground field. The example of $\QQ$ would again be a good landmark. From another point of view, Proposition \ref{BIJMAX} shows that the maximal spectrum is well behaved with respect to all $*$-algebras of finite $*$-type, and not only with respect to affine, i.e. special, ones, which are reduced. Along this line of thought, after some background on algebraic equivarieties we would naturally be led to an equiresidual version of the algebraic spaces, which would permit the use of infinitesimals in a mild formalism avoiding for the moment the need of an analogue or generalisation of scheme theory.\\
It is also possible to give back to first order logic what we borrowed and expressed here in the form of pure commutative algebra. We will consider this in \cite{PAG1}, which deals with \og positive algebraic geometry, an interplay between the present (equi)algebraic geometry, positive logic and quasivarieties (a \og subdoctrine" of first order logic), laying the foundation for algebraic geometry in fields considered in the light of model theory.
With some background on étale morphisms of affine equivarieties, we will hopefully build on this foundation in order to study a ubiquitary type of theories of fields which appear in connexion to number theory (real closed fields, p-adically closed fields, complete theories of pseudo-algebraically closed fields,...), and which have been recognised by McKenna in \cite{MK} and systematised by Bélair in \cite{BEL} thanks to the work of Robinson (\cite{ER}), joining forces with the tradition of coherent logic and connecting with topos theory. The archetypical example of pseudo-algebraically closed fields will fall into this field of investigation and we hope that the present work and some elements of this program will be of some use to the theory of \og Field Arithmetics" (\cite{FA}), where one's particular interest lies in algebraic geometry over many fields which are not algebraically closed.
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Home»Haircuts»Unique Try On A Haircut Pics Of Haircuts Style Unique Try On A Haircut Pics Of Haircuts Style By Gill Maureen On 8 December 2018 In Haircuts | No Comments Try on a new haircut or color with this app — InsiderBeautyBuzz from try on a haircut, source:insiderbeautybuzz.com Other Collections of Unique Try On A Haircut Pics Of Haircuts Style Tags:try a haircut on my picture, try a haircut on your photo free, try an anime haircut, try different haircuts online, try haircut in your face, try haircut male, try haircut photo, try haircut styles free, try on a haircut, try on a haircut app, try on a haircut free, try on haircut styles, try on haircuts for guys, try on haircuts online free, try on virtual haircut, try on wigs before haircut, try out a pixie haircut, try out haircut app, try your haircut, try your haircut online Fantasia Hairstyles Image Of Braided Hairstyles Style NEXT »Top Hairstyles for Short Black Natural Hair Pics Of Braided Hairstyles Style Next post link Leave a Reply Cancel reply Your email address will not be published. Required fields are marked *Comment Name * Email * Website
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Pamela Bethea
Leon High Tallahassee, FL Class of 1967
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| 236,708
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Artykuł
The paper presents analysis of the relationship between the way of pouring AlSi7Mg alloy into
moulds and misrun formation in the castings made with lost-wax casting method, with the use of
gypsum mass. The casting moulds have been filled with liquid metal with three various methods
(using gravity force, centrifugal force, and vacuum). Estimation of the quality of the obtained
castings (i.e. the degree of mould filling) allowed to assess the effectiveness of particular methods
of filling the moulds used for production of small-module castings. In result, it was found that the
highest mould filling degree is achieved with the use of vacuum technology.
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Hey Booklovers!
Happy Halloween! This is my final review for October 2018. Wow. What an year its been and I cant stop smiling thinking about all the good books I got to read this year including this one – The Perfect Family by Shalini Boland. Frankly am surprised theres no noise been made over this book yet. If there is one book I seriously recommend reading – its this one. Make no mistake, in one way or another I would love for you to read the books I have reviewed on my site but rarely would I say – READ THIS. So far I have felt that only for three other books this year, I guess.
