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e4bd14d888f03dfe292d6f94fd8b64c6bbd91d1e | subsection | 22 | 26 | Proof of Theorem | We now apply Lemma REF to find a
spanning oriented subgraph R^* of R which is a (robust) (\nu /24, 2\tau )-outexpander
and such that \delta ^0(R^*\cap H)\ge \eta k/16. Let H^*:=H\cap R^*.Our next aim is to modify the pure digraph G^{\prime } into a spanning oriented subgraph of G having minimum
semi-degree at least \et... | {
"cite_spans": []
} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
] | [
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5d3e4d873e2db44d7a71bbe5a4c62d25e2d7bdc3 | subsection | 23 | 26 | Proof of Theorem | Moreover,\delta ^0(G^*-V_0)\ge \frac{\eta m}{5} (1-\sqrt{\varepsilon })\delta ^0(H^*)-|V_0|
\ge \frac{\eta m}{5}\frac{\eta k}{17} -3 \sqrt{ \varepsilon } n \ge \frac{\eta ^2 n}{100}.We now modify G^* by altering the neighbours of the exceptional vertices:
For every x\in V_0 we select a set of \eta n/2 outneighbours of ... | {
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} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
] | [
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2cf19fe93b8d81dbf9bb61ae92efe0bf57a69550 | subsection | 24 | 26 | Hamilton cycles in regular tournaments | In this section we prove Conjecture REF for sufficiently large
regular tournaments. The following observation of Keevash and Sudakov will be useful for this.Proposition 17 Let 0<c<10^{-4} and let G be an oriented graph on n vertices such that \delta ^0 (G) \ge (1/2-c)n.
Then for any (not necessarily disjoint) S,T \sub... | {
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32d7eecb031513ceeac0e9cded18560989d70cd6 | subsection | 25 | 26 | Hamilton cycles in regular tournaments | So \delta ^0 (G^{\prime }) \ge \eta n -2\ge \eta n/2 and G^{\prime } is a robust (\nu /3, 2 \tau )-outexpander.
Thus by Theorem REF G^{\prime } contains a Hamilton cycle which corresponds to one in G. \squareDaniela Kühn, Deryk Osthus & Andrew TreglownSchool of MathematicsUniversity of BirminghamEdgbastonBirmingham... | {
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} | 0807.1827 | Hamiltonian degree sequences in digraphs | [
"Daniela Kühn",
"Deryk Osthus",
"Andrew Treglown"
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8bf786fcbe69a6938548a656e54dbd3c3f49e607 | abstract | 0 | 16 | Abstract | We discuss several PT-symmetric deformations of superderivatives. Based on
these various possibilities, we propose new families of complex PT-symmetric
deformations of the supersymmetric Korteweg-de Vries equation. Some of these
new models are mere fermionic extensions of the former in the sense that they
are formulate... | {
"cite_spans": []
} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
"Andreas Fring"
] | [
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99281c8a4be6a621694570212db7d10dd24f1dcd | subsection | 1 | 16 | Introduction | \mathcal {PT}-symmetry, that is the invariance under a simultaneous parity
transformation \mathcal {P}:x\rightarrow -x and time reversal \mathcal {T}:t\rightarrow -t, is a very desirable property to have in a physical model
without dissipation. For a Hamiltonian system it can be exploited to
guarantee the reality of th... | {
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"arxiv_i... | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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36eb9b6321b95e73c96cab21ad78667ffa25bbb5 | subsection | 2 | 16 | Body | Let us first fix our notations and recall some known facts about the
sKdV-equation. There exist various fermionic extensions of the KdV-equation
in terms of superfields, which are either supersymmetric or break
this symmetry , and are therefore mere fermionic
extensions. We take as a starting point the former case and ... | {
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"raw": "P. Mathieu, Supersymmetric extension of the Korteweg-de Vries equation, J. Math. Phys. 29, 2499–2506 (1988).",
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... | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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087aeab8b7ae61ebcc20413de7c18d8c52a2e117 | subsection | 3 | 16 | Body | We observe that the equation (REF ) remains invariant under the
following anti-linear symmetry transformation\mathcal {PT}:t\rightarrow -t,x\rightarrow -x,i\rightarrow -i,\Phi \rightarrow i\Phi ,D\rightarrow -iD.As a result of these properties of the superfield and superderivative we
deduce that the component fields an... | {
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576be2c29bb8ed0912b05c190753fb1cc8940253 | subsection | 4 | 16 | Body | By
construction H_{\varepsilon } is \mathcal {PT}-symmetric, but in addition
it is also supersymmetic, which is most easily verified for the component
version ()\mathcal {SUSY}:H_{\varepsilon }\rightarrow H_{\varepsilon }+\eta \int dx\partial _{x}\left( \xi u^{2}+\frac{i^{\varepsilon -1}}{1+\varepsilon }u_{x}^{\varepsi... | {
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... | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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08ab99062e19116506ec611a47e3477bd05fd84a | subsection | 5 | 16 | Body | With definition (REF ) we may then compute the corresponding
flow as\Phi _{t} &=&\left\lbrace \Phi (\mu ),H\right\rbrace =D\frac{\delta H}{\delta \Phi }=D\left[ \frac{\delta \int d\mu \mathcal {H}}{\delta \Phi }\right] , \\
&=&D\frac{\partial \mathcal {H}}{\partial \Phi }+D^{2}\frac{\partial \mathcal {H}}{\partial (D\P... | {
"cite_spans": []
} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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afb46fd0a9f17c562af915f888b3b57e7b72ceda | subsection | 6 | 16 | Deformed (super) derivatives | In the spirit of the construction in , we will define some
new superderivatives, which respect the \mathcal {PT}-transformation
properties (REF ). For this purpose we recall how to employ an
ordinary deformed derivative \partial _{x,\varepsilon } acting on some
arbitrary \mathcal {PT}-invariant function f(x)\partial _{... | {
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f6ad3ed3f98f8971ba15f5eb6aeb4c0c077064b7 | subsection | 7 | 16 | Deformed (super) derivatives | \mathcal {PT}: \partial _{x}^{n}\rightarrow (-1)^{n}\partial _{x}^{n} and \mathcal {PT}: \partial _{x,\varepsilon }^{n}\rightarrow (-1)^{n}\partial _{x,\varepsilon }^{n}, which gives rise to the simple construction principle: In a defining
equation of a particular model replace \partial _{x}^{n} by \partial _{x,\vareps... | {
"cite_spans": []
} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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fd8c8f673988f90e3a725af7228d9c1c403ae9ae | subsection | 8 | 16 | PT-symmetric superderivatives of bosonic-fermionic type | As a first option we define higher deformed superderivatives asD_{\varepsilon }^{2} &:&=D_{\varepsilon }D_{\varepsilon }, \\
D_{\varepsilon }^{n} &:&=D^{n-2}D_{\varepsilon }^{2}~~~~~~~~~~~~~~\ \ \text{for~}n>2.Accordingly the action on the superfield \Phi (x,\theta ) is computed toD_{\varepsilon }\Phi &=&\theta \partia... | {
"cite_spans": []
} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
"Andreas Fring"
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3cb77a3f131fdda25519a1897f2ad63ceb0fc60d | subsection | 9 | 16 | PT-symmetric superderivatives of fermionic type | Alternatively we may define\hat{D}_{\varepsilon }^{n}:=D^{n-1}D_{\varepsilon }~~~~~~~~~~~~~~\ \ \text{for~}n>1.in which case the action on the superfield \Phi (x,\theta ) gives\hat{D}_{\varepsilon }\Phi &=&\theta \partial _{x,\varepsilon }\xi +u, \\
\hat{D}_{\varepsilon }^{2}\Phi &=&\theta u_{x}+\partial _{x,\varepsilo... | {
"cite_spans": []
} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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007f31b8f36a0f597abceb0662f8fd99e75f6bea | subsection | 10 | 16 | PT-symmetric superderivatives of bosonic type | It is clear from the above discussion that the most interesting definitions
will be those just involving deformations of derivatives acting on the
bosonic fields. We may achieve this by defining\tilde{D}_{\varepsilon }^{2} &:&=D_{\varepsilon }D, \\
\tilde{D}_{\varepsilon }^{n} &:&=D^{n-2}D_{\varepsilon }^{2}~~~~~~~~~~~... | {
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} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
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1a6e35940f454e1c228ff393f83ce632f0290e72 | subsection | 11 | 16 | PT-symmetric superderivatives of bosonic type | We have \mathcal {PT}: D^{2}\Phi \rightarrow -iD^{2}\Phi , D^{3}\Phi \rightarrow -D^{3}\Phi and
therefore we may consistently define\check{D}_{\varepsilon }^{n} &:&=D^{n}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \text{for~}n=1,2 \\
\check{D}_{\varepsilon }^{3}\Phi &:&=-i(iD^{3}\Phi )^{\varepsilon }... | {
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} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
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4b605bf009b624a8073a64051f73ef92b5a7c055 | subsection | 12 | 16 | Construction of new models | We can replace the superderivatives by their deformed versions in various
different terms and in addition we may introduce different deformation
parameters in the higher order derivatives. In order to explore some of
these possibilities, let us first rewrite equation (REF ) as\Phi _{t}=-D^{6}\Phi +6D\Phi D^{2}\Phi +\la... | {
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9b2c9e0f4762071abf9e3c3d98b187b378bec8e5 | subsection | 13 | 16 | Construction of new models | Noting how a deformed derivative transforms under a supersymmetry
transformation\mathcal {SUSY} &:&\partial _{x,\varepsilon }u\rightarrow \partial _{x,\varepsilon }u+i\eta \varepsilon \partial _{x,\varepsilon -1}u\xi _{xx},\quad \\
&&\partial _{x,\varepsilon }^{3}u\rightarrow \partial _{x,\varepsilon }^{3}u+i\eta \vare... | {
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"Bijan Bagchi",
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] | [
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583b36d1eed9970523b8ac77ac305a68f3320a55 | subsection | 14 | 16 | Construction of new models | We find that (REF ) remains invariant under the
supersymmetry transformation (REF ), but () does not respect
it.Further interesting options are of course combinations of the above, such
for instance\Phi _{t}=-\check{D}_{\varepsilon }^{6}\Phi +6\tilde{D}_{\kappa }\Phi \tilde{D}_{\kappa }^{2}\Phi +\lambda \Phi \check{D}_... | {
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} | 10.1088/1751-8113/41/39/392004 | 0807.1828 | PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation | [
"Bijan Bagchi",
"Andreas Fring"
] | [
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9e2c0bf660b3e3a6efad94c68427208c87484c8b | subsection | 15 | 16 | Conclusion | We have discussed various possibilities to introduce \mathcal {PT}-symmetrically deformed superderivatives. The most interesting cases are
those just involving deformed derivatives acting on the bosonic field, i.e. \tilde{D}_{\varepsilon }^{n} and \check{D}_{\varepsilon }^{n} as defined
in () and (REF ), respectively. ... | {
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a1be8686afd057c515a9f5e79aea7628fa8a9e51 | abstract | 0 | 83 | Abstract | We define and study a series indexed by rooted trees and with coefficients in
Q(q). We show that it is related to a family of Lie idempotents. We prove that
this series is a q-deformation of a more classical series and that some of its
coefficients are Carlitz q-Bernoulli numbers. | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
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cc6ef1f9f30280b04166e145776e3ed6ecb7edd9 | subsection | 1 | 83 | Introduction | The aim of this article is to introduce and study a series \Omega _q
indexed by rooted trees, with coefficients that are rational functions
of the indeterminate q.The series \Omega _q is in fact an element of the group G_\mathsf {PL} of
formal power series indexed by rooted trees, which is associated to
the \operatorna... | {
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"Frédéric Chapoton"
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c237b113dd33823970524a43076bde939b30a804 | subsection | 2 | 83 | General setting | We will work over the field \mathbb {Q} of rational numbers and over the field
\mathbb {Q}(q) of fractions in the indeterminate q.We have tried to avoid using operads as much as possible, but this
language is needed at some points in this article. The reader may
consult , as references. The symbol \circ will
denote the... | {
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328b1cb6ba197216f9c1cb90e093fb3354fe3f9d | subsection | 3 | 83 | Pre-Lie algebras | Recall (see for instance ) that a pre-Lie
algebra is a vector space V endowed with a bilinear map \curvearrowleft
from V \otimes V to V satisfying the following axiom:(x \curvearrowleft y) \curvearrowleft z -x \curvearrowleft (y \curvearrowleft z)= (x \curvearrowleft z) \curvearrowleft y -x \curvearrowleft (z \curvear... | {
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"Frédéric Chapoton"
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feb1729f498eb5f55cb933c787b2a74960f4fe42 | subsection | 4 | 83 | Free pre-Lie algebras | The free pre-Lie algebras have a simple description using rooted
trees. Let us recall briefly this description and other properties.
Details can be found in .A rooted tree is a finite, connected and simply connected
graph, together with a distinguished vertex called the root. We will
picture rooted trees with their roo... | {
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"Frédéric Chapoton"
] | [
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0b21fedf995d1537f1fb1ebe679dcaf6aac2daad | subsection | 5 | 83 | Free pre-Lie algebras | In this basis of
{U}(\mathsf {PL}), there is a nice combinatorial description of the associative
product \star . Let F and F^{\prime } be forests in {U}(\mathsf {PL}). The product
F \star F^{\prime } is the sum of all possible forests, obtained from the
disjoint union of F and F^{\prime } by the addition of some edges ... | {
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"Frédéric Chapoton"
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61d2dc419c41fa3352425198195ab28cdca3bd5a | subsection | 6 | 83 | Free pre-Lie algebras | Let us now introduce a special element of G_{\mathsf {PL}}, for later use.
Let \exp ^* \in G_{\mathsf {PL}} be
\exp ^* = \includegraphics [height=5mm]{a0.eps}\curvearrowleft \left((\exp (\includegraphics [height=5mm]{a0.eps})-1) / \includegraphics [height=5mm]{a0.eps}\right).
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idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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149a35cdb788d0a16fec757851b84f383cf2293e | subsection | 7 | 83 | Free pre-Lie algebras | Proposition 3.1 There is a unique solution \Omega in \widehat{\mathsf {PL}}_\mathbb {Q} to the equation
\includegraphics [height=5mm]{a0.eps}\curvearrowleft \left(\frac{\Omega }{\exp (\Omega )-1}\right) = \Omega ,
where \frac{\Omega }{\exp (\Omega )-1} is in the completed enveloping
algebra \widehat{U}(\mathsf {PL})... | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
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06d5a6176684a5cb17f8e37a144006c02f5aea38 | subsection | 8 | 83 | Free pre-Lie algebras | Let \Omega be
any solution of (REF ). Let us write \Omega =\sum _{n\ge 1} \Omega _n where each \Omega _n is homogeneous of degree n.
