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2,700 | A new measure of growth for countable-dimensional algebras | math.RA | A new dimension function on countable-dimensional algebras (over a field) is
described. Its dimension values for finitely generated algebras exactly fill
the unit interval $[0,1]$. Since the free algebra on two generators turns out
to have dimension 0 (although conceivably some Noetherian algebras might have
positive d... | math |
2,701 | Orthomodularity in infinite dimensions; a theorem of M. Solèr | math.RA | Maria Pia Sol\`er has recently proved that an orthomodular form that has an
infinite orthonormal sequence is real, complex, or quaternionic Hilbert space.
This paper provides an exposition of her result, and describes its consequences
for Baer $\ast$-rings, infinite-dimensional projective geometries, orthomodular
latti... | math |
2,702 | A number of countable models of a countable supersimple theory | math.RA | In this paper, we prove the number of countable models of a countable
supersimple theory is either 1 or infinite. This result is an extension of
Lachlan's theorem on a superstable theory. | math |
2,703 | The relationship between two commutators | math.RA | We clarify the relationship between the linear commutator and the ordinary
commutator by showing that in any variety satisfying a nontrivial idempotent
Mal'cev condition the linear commutator is definable in terms of the
centralizer relation. We derive from this that abelian algebras are
quasi-affine in such varieties.... | math |
2,704 | Projectivity and isomorphisms of strictly simple algebras | math.RA | We describe a sufficient condition for the localization functor to be a
categorical equivalence. Using this result we explain how to simplify the test
for projectivity. This leads to a description of the strictly simple algebras
which are projective in the variety they generate. A byproduct of our efforts
is the result... | math |
2,705 | Almost orthogonal submatrices of an orthogonal matrix | math.RA | Let $A$ be an $n \times M$ matrix whose rows are orthonormal. Let $A_I$ be a
submatrix of $A$ whose column indexes belong to the set $I$. Given $\epsilon
>0$ we estimate the smallest cardinality of the set $I$, such that the operator
$A_I$ is an $\epsilon$-isometry. | math |
2,706 | Self-rectangulating varieties of type 5 | math.RA | We show that a locally finite variety which omits abelian types is
self-regulating if and only if it has a compatible semilattice term operation.
Such varieties must have a type-set {5}. These varieties are residually small
and, when they are finitely generated, they have definable principal
congruences. We show that i... | math |
2,707 | Tensor product representations for orthosymplectic Lie superalgebras | math.RA | We derive a general result about commuting actions on certain objects in
braided rigid monoidal categories. This enables us to define an action of the
Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the
action of the orthosymplectic Lie superalgebra $\spo(V)$ and the
orthosymplectic Lie color alg... | math |
2,708 | A note on Lascar strong types in simple theories | math.RA | Let T be a countable, small simple theory. In this paper, we prove for such
T, the notion of Lascar Strong type coincides with the notion of a strong
type,over an arbitrary set. | math |
2,709 | Modularity prevents tails | math.RA | We establish a direct correspondence between two congruence poroperties for
finite algebras. The first property is that minimal sets of type i omit tails.
The second property is that congruence lattices omit pentagons of type i. | math |
2,710 | Type 4 is not computable | math.RA | We extend a recent result of McKenzie, and show that it is an undecidable
problem to determine if 4 appears in the typeset of a finitely generated,
locally finite variety. | math |
2,711 | Closures in $\aleph_0$-categorical bilinear maps | math.RA | Alternating bilinear maps with few relations allow to define a combinatorial
closure similarly as in [2]. For the $\aleph_0$-categorical case we show that
this closure is part of the algebraic closure. | math |
2,712 | Rigid analytic flatificators | math.RA | Let K be an algebraically closed field endowed with a complete
non-archimedean norm. Let f:Y -> X be a map of K-affinoid varieties. We prove
that for each point x in X, either f is flat at x, or there exists, at least
locally around x, a maximal locally closed analytic subvariety Z in X
containing x, such that the base... | math |
2,713 | A theorem on spherically complete valued abelian groups | math.RA | We give a criterion for a group homomorphism on a valued abelian group to be
surjective and to preserve spherical completeness. We apply this to give a
criterion for the existence of integration on a valued differential field.