I decided to write a review right after I finished, which I usually don’t, in order to get over the fatigue of reading for so long. I like taking my time to read, savour the books especially if I like it. But this one was unstoppable. I started it on Monday and finished it last night. Thankfully it was a fast paced and the mystery gets deeper and deeper. Ok before I go on heres the gist of it:
Gemma Ballantine is a working mum with two adorable daughters, six and ten. She has a loving and devoted husband, Robert, but her work life is driving her bonkers and her husband doesn’t make much. No. She is the main breadwinner of the family. Naturally, her family life takes a backseat. So when she is coaxed to hiring a nanny cum tutor she feels everything is going to be perfect. The blurb is what got my attention – her younger daughter Katie goes missing in the beginning of the book and is soon found but things take a U-turn therein. I was curious as to what could be worser than that?
Soon things begin to go awry, missing keys, missing memory, horrible experiences and all this culminates to hiring a nanny that turns out to be life altering. What started great begins to feel like a mistake and soon Gemma finds herself in a devastating situation, but is it all real? – for a lack of better word. For sometime I really wondered if Gemma is crazy but later on I seriously wanted to punch the person behind all this – which brings me to the best part. You would never guess who – not from miles away. Thats what I loved about this book.
The thrillers I had read this year, most of them were mildly predictable. Maybe cause I hadn’t expected the obvious here but it was a pleasant surprise. The twist makes it even more delicious. I seriously love a book that leaves me conflicted yet satisfied. Shalini Boland has delivered a master stroke and I definitely want to check her other books.
Thank you to NetGalley and Bookouture for giving me this reader copy!
Publication date: 6th November, 2018.
Bibliogyan rating -> Mind = blown👍
4 thoughts on “The Perfect Family by Shalini Boland”
Great review, Vandana💝
LikeLiked by 1 person
Thank you shalini! 😃
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TITLE: Prove weighted ball in $l^2$ space is compact
QUESTION [4 upvotes]: Let $B \subseteq l^2$, $B=\left\{x\in l^2:\sum_{n\geq1}n|x_n|^2\leq1\right\}$, show that $B$ is compact.
My thought: $B$ is closed in $l^2$ which is complete. Then $B$ is complete. It suffices to show $B$ is totally bounded. I think we need to first get rid of the infinite tail sum, i.e., bound all sequences with balls centered at sequences that only have finitely many terms. And then find a ball cover for the finite sequences. But I don't know how to bound the tail sum and I'm stuck.
REPLY [2 votes]: Theorem. Let $p\in[1,+\infty]$ and $M\subseteq\ell_p$ be a bounded subset such that
$$
\lim\limits_{N\to\infty}\sup\{\Vert (0,0,\ldots,0,x_N,x_{N+1},\ldots) \Vert_p:x\in M\}=0
\tag{1}$$
then $M$ is totally bounded.
For the proof, see an answer of that question: How to show that this set is compact in $\ell^2$.
The theorem provides a characterization of totally bounded subsets of $\ell_p$.
Let’s apply it to $B$.
For $x\in B$, we have
$$\Vert x\Vert_2 = \sum_{n\ge1} \vert x_n \vert^2 \le \sum_{n\ge1} n\vert x_n \vert^2 \le 1$$ proving that $B$ in included in the closed ball centered on the origin with radius equal to one. Hence $B$ is bounded.
We’ll be done if we prove that $B$ satisfies condition $(1)$ of theorem above.
For $x \in B$
$$N\sum_{k\ge N} \vert x_k\vert^2 \le \sum_{k\ge N} k\vert x_k\vert^2 \le \sum_{k\ge 1} k\vert x_k\vert^2 \le 1$$
Therefore
$$\sup\{\Vert (0,0,\ldots,0,x_N,x_{N+1},\ldots) \Vert_2:x\in B\} \le 1/N$$ and condition $(1)$ is satisfied.
$B$ is totally bounded. And also complete as being a closed subset of a complete space.
Finally $B$ is compact.
| 3,645
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!
- Amika Un.Done Texture Spray || They sent the full size (5.3 oz) but it also comes in a mini size for $10 which would be perfect for travel. At first glance I thought this was a dry shampoo, but after a few sprays I was happy to find out it was actually a texture spray. Something that I don't use regularly - actually the only time I've used texture spray on my hair is when it's being done at the salon. The smell is amazing!! Both Amika products in the box smelled so good, I would purchase for that alone! It works really well at adding a little, you guessed it, texture to your hair both after a fresh blow dry & on second day hair! It didn't feel sticky or tacky like hairsprays can and it gives you the perfect amount of extra volume and texture. I used this past weekend with freshly dried hair & then also you can see the difference with slight waves in my hair to add more body. Verdict: Love the smell and would purchase the mini size for travel & on the random occasions I would use texture spray.