Then the homogeneous component of degree n of equation
(REF ) is
\includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _{n-1}=\sum _{k \ge 1} \frac{1}{k!} \sum _{\genfrac{}{}{0.0p... | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
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59656f4381ef535ee965d3b5dd8f3eb2bf67ea30 | subsection | 9 | 83 | Free pre-Lie algebras | By Proposition
REF , it is enough to prove that
\exp ^* (\Omega )=\includegraphics [height=5mm]{a0.eps},
because the image by \pi of the right-hand side of
(REF ) is \includegraphics [height=5mm]{a0.eps}.
But this amounts to say that \exp ^* is the inverse of \Omega in
the group G_\mathsf {PL}. This is nothing else t... | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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7024bd2bb3fcf5531317fbaecc2f40aec23f1039 | subsection | 10 | 83 | Free pre-Lie algebras | Then for n\ge 2, the homogeneous component of degree n of
equation (REF ) is
(q^n -1) \Omega _{q,n}= \includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _{q,n-1} - \sum _{k \ge 1} \frac{1}{k!} \sum _{\genfrac{}{}{0.0pt}1{m_1\ge 1,\dots ,m_k\ge 1,\ell \ge 1}{m_1+\dots +m_k+\ell =n}} q^\ell ((\Omega _{q,\ell } ... | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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fe8bd8f37ae30c988cbf77fa1c7a82ac092aa672 | subsection | 11 | 83 | Free pre-Lie algebras | Proof.
Let us compute the right-hand side of Eq. (REF ), using Eq.
(REF ) for \Omega _q, written as
\Omega _q + \includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _q +(q-1)\, \includegraphics [height=5mm]{a0.eps}= \sum _{n \ge 1} \frac{1}{(n-1)!}\mathtt {Crl}^\natural _{n} \circ _\natural \Omega _q[q].
One g... | {
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
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8e0adbd5e5cae2fb46e60702cc38be8537cf9acd | subsection | 12 | 83 | Free pre-Lie algebras | Dendriform algebra
Recall that a dendriform algebra (notion due to Loday, see
) is a vector space V endowed with two bilinear maps
\succ and \prec from V \otimes V to V satisfying the following
axioms:
x \prec ( y \prec z)+x \prec (y \succ z) &= (x \prec y) \prec z,\\
x \succ ( y \prec z) &= (x \succ y) \prec z,\\
... | {
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18438e567bce501fee859c8050ff677b1dce2fa4 | subsection | 13 | 83 | Free pre-Lie algebras | Let L=\sum _{n
\ge 1}L_n be the unique solution in \widehat{\operatorname{Dend}} to the equation
L=\includegraphics [height=3mm]{a1.eps}+L \succ \includegraphics [height=3mm]{a1.eps}=(1+L) \succ \includegraphics [height=3mm]{a1.eps},
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
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22d24797045bd4a626ca56528ba840e9d31e73ca | subsection | 14 | 83 | Free pre-Lie algebras | Then
\mathtt {Lnr}^\flat =\includegraphics [height=5mm]{a0.eps}{}^\flat + \includegraphics [height=5mm]{a0.eps}\curvearrowleft \mathtt {Lnr}^\flat ,
as one can easily check.
These relations can be taken as definitions of the elements E and
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} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
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"Frédéric Chapoton"
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... | |
1bbfb34687894f458cc865599a357b19824d3edb | subsection | 15 | 83 | Free pre-Lie algebras | Proof.
This was proved in , , .
Proposition 5.5
The image of \sum _{\ell \ge 0} \sum _{n\ge 0}
\frac{(-1)^\ell }{n!}\mathtt {Frk}^\natural _{\ell ,n} by \varphi is
(1+R) * \includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L}),
where \includegraphics [height=3mm]{a1.eps}^\natural is the planar binary tr... | {
"cite_spans": [
{
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"doi": "",
"end": 31,
"openalex_id": "",
"raw": "M. Ronco. A Milnor-Moore theorem for dendriform Hopf algebras. C. R. Acad. Sci. Paris Sér. I Math., 332(2):109–114, 2001.",
"source_ref_id": "2f751b23d5e59a69f7583a359abf9cecc2e21775",
"s... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.06177118420600891,
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0... | |
fb55a5cf7ee415708b57b49504e0717c5830860d | subsection | 16 | 83 | Free pre-Lie algebras | One gets, using the dendriform axioms,
(1+R) \succ \includegraphics [height=3mm]{a1.eps}^\natural \prec (1-\widetilde{L}) + \includegraphics [height=3mm]{a1.eps}\prec ((1+R)*
\includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L})) - ((1+R)* \includegraphics [height=3mm]{a1.eps}^\natural *
(1-\widetilde{L})... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.03985307365655899,
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-0.006305238232016563,... | |
976054a8bf89d8d76717d4aec5ed2efd0f419800 | subsection | 17 | 83 | Free pre-Lie algebras | One can now deduce a useful functional equation for the image of
\Omega _q by \varphi , using only the associative product * of
\operatorname{Dend}.
Proposition 5.6
The series \varphi (\Omega _q) is the unique solution in \widehat{\operatorname{Dend}} of
\varphi (\Omega _q)=(1-\widetilde{L})^{-1} * \varphi (\Omega _q... | {
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"raw": "I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon. Noncommutative symmetric functions. Adv. Math., 112(2):218–348, 1995.",
"source_ref_id": "ad9aac43f1c305ecac0e1... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
87b5a243b517564de13916d1c8f17467e473df37 | subsection | 18 | 83 | Free pre-Lie algebras | Indeed, one has
1+L=\sum _{n\ge 0} \theta (S_n)\quad \text{and}\quad E=\sum _{n\ge 1} n\theta ( S_n).
Therefore
B=\sum _{n \ge 1} \theta (\Psi _n).
We need to introduce the following notations.
The leaves of a planar binary tree with n vertices are labelled from
0 to n from left to right. The leaves with labels di... | {
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{
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"doi": "",
"end": 1430,
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"raw": "D. Krob, B. Leclerc, and J.-Y. Thibon. Noncommutative symmetric functions. II. Transformations of alphabets. Internat. J. Algebra Comput., 7(2):181–264, 1997.",
"source_ref_id": "c4f362d0418d... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.... | |
715a0ea3dea45d61e1a065334577abf4c249c776 | subsection | 19 | 83 | Free pre-Lie algebras | Indeed, by Proposition REF , one has
\sum _{n\ge 1}\varphi (\widetilde{\Omega _q}) =(1+L)^{-1} *(
\varphi (\widetilde{\Omega _q})[q]) *(1+L) +(1-q) \sum _{n\ge 1} \varphi (\mathtt {Lnr}_n).
Then using Prop. REF and Eq. (REF ), one gets that
\theta ((1-q)\Psi (\frac{A}{1-q})) and \varphi (\widetilde{\Omega _q}) satisf... | {
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"doi": "",
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"source_ref_id": "c4f362d0418d4... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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1c7b14e125fa9180010f3ec3fbae8faf3e0f8b29 | subsection | 20 | 83 | Free pre-Lie algebras | Note that the
expected denominator of \Omega _{q,n} (from recursion
(REF )) is the product \prod _{d=2}^{n}(q^d-1). Let
\Phi _d be the d^{th} cyclotomic polynomial.