Further, we give a criterion for a sum of spherically complete subgroups of a
valued abelian ... | math |
2,714 | Z_8 is not dualizable | math.RA | In this paper we show that Z_8 does not admit a natural duality. In fact, we
show that 2Z_8 = {2, 4, 6, 8 | +,.} is not dualizable, and this will imply that
the original ring is not dualizable, either. As a corollary we show that
Sindi's conjecture does not hold. Our technique will be similar to one due to
Quackenbush ... | math |
2,715 | The Pfaffian closure on an o-minimal structure | math.RA | Every o-minimal expansion R-tilde of the real field has an o-minimal
expansion P(R-tilde) in which the solutions to Pfaffian equations with
definable C^1 coefficients are definable. | math |
2,716 | Complexity problems associated with matrix rings, matrix semigroups and Rees matrix semigroups | math.RA | Complexity problems associated with finite rings and finite semigroups,
particularly semigroups of matrices over a field and the Rees matrix
semigroups, are examined. Let M_nF be the ring of n x n matrices over the
finite field F and let T_nF be the multiplicative semigroup of n x n matrices
over the finite field F. It... | math |
2,717 | Primitive representations of finite semigroups I | math.RA | Primitive representations of finite groups as well as primitive finite groups
were classified in the O'Nan-Scott Theorem. In this paper we classify faithful
finite primitive semigroup representations. To each finite primitive
representation, we associate an invariant, a finite dimensional matirix with
entries in a prim... | math |
2,718 | A finite basis theorem for residually finite, congruence meet-semidistributive varieties | math.RA | We derive a Mal'cev condition for congruence meet-semidistributivity and then
use it to prove two theorems. Theorem A: if a variety in a finite language is
congruence meet-semidistributive and residually less than some finite cardinal,
then it is finitely based. Theorem B: there is an algorithm which, given m<w
and a f... | math |
2,719 | Associative algebras satisfying a semigroup identity | math.RA | Denote by (R,.) the multiplicative semigroup of an associative algebra R over
an infinite field, and let (R,*) represent R when viewed as a semigroup via the
circle operation x*y=x+y+xy. In this paper we characterize the existence of an
identity in these semigroups in terms of the Lie structure of R. Namely, we
prove t... | math |
2,720 | Bounds on norms of compound matrices and on products of eigenvalues | math.RA | An upper bound on operator norms of compound matrices is presented, and
special cases that involve the $\ell_1$, $\ell_2$ and $\ell_\infty$ norms are
investigated. The results are then used to obtain bounds on products of the
largest or smallest eigenvalues of a matrix. | math |
2,721 | On matrices for which norm bounds are attained | math.RA | Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$
columns by the H\"older $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or
$C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio
$\|A\|_{r,s}/\|A\|_{p,q}$. In this paper we study the classes of matrices for
which the... | math |
2,722 | On the $n$-ary algebras, semigroups and their universal covers | math.RA | For any $n$-ary associative algebra we construct a $\Z_{n-1}$ graded algebra,
which is a universal object containing the $n$-ary algebra as a subspace of
elements of degree 1. Similar construction is carried out for semigroups. | math |
2,723 | Modules Whose Small Submodules Have Krull Dimension | math.RA | The main aim of this paper is to show that an AB5*-module whose small
submodules have Krull dimension has a radical having Krull dimension. The proof
uses the notion of dual Goldie dimension. | math |
2,724 | On Semilocal Modules and Rings | math.RA | It is well-known that a ring R is semiperfect if and only if R as a left (or
as a right) R-module is a supplemented module. Considering weak supplements
instead of supplements we show that weakly supplemented modules M are semilocal
(i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if R
as a lef... | math |
2,725 | Hermitian Positive Semidefinite Matrices Whose Entries Are 0 Or 1 in Modulus | math.RA | We show that a matrix is a Hermitian positive semidefinite matrix whose
nonzero entries have modulus 1 if and only if it similar to a direct sum of all
$1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular,
the only such nonsingular matrix is the identity matrix and the only such
irreducible ma... | math |
2,726 | The Octonionic Eigenvalue Problem | math.RA | We discuss the eigenvalue problem for 2x2 and 3x3 octonionic Hermitian
matrices. In both cases, we give the general solution for real eigenvalues, and
we show there are also solutions with non-real eigenvalues. | math |
2,727 | Principal pivot transforms: properties and applications | math.RA | The principal pivot transform (PPT) of a matrix A partitioned relative to an
invertible leading principal submatrix is a matrix B such that
A [x_1^T x_2^T]^T = [y_1^T y_2^T]^T
if and only if
B [y_1^T x_2^T]^T = [x_1^T y_2^T]^T,
where all vectors are partitioned conformally to A. The purpose of this paper
is to ... | math |
2,728 | Finding Octonionic Eigenvectors Using Mathematica | math.RA | The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some
surprises, which we have reported elsewhere. In particular, the eigenvalues
need not be real, there are 6 rather than 3 real eigenvalues, and the
corresponding eigenvectors are not orthogonal in the usual sense. The
nonassociativity of the octon... | math |
2,729 | Polycyclic-by-finite group algebras are catenary | math.RA | We show that group algebras kG of polycyclic-by-finite groups G, where k is a
field, are catenary: Given prime ideals P and P' of kG, with P contained in P',
all saturated chains of primes between P and P' have the same length. | math |