- Amika Nourishing Hair Mask || I've really gotten into hair masks especially since adding balayage to my hair - I want to make sure that it stays healthy - especially the ends! I've used it as both a conditioner and as a hair mask - which the directions say you can do. When using it as a mask I definitely see more of a difference, but you don't have to leave it on for more than 5 minutes. This smells amazing and even the next day, the smell stays with your hair! I saw the biggest difference when I had to blow dry my hair right after using the mask - it was incredibly smooth and silky, which is all I ever really want from my hair! It does come in a 2 oz. mini size, which is the one that I received and again is $10! With only using it one to two times a week and a little dime size amount going a long way, I'll be using this for a while! Verdict: Will definitely purchase the mini size because I don't need to use it that often.
- Grande Lips Hydrating Lip Plumper || Every lip plumper I have ever tried stung my lips so badly that I ended up having to remove it before seeing any results. The finish I received was just the "high clear gloss" but it does come in 13 different shades. There is a slight tingle about a minute after applying, but it's not painful & after that time you don't really feel anything else. I had a really hard time seeing a difference in my lips getting plumped up after using it and I even applied more than what I would qualify to be a normal amount. You're supposed to apply twice a day for 30 days and it will increase the softness and even hydration of your lips - can't say that I've seen anything other than my lips feeling softer. I really liked the sleek gold packaging & the applicator tip as well as the click applicator at the bottom. Verdict: I'm not a huge fan of lip plumpers so I will not purchase.
- Dr. Brandt No More Baggage De-Puffing Eye Gel || I've tried other Dr. Brandt products before, but never in a full size version. This is expensive and is supposed to take the appearance of bags under your eyes away. The catch, it's pretty difficult to apply & if not blended correctly you really can't wear it with your concealer, but using it on it's own does not eliminate the bags. The work around I found was applying an eye moisturizer on top of the eye gel after it had dried for a minute or so. I knew something was working under my eyes because I was using less and less concealer. The area where you apply the gel does get a bit tighter so to me it seemed like it was working. I'd be a little hesitant to say this product was the reason that my under eye area has started to appear firmer because I've been very regimented with my eye care routine recently, but I do think it may have helped a bit. Verdict: I've found that using eye gel patches works just as well and is a bit easier on the wallet so I wouldn't purchase this.
- Lisse Luxe Hair Towel by Aquis || If you tell me something is going to cut my drying time in half, I'm basically sold. If a product can save me time - I'm willing to test it out! What I didn't realize was that regular towels don't protect your hair against breakage, but the fibers that make up the towel don't grab at your hair because there isn't friction like you get with regular towels. More often than not I wash my hair at night, so I don't really care how long it takes my hair to dry. However, when I'm on a tight schedule I wrap my hair in the towel and go about doing my makeup and then when I'm ready to dry my hair I add in a little heat protector spray (which also helps to cut down on drying time) and I'm done in about 7 minutes vs. 15 before! Verdict: I'm not sure I need more than one of these towels in my beauty arsenal, but I know that they would make a really great holiday gift - so that's going on the short list of things to by for the ladies in my family!
I am in need of a good hair mask. Might be fun to try the mini for sure! Not a fan of lip plumpers either but the packaging is so cute!! Happy Wednesday gorgeous!!
That towel sounds awesome - drying my hair is one of my least favorite things to do when getting ready, so anything to lessen that time would be great!
That hair towel sounds really interesting! My hair is really thick and holds water really well so it usually takes me a while to dry my hair--even with it being short. When you wash your hair at night do you sleep with it wet?
Sarah at MeetTheShaneyfelts
I love the turban! I use mine regularly and do think it makes a difference. I would also be interested in the texture spray, my hair is so fine it can use a little texture!
Brooke
pumps and push-ups
These products sound great!! I love the Amika hair brand. Will have to try out the texture spray!
Katherine |
Such neat products! I'd love to try the texture spray!