Proposition 6.2 The common denominator of the coefficients of the element
\Omega _{q,n} divides the product \prod _{d=2}^{n} \Phi _d.
Proof.
For the imag... | {
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{
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"doi": "",
"end": 489,
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"raw": "V. J. W. Guo and J. Zeng. Some arithmetic properties of the q-Euler numbers and q-Salié numbers. European J. Combin., 27(6):884–895, 2006.",
"source_ref_id": "6dc09d761e85a19018791d725a3fb5921... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.... | |
4f57c8183bf019ebc711dc922f5396810947b06f | subsection | 21 | 83 | Free pre-Lie algebras | More
precisely, the inverse of \sum _{n \ge 1} \frac{1}{(n-1)!}\mathtt {Crl}_{n}
in the group of characters of the Connes-Kreimer Hopf algebra was
shown there to be
\sum _T \frac{(-1)^{\#T-1}}{\operatorname{aut}(T)} T,
where \operatorname{aut}(T) is the cardinal of the automorphism group of the
rooted tree T. But it ... | {
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{
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"doi": "",
"end": 439,
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"raw": "F. Chapoton and M. Livernet. Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices, (8):395–408, 2001.",
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... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-... | |
12161d2e8c91fa24f921248794564bfd0bf4c3ba | subsection | 22 | 83 | Free pre-Lie algebras | The underlying vector space is therefore identified
with \mathbb {Q}[x] and the pre-Lie product is
x^p \curvearrowleft x^q=
{\left\lbrace \begin{array}{ll}
x^{p+1} \quad \text{if}q=0,\\
0 \quad \text{else.}
\end{array}\right.}
It is known (see ) that the image of \Omega is the
generating function \frac{x}{\exp (x)-1}... | {
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{
"arxi... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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db841af54c658418ad9a0a3ce0d99e42f00f0098 | subsection | 23 | 83 | Free pre-Lie algebras | First terms of some expansions
\Omega =\includegraphics [height=5mm]{a0.eps}-\frac{1}{2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{3}
\includegraphics [height=5mm]{a110.eps}+\frac{1}{12}\includegraphics [height=5mm]{a200.eps} -\frac{1}{4}
\includegraphics [height=5mm]{a1110.eps}-\frac{1}{12}\includegraphics [heig... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.0018190115224570036,
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-... | |
6b69150f8d69faa161bd1f1c9a663e7e327396ea | subsection | 24 | 83 | Free pre-Lie algebras | \Omega _q=\includegraphics [height=5mm]{a0.eps}-\frac{1}{\Phi _2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{\Phi _3}
\includegraphics [height=5mm]{a110.eps}+\frac{q}{2 \,\Phi _2 \Phi _3}\includegraphics [height=5mm]{a200.eps} \\
-\frac{1}{\Phi _2 \Phi _4}
\includegraphics [height=5mm]{a1110.eps}-\frac{q}{2\,\Phi _... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
34e1f94167af23475a4c7ef727a4e3504a828ff0 | subsection | 25 | 83 | The classical case | Let us start by recalling the definition of a classical element
\Omega of \widehat{\mathsf {PL}} with rational coefficients. It was considered under
the name of \log ^* in and has been since studied in
, , , .Proposition 3.1 There is a unique solution \Omega in \widehat{\mathsf {PL}}_\mathbb {Q} to the equation\include... | {
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"ar... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.061722662299871445,
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278e27e730476b0ed05281da75026eedcf123a3d | subsection | 26 | 83 | The classical case | First, by right action on (REF ) by \Omega , one can see
that the unique solution \Omega of (REF ) is indeed a
solution of (REF ).
Let us now prove uniqueness of a non-zero solution. Let \Omega be
any solution of (REF ). Let us write \Omega =\sum _{n\ge 1} \Omega _n where each \Omega _n is homogeneous of degree n.
Then... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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d74afb8c9010939ee741762b5e12df2d1cf61edd | subsection | 27 | 83 | The classical case | By Proposition
REF , it is enough to prove that
\exp ^* (\Omega )=\includegraphics [height=5mm]{a0.eps},
because the image by \pi of the right-hand side of
(REF ) is \includegraphics [height=5mm]{a0.eps}.
But this amounts to say that \exp ^* is the inverse of \Omega in
the group G_\mathsf {PL}. This is nothing else t... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-... | |
72f95853f21faf1099ebec02139aa74ef9be74de | subsection | 28 | 83 | The classical case | Then for n\ge 2, the homogeneous component of degree n of
equation (REF ) is
(q^n -1) \Omega _{q,n}= \includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _{q,n-1} - \sum _{k \ge 1} \frac{1}{k!} \sum _{\genfrac{}{}{0.0pt}1{m_1\ge 1,\dots ,m_k\ge 1,\ell \ge 1}{m_1+\dots +m_k+\ell =n}} q^\ell ((\Omega _{q,\ell } ... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.03317900002002716,
0.006... | |
1e52acc43d94ce15c5591f3c2e86db17ac0f4c12 | subsection | 29 | 83 | The classical case | Proof.
Let us compute the right-hand side of Eq. (REF ), using Eq.
(REF ) for \Omega _q, written as
\Omega _q + \includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _q +(q-1)\, \includegraphics [height=5mm]{a0.eps}= \sum _{n \ge 1} \frac{1}{(n-1)!}\mathtt {Crl}^\natural _{n} \circ _\natural \Omega _q[q].
One g... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.02... | |
0ade8383fb76647a4a441b7e2fe4a0a81d1f3e75 | subsection | 30 | 83 | The classical case | Dendriform algebra
Recall that a dendriform algebra (notion due to Loday, see
) is a vector space V endowed with two bilinear maps
\succ and \prec from V \otimes V to V satisfying the following
axioms:
x \prec ( y \prec z)+x \prec (y \succ z) &= (x \prec y) \prec z,\\
x \succ ( y \prec z) &= (x \succ y) \prec z,\\
... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 680,
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"raw": "J.-L. Loday. Dialgebras. In Dialgebras and related operads, volume 1763 of Lecture Notes in Math., pages 7–66. Springer, Berlin, 2001.",
"source_ref_id": "2df386aef333e34c72fbe8cf1ad4d5d294f5f... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0... | |
ad00e34f119a7b6486c3c148acc4e90640c91805 | subsection | 31 | 83 | The classical case | Let L=\sum _{n
\ge 1}L_n be the unique solution in \widehat{\operatorname{Dend}} to the equation
L=\includegraphics [height=3mm]{a1.eps}+L \succ \includegraphics [height=3mm]{a1.eps}=(1+L) \succ \includegraphics [height=3mm]{a1.eps},
and let R=\sum _{n\ge 1} R_n be the unique solution in \widehat{\operatorname{Dend}... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.013929087668657303... | |
f4d7fa20815e10498477f89ae204b0e9730d0488 | subsection | 32 | 83 | The classical case | Then
\mathtt {Lnr}^\flat =\includegraphics [height=5mm]{a0.eps}{}^\flat + \includegraphics [height=5mm]{a0.eps}\curvearrowleft \mathtt {Lnr}^\flat ,
as one can easily check.
These relations can be taken as definitions of the elements E and
B^\flat of \widehat{\operatorname{Dend}}. One can forget the marking \flat in... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.02650153450667858,
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... | |
091c8fe3780da773508e9b2fd3ce0eab8d342256 | subsection | 33 | 83 | The classical case | Proof.