2,730 | On the Splitting of the Dual Goldie Torsion Theory | math.RA | The splitting of the Goldie (or singular) torsion theory has been extensively
studied. Here we determine an appropriate dual Goldie torsion theory, discuss
its splitting and answer in the negative a question proposed by Ozcan and
Harmanci as to whether the splitting of the dual Goldie torsion theory implies
the ring to... | math |
2,731 | Supersimple fields and division rings | math.RA | It is proved that any supersimple field has trivial Brauer group, and more
generally that any supersimple division ring is commutative. As prerequisites
we prove several results about generic types in groups and fields whose theory
is simple. | math |
2,732 | Lascar and Morley ranks differ in differentially closed fields | math.RA | We note here, in answer to a question of Poizat, that the Morley and Lascar
ranks need not coincide in differentially closed fields. We approach this
through the (perhaps) more fundamental issue of the variation of Morley rank in
families. | math |
2,733 | On finite homomorphic images of the multiplicative group of a division algebra | math.RA | This paper, together with a forthcoming paper by the author and Seitz, proves
the Margulis-Platonov conjecture concerning the normal subgroup structure of
algebraic groups over number fields, in the case of inner forms of anisotropic
groups of type $A_n$. | math |
2,734 | Growth and Relations in Graded Rings | math.RA | Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal
generated by a set $\alpha$ of homogeneous elements, and B = A/I. In this note,
some inequalities between Hilbert series of algebras $A,B$ and the number of
elements of the set $\alpha$ are announced. As in the Golod--Shafarevich
inequality as i... | math |
2,735 | One-sided Noncommutative Groebner Bases with Applications to Green's Relations | math.RA | Standard noncommutative Gr\"obner basis procedures are used for computing
ideals of free noncommutative polynomial rings over fields. This paper
describes Gr\"obner basis procedures for one-sided ideals in finitely presented
noncommutative algebras over fields. The polynomials defining a $K$-algebra $A$
as a quotient o... | math |
2,736 | Enumeration of Finite Rings with Jacobson Radical of Cube Zero | math.RA | In [1], finite associative rings wih identity and such that the set of all
zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and
square non-zero, were constructed for all the characteristics. These rings are
completely primary and we call them rings with property(T). In this paper, we
associate ... | math |
2,737 | The graded version of Goldie's Theorem | math.RA | The analogue of Goldie's Theorem for prime rings is proved for rings graded
by abelian groups, eliminating unnecessary additional hypotheses used in
earlier versions. | math |
2,738 | Generalized Selective Modal Analysis | math.RA | A new approach which generalizes the Selective Modal Analyis (SMA) and
algorithms based upon it for solving the generalized eigenvalue problem is
described. This approach allows for the systematic consideration of physical
properties of the system under study. Two small application cases demonstrate
the capabilities of... | math |
2,739 | Graded Lie Algebras of Maximal Class II | math.RA | We describe the isomorphism classes of infinite-dimensional graded Lie
algebras of maximal class, generated by elements of weight one, over fields of
odd characteristic. | math |
2,740 | Graded Lie algebras of maximal class IV | math.RA | We describe the isomorphism classes of certain infinite-dimensional graded
Lie algebras of maximal class, generated by an element of weight one and an
element of weight two, over fields of odd characteristic. | math |
2,741 | Nahm Algebras | math.RA | Given a Lie algebra $\frak{g}$, the \emph{Nahm algebra} of $\frak{g}$ is the
vector space $\frak{g}\times \frak{g}\times \frak{g}$ with the natural
commutative, nonassociative algebra structure associated with the Nahm
equations $\dot{x} = [y,z]$, $\dot{y} = [z,x]$, $\dot{z} = [x,y]$. Motivated by
potential application... | math |
2,742 | Hecke Algebras, SVD, and Other Computational Examples with {\sc CLIFFORD} | math.RA | {\sc CLIFFORD} is a Maple package for computations in Clifford algebras $\cl
(B)$ of an arbitrary symbolic or numeric bilinear form B. In particular, B may
have a non-trivial antisymmetric part. It is well known that the symmetric part
g of B determines a unique (up to an isomorphism) Clifford structure on
$\cl(B)$ whi... | math |
2,743 | FP-injective and weakly quasi-Frobenius rings | math.RA | The classes of FP-injective and weakly quasi-Frobenius rings are
investigated. The properties for both classes of rings are closely linked with
embedding of finitely presented modules in fp-flat and free modules
respectively. Using these properties, we describe the classes of coherent CF
and FGF-rings. Moreover, it is ... | math |
2,744 | Index of Hadamard multiplication by positive matrices II | math.RA | Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal
index of A: $I(A) = \max \{\lambda \ge 0 : A\circ B \ge \lambda B$ for all
$0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} =
A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the
N-index of A: $I... | math |
2,745 | Bass's Work in Ring Theory and Projective Modules | math.RA | The early papers of Hyman Bass in the late 50s and the early 60s leading up
to his pioneering work in algebraic K-theory have played an important and very
special role in ring theory and the theory of projective (and injective)
modules. In this article, we give a general survey of Bass's fundamental
contributions in th... | math |
2,746 | Interpolation in ortholattices | math.RA | If L is a complete ortholattice, f any partial function from L^n to L, then
there is a complete ortholattice L* containing L as a subortholattice, and an
ortholattice polynomial with coefficients in L* which represents f on L^n.