All of these products sound great, but I will definitely be looking into that hair mask! I am always looking for ways to protect my hair and add moisture, and that one sounds perfect!
xo, Whitney and Blaire
Peaches In A Pod
Love these reviews. I've not done a hair mask in ages!! TIme to remedy that
Aside from Dr. Brandt I haven't heard of most of these brands before! I am always on the hunt for a good hair mask so I might check this one out!! I totally feel ya on not being able to use up all the sample-sized products--collectively they end up taking up so much room!
Rosy Outlook
I've tried that mask before and it smells SO good! I really feel like plumpers in general just don't work. I've tried a few and never really seen any noticeable results. Especially not for the price! <3, Pamela Sequins & Sea Breezes
Hmmm, Ineed to try hair masks again. With shorter hair I've let them kind of go by the wayside. Not a fan of plumpers either, love a good eye patch though.
I have heard great things about Amika products and it's good to know the smell is worth it (it matters way too much to me!) I just got one of those hair towels to try, hopefully it will help with the post-baby breakage at least!
oh what a fun box! the only one i still get nowadays is the sephora box. this one looks awesome. that texturizing spray looks fun. i have the oribe one and i love it but this would be a better price point. i have a hair towel i got in a box and i think i'm insane bc i am not sure it does anything super fantastic ha.
xoxo cheshire kat
What a fun box to have!!
Thank you for the review, I will definitely have to try some of these products, especially the hair mask!
xx, Elise
Love that this box has the full size of the items to try out, because I cancelled the Sephora Play box for the same reason. I definitely need to try that hair mask! <3
Green Fashionista
I love a good texture spray- they're great! I have yet to find a lip plumper that I enjoy- such a bummer!
As much as I love getting good sized samples of products, once you get a lot they really do get overwhelming and you don't get a chance to try everything. The hair mask sounds really nice and I'm definitely intrigued by that towel.
Oh that hair mask sounds interesting. I love my play box subscription! Although, I do get overwhelmed with all the samples. When I find myself with so many, I just throw a few into packages when I sell an item on Poshmark. It's like a little free gift with a purchase. haha
I am obsessed with that hair towel! It really works wonders. I totally agree the smell of those Amika products are incredible and I love that hair mask!
lip plumpers freak me out - I mean, should things really make a stinging feeling & make things engorge? That doesnt seem healthy
That hair mask sounds fabulous, I need to look in to it. I cancelled my beauty box last year when I got pregnant because I was getting a lot of products I couldn't use at the time, and now that I can use them again...I haven't really missed getting the boxes really. Though, getting these Influenster boxes is really nice!!! I always get surveys for them but rarely ever actually receive one. Boo! That hair towel looks fun!
Hmmm... drying towel is totally tempting!! Anything to save some time these days. I'll be purchasing!! XOXO, R
The only box I'm subscribed to now is Boxycharm, mainly because you get full size products and they are always high-end, sometimes even luxury brands, for only $21! And if I don't use them, they go into the future giveaway pile. LOL!
I love Amika products and have a few of their things in mini's because I don't use a ton of hair products.
xo, Lily
Beauty With Lily
Those beauty boxes are fun when you are in need of something new and different!
That's great to know about this Amika texture spray. I have been looking for a spray to use on occasion and I'm so glad to hear that you loved the smell! Will definitely have to check this out. Great review, Biana!
xo, Rachel
A Blonde's Moment
I've never heard of this beauty box before. I sub to BoxyCharm and LOVE it! Thanks for sharing this with us!
I've heard great things about that Amika Texture Spray. It's been sitting in my Amazon cart forever so I probably just need to try it!
I cut my subscription boxes for the same reason, but every once in a while it's nice to get something like this! Thanks for the info.
Such a fun box of treats. I only use texture spray at the salon and this one sounds like it might need to be purchased. I have one of those towels (but not that brand) and never use it, I need to try it. Good to know about the eye gel packs.
I got this same box and I have the same opinions as you do! My favorite so far is the towel and the hair mask!
The pattern on the bottle of that texture spray - I love it, things like that can sell me! I need to look into that towel - anything to cut down on time!
Good to know about the hair towels because I 100% of the time wrap my hair up before blow drying. Sounds like you got some amazing products to test in this VoxBox. I've never tried any of Amika's products but the travel sizes sound perfect!