This was proved in , , .
Proposition 5.5
The image of \sum _{\ell \ge 0} \sum _{n\ge 0}
\frac{(-1)^\ell }{n!}\mathtt {Frk}^\natural _{\ell ,n} by \varphi is
(1+R) * \includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L}),
where \includegraphics [height=3mm]{a1.eps}^\natural is the planar binary tr... | {
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"doi": "",
"end": 31,
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"s... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0... | |
d6796e6d571e238b74885f29f648c196a9bac6d1 | subsection | 34 | 83 | The classical case | One gets, using the dendriform axioms,
(1+R) \succ \includegraphics [height=3mm]{a1.eps}^\natural \prec (1-\widetilde{L}) + \includegraphics [height=3mm]{a1.eps}\prec ((1+R)*
\includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L})) - ((1+R)* \includegraphics [height=3mm]{a1.eps}^\natural *
(1-\widetilde{L})... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.014708408154547215,
-0.006305238232016563,... | |
de8f52badc6c20aed44567a7fe37ccbb224ea4a7 | subsection | 35 | 83 | The classical case | One can now deduce a useful functional equation for the image of
\Omega _q by \varphi , using only the associative product * of
\operatorname{Dend}.
Proposition 5.6
The series \varphi (\Omega _q) is the unique solution in \widehat{\operatorname{Dend}} of
\varphi (\Omega _q)=(1-\widetilde{L})^{-1} * \varphi (\Omega _q... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1238,
"openalex_id": "",
"raw": "I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J.-Y. Thibon. Noncommutative symmetric functions. Adv. Math., 112(2):218–348, 1995.",
"source_ref_id": "ad9aac43f1c305ecac0e1... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
cfb1e0a0ae3727ae3e03fdcf8653b43a2b346a4e | subsection | 36 | 83 | The classical case | Indeed, one has
1+L=\sum _{n\ge 0} \theta (S_n)\quad \text{and}\quad E=\sum _{n\ge 1} n\theta ( S_n).
Therefore
B=\sum _{n \ge 1} \theta (\Psi _n).
We need to introduce the following notations.
The leaves of a planar binary tree with n vertices are labelled from
0 to n from left to right. The leaves with labels di... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1430,
"openalex_id": "",
"raw": "D. Krob, B. Leclerc, and J.-Y. Thibon. Noncommutative symmetric functions. II. Transformations of alphabets. Internat. J. Algebra Comput., 7(2):181–264, 1997.",
"source_ref_id": "c4f362d0418d... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.... | |
0b10b621fe20c1ae53f5f1adb633121444a005d9 | subsection | 37 | 83 | The classical case | Indeed, by Proposition REF , one has
\sum _{n\ge 1}\varphi (\widetilde{\Omega _q}) =(1+L)^{-1} *(
\varphi (\widetilde{\Omega _q})[q]) *(1+L) +(1-q) \sum _{n\ge 1} \varphi (\mathtt {Lnr}_n).
Then using Prop. REF and Eq. (REF ), one gets that
\theta ((1-q)\Psi (\frac{A}{1-q})) and \varphi (\widetilde{\Omega _q}) satisf... | {
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{
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"doi": "",
"end": 843,
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"source_ref_id": "c4f362d0418d4... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
0135c7d608bd79de082691a7e5256350eebef4bb | subsection | 38 | 83 | The classical case | Note that the
expected denominator of \Omega _{q,n} (from recursion
(REF )) is the product \prod _{d=2}^{n}(q^d-1). Let
\Phi _d be the d^{th} cyclotomic polynomial.
Proposition 6.2 The common denominator of the coefficients of the element
\Omega _{q,n} divides the product \prod _{d=2}^{n} \Phi _d.
Proof.
For the imag... | {
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{
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"doi": "",
"end": 489,
"openalex_id": "",
"raw": "V. J. W. Guo and J. Zeng. Some arithmetic properties of the q-Euler numbers and q-Salié numbers. European J. Combin., 27(6):884–895, 2006.",
"source_ref_id": "6dc09d761e85a19018791d725a3fb5921... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.... | |
6493eadfcc1bb8e98618f160e4ec264a763f8968 | subsection | 39 | 83 | The classical case | More
precisely, the inverse of \sum _{n \ge 1} \frac{1}{(n-1)!}\mathtt {Crl}_{n}
in the group of characters of the Connes-Kreimer Hopf algebra was
shown there to be
\sum _T \frac{(-1)^{\#T-1}}{\operatorname{aut}(T)} T,
where \operatorname{aut}(T) is the cardinal of the automorphism group of the
rooted tree T. But it ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 439,
"openalex_id": "",
"raw": "F. Chapoton and M. Livernet. Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices, (8):395–408, 2001.",
"source_ref_id": "07ceb79c79c0c67a8f480d65eb0ca29bd01ad103",
... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-... | |
ef126669fd901c0717d79cb446d018913dedbcc8 | subsection | 40 | 83 | The classical case | The underlying vector space is therefore identified
with \mathbb {Q}[x] and the pre-Lie product is
x^p \curvearrowleft x^q=
{\left\lbrace \begin{array}{ll}
x^{p+1} \quad \text{if}q=0,\\
0 \quad \text{else.}
\end{array}\right.}
It is known (see ) that the image of \Omega is the
generating function \frac{x}{\exp (x)-1}... | {
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{
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"doi": "",
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"raw": "F. Chapoton. Rooted trees and an exponential-like series. arXiv.org:math/0209104, 2002.",
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{
"arxi... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.017411552369594574,... | |
2a2a934e0d34853ea9df200307f830038127c81c | subsection | 41 | 83 | The classical case | First terms of some expansions
\Omega =\includegraphics [height=5mm]{a0.eps}-\frac{1}{2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{3}
\includegraphics [height=5mm]{a110.eps}+\frac{1}{12}\includegraphics [height=5mm]{a200.eps} -\frac{1}{4}
\includegraphics [height=5mm]{a1110.eps}-\frac{1}{12}\includegraphics [heig... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.0018190115224570036,
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0.017077915370464325,
-0.03244651108980179,
-... | |
85dd9eb88f4f05dc466ed8d00a54b227dbbd2898 | subsection | 42 | 83 | The classical case | \Omega _q=\includegraphics [height=5mm]{a0.eps}-\frac{1}{\Phi _2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{\Phi _3}
\includegraphics [height=5mm]{a110.eps}+\frac{q}{2 \,\Phi _2 \Phi _3}\includegraphics [height=5mm]{a200.eps} \\
-\frac{1}{\Phi _2 \Phi _4}
\includegraphics [height=5mm]{a1110.eps}-\frac{q}{2\,\Phi _... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.06056807190179825,
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0.018255900591611862,
-0.006865042727440596,
... | |
ac7af0829983ce9843cb8aa2eafdcfd00c474f0c | subsection | 43 | 83 | The quantum case | We will introduce now an element \Omega _q in \widehat{\mathsf {PL}} with coefficients
in \mathbb {Q}(q). We will show later that this is a q-deformation of \Omega .If A=\sum _{n\ge 1} A_n is an element of \widehat{\mathsf {PL}}, let A[q] be the
q-shift of A defined byA[q]=\sum _{n\ge 1} q^n A_n.Proposition 4.1
There ... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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1901811e07fef1e4ce869c41db65e8eb13eb1029 | subsection | 44 | 83 | The quantum case | (REF ) and Prop. REF .