Such a good review girl! I love hair mask and may need to get the travel size one to test out!
Love the honest review! So many goodies! I kind of want to try the lip plumping gloss!
<3 Shannon
Upbeat Soles
I only have the Sephora Box subscription, but I definitely find myself not being able to use all of the products that it comes with. This Luxe Box sounds like a great deal, especially that texturing spray!
Ohh looks like you got so many great goodies! Love a good subscription box!
~Samantha
I always love a box of goodies!!
Doesn't the Amika mask smell AMAZING?! All of their products smell like this and I loooooove it!
-Ashley
Le Stylo Rouge
I feel like I need some de-puffy-eye things and a hair towel now... but that may be because I'm tired and winter is coming... so cutting down the hair drying time sounds like a REALLY good idea!
This box was such a winner! I agree with all your reviews - I had wanted to try the Amika texture spray for a long time, so I was thrilled that we got a full size bottle.
Nice read dear. I really love how you do your reviews. Sounds really genuine and informative. Keep it up dear!
Jessica | notjessfashion.com
| 41,593
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Serves: 4
Prep: 10min
Cooking: 30min
This is a very typical Italian family dish when the aubergines are in season, perfectly accompanied by a mozzarella salad starter. Enjoyed both by children and adults because of its fragrant smell and unique taste. Prepare it for your family for a Sunday meal: I know they'll love it too!
Ingredients
- 400g MAMMA FLORA Mezzi Paccheri
- 300 gr of cherry tomatoes or 2 jars of MAMMA FLORA handmade tomato sauce
- 1 tbsp of extra virgin olive oil
- 1 aubergine
- 1 1/2 cup of frying oil
- 1 garlic clove
- basil
- salt
Tools:
- 1 Pot
- 2 Pans
- Kitchen paper
Method:
- Let’s begin by washing the aubergines and drying them. Cut off the tops and bottoms and quarter the eggplant, trim off the seedy centre and dice the eggplant
- Pour the frying oil in a big pan on medium from absorbing the oil
- Add some aubergines in the pan, not too many, if the pan is not big enough for all, then repeat the process multiple times until all aubergines are fried. Fry until golden brown, it should take around 5-7min per time
- Prepare a plate with kitchen paper to arrange the aubergines once ready and to absorb the extra oil
- In the meantime set a pot filled with water on the stove, and bring to boil, salt the water once the water is boiling and cook the MAMMA FLORA Mezzi paccheri for the time provided on the pack or till al dente.
- Cut the cherry tomatoes in halves. In a pan add pour the extra virgin olive oil with a garlic clove and gently heat, once the oil is warm then add the cherry tomatoes, add salt as per taste and cover for 7min until the tomatoes are slightly cooked. If you are using MAMMA FLORA tomato sauce you can skip this step.
- Once all the aubergines are fried, add them in the pan with the cherry tomatoes and stir together for a couple of minutes, add some basil leaves. Or otherwise add the MAMMA FLORA tomato sauce and stir.
- Once the pasta is ready, drain it and add it to the aubergine sauce. Finish the cooking for another couple of minutes to let the pasta absorb all the sauce.
- Add some parmesan and enjoy.
| 261,693
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| 414,350
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Not All Visitor Information Centers Are Alike May 10, 2011Posted by John Stephens in : News & Events , add a comment
Visitor centers in towns accustomed to hosting a large number of tourists can be a common scene – especially in towns and cities with a large timeshare presence. However, as many frequent travelers have discovered, not all visitor information centers are alike.
Some of these centers are really nothing more that cleverly disguised, off-premise sales offices for timeshare resorts looking to sell timeshare to visitors by bringing them in to tour their facilities.
Such is the case in Charleston, South Carolina, where a new city council ordinance has been written to try and differentiate the city-run centers from the ones that act as lead-generation facilities for timeshares.
While the Charleston City Council unanimously agreed on how to deal with timeshare solicitors in downtown, it’s not a done deal yet.
Mayor Joe Riley says visitors complain about overly persistent timeshare operators and that it’s a threat to a city known for its southern hospitality.
He says local businesses are important to the city’s economy, but misleading practices are not.