Let \mathtt {Frk}^\natural _{\ell ,n} be the rooted tree with a linear trunk
of \ell vertices, a vertex \natural on top of this trunk and a
corolla with n leaves on top of the vertex \natural , see Fig.
REF . We will call this a fork. One has
\mathtt {Frk}_{\ell ,n}=\mathtt {Lnr}_{\ell +1}^\flat... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.... | |
8f9d372e3d3206bef57bde3c261a338f3b7d181c | subsection | 45 | 83 | The quantum case | As \mathtt {Lnr}^\flat _{\ell } \circ _\natural \includegraphics [height=5mm]{a0.eps}=\mathtt {Lnr}_{\ell } and
\mathtt {Lnr}^\flat _\ell \circ _\flat (\includegraphics [height=5mm]{a0.eps}\curvearrowleft \Omega _q)=\mathtt {Lnr}_{\ell +1}^\flat \circ _\flat \Omega _q, the
two right-most terms cancels, and the sum simp... | {
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{
"arxiv_id": "",
"doi": "",
"end": 1843,
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"raw": "J.-L. Loday. Dialgebras. In Dialgebras and related operads, volume 1763 of Lecture Notes in Math., pages 7–66. Springer, Berlin, 2001.",
"source_ref_id": "2df386aef333e34c72fbe8cf1ad4d5d294f5... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.0... | |
cd1f087ea1c8cecc0918ee98e89ac7c6b6308eec | subsection | 46 | 83 | The quantum case | REF . In particular, the free dendriform algebra on one
generator, denoted by \operatorname{Dend}, has a basis indexed by planar binary
trees. This is a graded vector space, the degree \# t of a planar
binary tree t being the number of its inner vertices.
There is a unique morphism \varphi of pre-Lie algebras from \mat... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0... | |
2c327ef65ba403598eec88fed4d38a6fe169a5f5 | subsection | 47 | 83 | The quantum case | (REF ), this becomes
1+R-\includegraphics [height=3mm]{a1.eps}\prec (1+R)+\widetilde{L} \succ \includegraphics [height=3mm]{a1.eps}\prec (1+R) -\widetilde{L} \succ R.
The last two terms cancel by Eq. (REF ) and one gets
1+R-\includegraphics [height=3mm]{a1.eps}\prec (1+R),
which is just 1, again by Eq. (REF ).
Equ... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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eeb94e376dc18a625cbc08f85718df6efc2237e4 | subsection | 48 | 83 | The quantum case | Using Eq. (REF ) and the dendriform axioms, this becomes
L+L\prec \includegraphics [height=3mm]{a1.eps}+((1+L)*B) \succ \includegraphics [height=3mm]{a1.eps}- (1+L) \succ \includegraphics [height=3mm]{a1.eps}\prec B +L
\prec (B \succ \includegraphics [height=3mm]{a1.eps}- \includegraphics [height=3mm]{a1.eps}\prec B).... | {
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"... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
ea3345bc9da883bee50e9631dd215a23d3ffec64 | subsection | 49 | 83 | The quantum case | Then one gets, by expanding again,
(1+R) \succ \includegraphics [height=3mm]{a1.eps}^\natural \prec (1-\widetilde{L}) + (R \prec \includegraphics [height=3mm]{a1.eps}^\natural ) \prec (1-\widetilde{L}) - ((1+R)* \includegraphics [height=3mm]{a1.eps}^\natural ) \succ \widetilde{L}.
Using the dendriform axioms, this is... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.014657292515039444,
-0... | |
1746f2ad934671b229e6a5fad54e9ebe2412ebe6 | subsection | 50 | 83 | The quantum case | By Lemma REF , one has
\widetilde{B}^\flat =\includegraphics [height=3mm]{a1.eps}^\flat -\widetilde{B}^\flat \succ \includegraphics [height=3mm]{a1.eps}+\includegraphics [height=3mm]{a1.eps}\prec \widetilde{B}^\flat , hence
D^{\prime }=(1+R) \succ \includegraphics [height=3mm]{a1.eps}^\natural \prec (1-\widetilde{L})+... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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1c0437c3673ea213e812b84fec01cc3322f09fbd | subsection | 51 | 83 | The quantum case | Explicit formula
We will prove in this section that the image of \Omega _q by \varphi
coincides (in some sense) with a known family of Lie idempotents, and
has an explicit description using q-binomial coefficients, descents
and major indices of planar binary trees. To obtain this description,
we use a result on noncom... | {
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idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.0... | |
9ff90fb69a9ebc466d316fa5a7cf2388c4934c0b | subsection | 52 | 83 | The quantum case | Proposition 5.8
The image by \theta of R_I is the sum
\sum _{\genfrac{}{}{0.0pt}1{\# t=n}{ D(t)=D(I)}} t
of all planar binary trees with n vertices and descent set D(I).
Proof.
This is a well-known property of the injection of \textbf {Sym} in \operatorname{Dend}.
In , elements \Psi _n(\frac{A}{1-q}), for n\ge 1,... | {
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idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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5c74a613bda177e8767a1e6231f7a265c2f9d80a | subsection | 53 | 83 | The quantum case | The Theorem 6.11 of tells that the element
(1-q)\Psi _n(\frac{A}{1-q}) is
\frac{1}{[n]_q} \sum _{|I|=n} (-1)^{d(I)}
\genfrac[]{0.0pt}1{n-1}{d(I)}_q^{-1} q^{\operatorname{maj}(I)-\binom{d(I)+1}{2}} \, R_I.
By Prop. REF , the image by \theta of this formula is
(-1)^{n-1} \varphi (\Omega _{q,n}). By Prop. REF , this bec... | {
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"doi": "",
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"source_ref_id": "c4f362d0418d4... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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b899f41ed67216e4e17a33979e70ffd7a7bc75cb | subsection | 54 | 83 | The quantum case | Let \omega _{q,T}
be the coefficient of the rooted tree T in the expansion of
\Omega _q. We will call valuation at q=\infty the smallest
exponent in the formal Laurent expansion in powers of q^{-1} of an
element of \mathbb {Q}(q).
Proposition 6.3 The valuation of \omega _{q,T} at q=\infty is at least \# T-1.
Proof.
Th... | {
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"Frédéric Chapoton"
] | [
"math.QA"
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22d56772999cc4c9b282a5b633024465295dd98d | subsection | 55 | 83 | The quantum case | This extends uniquely
to a morphism from \widehat{\mathsf {PL}} to the algebra \mathbb {Q}[[x]]_+ of formal power
series in x without constant term.
One can show that this morphism send the linear trees \mathtt {Lnr}_n with
n vertices to the monomials x^n and all others trees to 0.
It is known (see ) that the image of ... | {
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"doi": "",
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"ar... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0c41e9e61e06b080ac9a1b65f2c267aa451cdc2a | subsection | 56 | 83 | The quantum case | We describe it here only as a side remark, as the
image of \Omega _q seems to have no special property.
Consider the the vector space V=\mathbb {Q}[x]_+, endowed with the following
pre-Lie product:
(f \curvearrowleft g)=x f \, \partial _x g.
Then there is a unique morphism from \mathsf {PL} to V sending \includegraph... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.01388897188007831... | |
b19b519f1ca518622af6209e1c263b819ab7b084 | subsection | 57 | 83 | The quantum case | \Omega _q=\includegraphics [height=5mm]{a0.eps}-\frac{1}{\Phi _2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{\Phi _3}
\includegraphics [height=5mm]{a110.eps}+\frac{q}{2 \,\Phi _2 \Phi _3}\includegraphics [height=5mm]{a200.eps} \\
-\frac{1}{\Phi _2 \Phi _4}
\includegraphics [height=5mm]{a1110.eps}-\frac{q}{2\,\Phi _... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
97ce5fe0018d82aa09c65c26a736be6fef0efa65 | subsection | 58 | 83 | Image in the free dendriform algebra | We describe in this section the image of \Omega _q by the usual
morphism from the free pre-Lie algebra to the free dendriform algebra.
We show that this image is related to a family of Lie idempotents in
the descent algebras of the symmetric groups. One deduces from that a
nice explicit formula, that will be used later... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.038324788212776184,
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0.0023781354539096355,
0... | |
366e73edb9851143d9c470570cc41f689e93a36b | subsection | 59 | 83 | Dendriform algebra | Recall that a dendriform algebra (notion due to Loday, see
) is a vector space V endowed with two bilinear maps
\succ and \prec from V \otimes V to V satisfying the following
axioms:x \prec ( y \prec z)+x \prec (y \succ z) &= (x \prec y) \prec z,\\
x \succ ( y \prec z) &= (x \succ y) \prec z,\\
x \succ (y \succ z) &=... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 648,
"openalex_id": "",
"raw": "J.-L. Loday. Dialgebras. In Dialgebras and related operads, volume 1763 of Lecture Notes in Math., pages 7–66. Springer, Berlin, 2001.",
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idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0... | |
173d9ec6fb7ea4b07d099360bde430a723b1b148 | subsection | 60 | 83 | Dendriform algebra | Let L=\sum _{n
\ge 1}L_n be the unique solution in \widehat{\operatorname{Dend}} to the equationL=\includegraphics [height=3mm]{a1.eps}+L \succ \includegraphics [height=3mm]{a1.eps}=(1+L) \succ \includegraphics [height=3mm]{a1.eps},and let R=\sum _{n\ge 1} R_n be the unique solution in \widehat{\operatorname{Dend}} toR... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.013100605458021164... | |
161912da2d3982eb0de4d0aaf5a8eb2a60c9262f | subsection | 61 | 83 | Dendriform algebra | Then
\mathtt {Lnr}^\flat =\includegraphics [height=5mm]{a0.eps}{}^\flat + \includegraphics [height=5mm]{a0.eps}\curvearrowleft \mathtt {Lnr}^\flat ,
as one can easily check.
These relations can be taken as definitions of the elements E and
B^\flat of \widehat{\operatorname{Dend}}. One can forget the marking \flat in... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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... | |
42e56df4fd2d383757df08a5ec7d91d820542d34 | subsection | 62 | 83 | Dendriform algebra | Proof.
This was proved in , , .
Proposition 5.5
The image of \sum _{\ell \ge 0} \sum _{n\ge 0}
\frac{(-1)^\ell }{n!}\mathtt {Frk}^\natural _{\ell ,n} by \varphi is
(1+R) * \includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L}),
where \includegraphics [height=3mm]{a1.eps}^\natural is the planar binary tr... | {
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"doi": "",
"end": 31,
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"raw": "M. Ronco. A Milnor-Moore theorem for dendriform Hopf algebras. C. R. Acad. Sci. Paris Sér. I Math., 332(2):109–114, 2001.",
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"s... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0... | |
5a862ad3d0766763d58b1ea56636481c3f562912 | subsection | 63 | 83 | Dendriform algebra | One gets, using the dendriform axioms,
(1+R) \succ \includegraphics [height=3mm]{a1.eps}^\natural \prec (1-\widetilde{L}) + \includegraphics [height=3mm]{a1.eps}\prec ((1+R)*
\includegraphics [height=3mm]{a1.eps}^\natural * (1-\widetilde{L})) - ((1+R)* \includegraphics [height=3mm]{a1.eps}^\natural *
(1-\widetilde{L})... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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9a4105b62461737200c3d88ace005b87a31e878b | subsection | 64 | 83 | Dendriform algebra | One can now deduce a useful functional equation for the image of
\Omega _q by \varphi , using only the associative product * of
\operatorname{Dend}.
Proposition 5.6
The series \varphi (\Omega _q) is the unique solution in \widehat{\operatorname{Dend}} of
\varphi (\Omega _q)=(1-\widetilde{L})^{-1} * \varphi (\Omega _q... | {
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idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
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75208e81f1219d8cd63f77ee9b76d9bce1c4ca08 | subsection | 65 | 83 | Dendriform algebra | Indeed, one has
1+L=\sum _{n\ge 0} \theta (S_n)\quad \text{and}\quad E=\sum _{n\ge 1} n\theta ( S_n).
Therefore
B=\sum _{n \ge 1} \theta (\Psi _n).
We need to introduce the following notations.
The leaves of a planar binary tree with n vertices are labelled from
0 to n from left to right. The leaves with labels di... | {
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"source_ref_id": "c4f362d0418d... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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851528f07119eff22d9b0da8430631213c89e1c0 | subsection | 66 | 83 | Dendriform algebra | Indeed, by Proposition REF , one has
\sum _{n\ge 1}\varphi (\widetilde{\Omega _q}) =(1+L)^{-1} *(
\varphi (\widetilde{\Omega _q})[q]) *(1+L) +(1-q) \sum _{n\ge 1} \varphi (\mathtt {Lnr}_n).
Then using Prop. REF and Eq. (REF ), one gets that
\theta ((1-q)\Psi (\frac{A}{1-q})) and \varphi (\widetilde{\Omega _q}) satisf... | {
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"source_ref_id": "c4f362d0418d4... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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917a938ed5d406f4f31a1b252afc7c4bbec19a1c | subsection | 67 | 83 | Dendriform algebra | Note that the
expected denominator of \Omega _{q,n} (from recursion
(REF )) is the product \prod _{d=2}^{n}(q^d-1). Let
\Phi _d be the d^{th} cyclotomic polynomial.
Proposition 6.2 The common denominator of the coefficients of the element
\Omega _{q,n} divides the product \prod _{d=2}^{n} \Phi _d.
Proof.