As expected, a marketing representative for a downtown timeshare says they’re trying to help the city and visitors. No surprise that they would take the “we’re all about servicing the guest” tact. However, anyone who has encountered these “information centers” knows that they can be a grueling experience, especially if all you’re looking for is a map and a point in the right direction.
With the new regulation, signs will have to label such facilities as not affiliated with the city of Charleston.
The city council hopes there will be a final reading of this ordinance sometime this week, and then timeshare operators will have to abide by the regulation as law. The mayor says that can happen as early as this summer.
Of course, people purposely looking for South Carolina timeshare or timeshare in any other part of the world can always come to our BuyaTimeshare.com website and find the right deal without all the hassle of dealing with the resort sales process. Not to mention the overly inflated prices that they charge at the resorts. Take a look by clicking here.
| 377,268
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TITLE: A question concerning the representation of an Abel group..
QUESTION [0 upvotes]: Let $K$ be an algebraically closed field. Let $A$ be an Abel group and $L$ be an irreducible $KA$-module...
I read in a paper that "Since $K$ is algebraically closed and $A$ is abelian, we have $dim_KL=1$"...
I know that the dimension of the representation of an Abel group is 1.
My question is that why the dimension of the irreducible $KA$-module $L$ depends on the condition that $K$ is a algebraically closed field.
Thanks to every one!!
REPLY [1 votes]: When $K$ is algebraically closed and $A$ acts on a vector space $V$ by linear operators, one can find a simultaneous eigenvector for the elements of $A$. This, of course, depends on the characteristic polynomials for the elements of $A$ to split (which is where algebraically closed comes in).
To see what fails outside of the algebraically closed setting it is useful to look at a concrete example. Let $C_3=\langle c\mid c^3=1\rangle$ be the cyclic group of order 3. Then, $\mathbb{Q}C_3$ acts on $V=\mathbb{Q}C_3$ by left multiplication (the regular representation). There is a 1-dimensional submodule $X=span\{1+c+c^2\}$. Let $\alpha=1-c$ and $\beta=c-c^2$. Then, the 2-dimensional compliment $Y=span\{\alpha,\beta\}$ of $X$ is also $\mathbb{Q}C_3$-stable since
$$c.\alpha=\beta\;\;\;\mbox{and}\;\;\;c.\beta=-(\alpha+\beta).$$
Assume $Y$ is not irreducible, and argue for a contradiction. By assumption, $Y$ contains a 1-dimensional $\mathbb{Q}C_3$-stable subspace $$Z=span\{a\alpha+b\beta\}=\{\lambda(a\alpha+b\beta)\mid\lambda\in\mathbb{Q}\}.$$
Then, we must have $c.(a\alpha+b\beta)=\mu(a\alpha+b\beta)$ for some $\mu\in\mathbb{Q}$ with $\mu^3=1$ (since $c^3=1$). This forces $\mu=1$.
However, using the formulas for the action of $c$ on $\alpha$ and $\beta$ above, we have
$$c.(a\alpha+b\beta)=-b\alpha+(a-b)\beta.$$
This means that $a=-b$ and $a-b=b$. Hence $2b=-b$, which implies $b=0$ and $a=0$. This is the contradiction we want.
Note that the problem above is that the only possible eigenvalue for $c$ in $\mathbb{Q}$ is $1$. The characteristic polynomial for the action of $c$ on $V$ is $x^3-1$ which has roots $1, \omega:=e^{2\pi i/3}$, and $\omega^2=e^{4\pi i/3}$ in $\mathbb{C}$. If we extend scalars in the example above to $\mathbb{Q}(\omega)$, then $Y$ does contain two 1-dimensional submodules. One is spanned by $1+\omega c+\omega^2 c^2$ and the other is spanned by $1+\omega^2 c+ \omega c^2$. The point is: the characteristic polynomial splits in $\mathbb{Q}(\omega)$.
| 24,086
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Pumpkin pie's the name of the game when Bobby faces an award-winning baker.
A topping of crushed gingersnaps adds spicy crunch to this pumpkin mousse.
Ellie Krieger makes pumpkin flan for a lighter alternative to pumpkin pie.
Aida's Pecan Pumpkin Crunch is a crustless alternative to pumpkin pie.
Stuffed French toast and pumpkin spice doughnuts turn breakfast into art..
| 69,269
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