For the imag... | {
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"source_ref_id": "6dc09d761e85a19018791d725a3fb5921... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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-0.... | |
150e8bec5545f5adea40f23f7587734b59578061 | subsection | 68 | 83 | Dendriform algebra | More
precisely, the inverse of \sum _{n \ge 1} \frac{1}{(n-1)!}\mathtt {Crl}_{n}
in the group of characters of the Connes-Kreimer Hopf algebra was
shown there to be
\sum _T \frac{(-1)^{\#T-1}}{\operatorname{aut}(T)} T,
where \operatorname{aut}(T) is the cardinal of the automorphism group of the
rooted tree T. But it ... | {
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{
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"raw": "F. Chapoton and M. Livernet. Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices, (8):395–408, 2001.",
"source_ref_id": "07ceb79c79c0c67a8f480d65eb0ca29bd01ad103",
... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.06415395438671112,
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-... | |
59ba6dab6052f41f5f18324bf493737bfbc7ab7b | subsection | 69 | 83 | Dendriform algebra | The underlying vector space is therefore identified
with \mathbb {Q}[x] and the pre-Lie product is
x^p \curvearrowleft x^q=
{\left\lbrace \begin{array}{ll}
x^{p+1} \quad \text{if}q=0,\\
0 \quad \text{else.}
\end{array}\right.}
It is known (see ) that the image of \Omega is the
generating function \frac{x}{\exp (x)-1}... | {
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{
"arxi... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.08692042529582977,
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-0.017411552369594574,... | |
47a5fba53d24a399d262087c69f9513c625b53cb | subsection | 70 | 83 | Dendriform algebra | First terms of some expansions
\Omega =\includegraphics [height=5mm]{a0.eps}-\frac{1}{2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{3}
\includegraphics [height=5mm]{a110.eps}+\frac{1}{12}\includegraphics [height=5mm]{a200.eps} -\frac{1}{4}
\includegraphics [height=5mm]{a1110.eps}-\frac{1}{12}\includegraphics [heig... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.0018190115224570036,
0.003065706929191947,
-0.01689477264881134,
-0.004864687565714121,
-0.031500283628702164,
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0.03272122144699097,
0.01874144747853279,
0.034735776484012604,
-0.014597877860069275,
0.004418280906975269,
0.017077915370464325,
-0.03244651108980179,
-... | |
2dff5a34520fdf96f79fd7df8c5b7d48f6075328 | subsection | 71 | 83 | Dendriform algebra | \Omega _q=\includegraphics [height=5mm]{a0.eps}-\frac{1}{\Phi _2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{\Phi _3}
\includegraphics [height=5mm]{a110.eps}+\frac{q}{2 \,\Phi _2 \Phi _3}\includegraphics [height=5mm]{a200.eps} \\
-\frac{1}{\Phi _2 \Phi _4}
\includegraphics [height=5mm]{a1110.eps}-\frac{q}{2\,\Phi _... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.06056807190179825,
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0.011913653463125229,
-0.011119918897747993,
0.018255900591611862,
-0.006865042727440596,
... | |
068d95624fe2a6c52b061e8febbe2b82e3a122d5 | subsection | 72 | 83 | Arithmetic properties | In this section, we obtain some properties of the denominators in
\Omega _q and consider what happens when q is specialized to 1,0
and \infty . | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.08326122164726257,
0.0392194539308548,
-0.0060278926976025105,
-0.003004408674314618,
-0.03415297344326973,
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0.0011779189808294177,
-0.0060698590241372585,
-0.003717836458235979,
... | |
6b4d4a961c0734335ea9219180c7779375c684c4 | subsection | 73 | 83 | Body | Let us first note that the morphism \varphi from \widehat{\mathsf {PL}} to the
completed free dendriform algebra \widehat{\operatorname{Dend}} is defined over \mathbb {Q} and
injective. Hence one can deduce results on \Omega _q from results on
its image by \varphi .Proposition 6.1 The series \Omega _q is regular at q=1... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1459,
"openalex_id": "",
"raw": "V. J. W. Guo and J. Zeng. Some arithmetic properties of the q-Euler numbers and q-Salié numbers. European J. Combin., 27(6):884–895, 2006.",
"source_ref_id": "6dc09d761e85a19018791d725a3fb592... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.05211424455046654,
0.04018411040306091,
-0.0361260324716568,
-0.046805184334516525,
0.01007653959095478,
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0.01858171820640564,
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-0.004531010054051876,
-0.009062020108103752,
0.023... | |
e3f9cbb9b0bd35f02e3552de371d77f3ba2393f8 | subsection | 74 | 83 | Body | Hence the
valuation of \Omega _{q,n} is at least n-1.
Hence there exists a limit \Omega _\infty for \Omega _q[q]/q when q goes to
\infty and the limit of \Omega _q/q is zero.
The equation (REF ), divided by q, becomes at q=\infty ,
\Omega _\infty \curvearrowleft \exp (\Omega ) = \includegraphics [height=5mm]{a0.eps}.... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 589,
"openalex_id": "",
"raw": "F. Chapoton and M. Livernet. Relating two Hopf algebras built from an operad. Int. Math. Res. Not. IMRN, (24):Art. ID rnm131, 27, 2007.",
"source_ref_id": "12dcbd1f853c1904111aa4d4818406e4eaab... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.05570404231548309,
0.0199771486222744,
-0.03378869965672493,
-0.01625337079167366,
0.006432672496885061,
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0.0013830626849085093,
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-0.015932882204651833,
-0.010125926695764065,
-0... | |
90e1eacf78953d34d356a9c6f5253643306c979b | subsection | 75 | 83 | Body | (REF ) that the image of \Omega _q is
the q-logarithm defined by
\log _q(x)=\sum _{n\ge 1} \frac{(-1)^{n-1}}{[n]_q}x^n,
which is the unique solution to the functional equation
x \log _q(qx)=x \log _q(x)+(q-1)\,x-\log _q(qx)+\log _q(x).
Morphism for corollas
As shown in , the subspace of \mathsf {PL} spanned by tre... | {
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{
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},
{
"ar... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.08385895937681198,
0.01570066623389721,
-0.02006450667977333,
-0.027556272223591805,
0.0053937663324177265,
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0.01701286993920803,
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-0.005496759433299303,
0.007774042431265116,
0... | |
fb46b9db1f79493deccbf762c1c2a5f1dba85eea | subsection | 76 | 83 | Body | The proof is just a check
that this sum-of-coefficients map defines a morphism of pre-Lie
algebra from \mathsf {PL} to V.
First terms of some expansions
\Omega =\includegraphics [height=5mm]{a0.eps}-\frac{1}{2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{3}
\includegraphics [height=5mm]{a110.eps}+\frac{1}{12}\incl... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.026188934221863747,
-0.007596469484269619,
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-0.02119838446378708,
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-0.0005532336072064936,
-0.0031477082520723343,
0.01419330295175314,
-0.038123518228530884... | |
810d2a8071f04770b93a9d69b4cd5bd1da56e7c5 | subsection | 77 | 83 | Body | \Omega _q=\includegraphics [height=5mm]{a0.eps}-\frac{1}{\Phi _2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{\Phi _3}
\includegraphics [height=5mm]{a110.eps}+\frac{q}{2 \,\Phi _2 \Phi _3}\includegraphics [height=5mm]{a200.eps} \\
-\frac{1}{\Phi _2 \Phi _4}
\includegraphics [height=5mm]{a1110.eps}-\frac{q}{2\,\Phi _... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.06056807190179825,
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0.011913653463125229,
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0.018255900591611862,
-0.006865042727440596,
... | |
d3af240b3e99da12a782cb681a28bd56dbf5e0e6 | subsection | 78 | 83 | Morphisms and images | In this section, we consider two quotients of the free pre-Lie algebra
\mathsf {PL} and the images of \Omega _q in these quotients. We will use
some results of . | {
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{
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"doi": "",
"end": 161,
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"start": 132
}
]
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
-0.033751480281353,
0.024276042357087135,
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-0.0... | |
3daeb19bb2751f94e7d028a2c46ccf1a25e51211 | subsection | 79 | 83 | Morphism to the free associative algebra | Consider the free (non-unital) associative algebra on one generator
x, denoted by \mathbb {Q}[x]_+. As the associative product is also a
pre-Lie product, there exists a unique morphism of pre-Lie algebras
from \mathsf {PL} to \mathbb {Q}[x]_+ sending \includegraphics [height=5mm]{a0.eps} to x. This extends uniquely
to ... | {
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"start": 576
}
]
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
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0.007596573326736689,
0.010288933292031288,
-0.03642694279551506,
-0.013171969912946224,
... |
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