Iterating this construction long enough yields a complete ortholattice in
which every fun... | math |
2,747 | On the degrees of irreducible representations of Hopf algebras | math.RA | Let H denote a semisimple Hopf algebra over an algebraically closed field k
of characteristic 0. We show that the degree of any irreducible representation
of H whose character belongs to the center of H^* must divide the dimension of
H . | math |
2,748 | Matrix Representations of Octonions and Their Applications | math.RA | As is well-known, the real quaternion division algebra $ {\cal H}$ is
algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division
octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix
algebras over the real number field ${\cal R}$, because ${\cal O}$ is a
non-associative alge... | math |
2,749 | Rank Equalities Related to Generalized Inverses of Matrices and Their Applications | math.RA | This paper is divided into two parts. In the first part, we develop a general
method for expressing ranks of matrix expressions that involve Moore-Penrose
inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose
inverses of matrices. Through this method we establish a variety of valuable
rank equali... | math |
2,750 | Matrix Theory over the Complex Quaternion Algebra | math.RA | We present in this paper some fundamental tools for developing matrix
analysis over the complex quaternion algebra. As applications, we consider
generalized inverses, eigenvalues and eigenvectors, similarity, determinants of
complex quaternion matrices, and so on. | math |
2,751 | Finite Groups Embeddable in Division Rings | math.RA | Finite groups that are embeddable in the multiplicative groups of division
rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has
characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic
groups, according to a theorem of I. N. Herstein. Thus, the only interesting
case is when... | math |
2,752 | Jonssons's theorem in non-modular varieties | math.RA | A version of Jonsson's theorem, as previously generalized, holds in
non-modular varieties. | math |
2,753 | Regular coverings in filter and ideal lattices | math.RA | The Dedekind-Birkhoff theorem for finite-height modular lattices has
previously been generalized to complete modular lattices using the theory of
regular coverings. In this paper, we investigate regular coverings in lattices
of filters and lattices of ideals, and the regularization strategy--embedding
the lattice into ... | math |
2,754 | Abelian extensions of algebras in congruence-modular varieties | math.RA | We define abelian extensions of algebras in congruence-modular varieties. The
theory is sufficiently general that it includes, in a natural way, extensions
of R-modules for a ring R. We also define a cohomology theory, which we call
clone cohomology, such that the cohomology group in dimension one is the group
of equiv... | math |
2,755 | Algebras with a compatible uniformity | math.RA | Given a variety of algebras V, we study categories of algebras in V with a
compatible structure of uniform space. The lattice of compatible uniformities
of an algebra, Unif A, can be considered a generalization of the lattice of
congruences Con A. Mal'cev properties of V influence the structure of Unif A,
much as they ... | math |
2,756 | Octonionic Hermitian Matrices with Non-Real Eigenvalues | math.RA | We extend previous work on the eigenvalue problem for Hermitian octonionic
matrices by discussing the case where the eigenvalues are not real, giving a
complete treatment of the 2x2 case, and summarizing some prelimenary results
for the 3x3 case. | math |
2,757 | Algebras without noetherian filtrations | math.RA | We provide examples of finitely generated noetherian PI algebras for which
there is no finite dimensional filtration with a noetherian associated graded
ring; thus we answer negatively a question raised by M. Lorenz. | math |
2,758 | Nonfiliform characteristically nilpotent Lie algebras | math.RA | We construct large families of characteristically nilpotent Lie algebras by
considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and
which arises as a central extension fo the filiform Lie algebra L_{n}. By
studying the graded cohomology spaces we obtain that the sill algebras are
isomorphic to the n... | math |
2,759 | Computing homomorphisms between holonomic D-modules | math.RA | Let K be a subfield of the complex numbers, and let D be the Weyl algebra of
K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic
left D-modules we present an algorithm that computes explicit generators for
the finite dimensional vector space hom_D(M,N). This enables us to answer
algorithmically ... | math |
2,760 | On k-abelian, p-filiform Lie algebras | math.RA | We classify the (n-5)-filiform Lie algebras which have the additional
property of a non-abelian derived subalgebra. We show that this property is
strongly related with the structure of the Lie algebra of derivations;
explicitely we show that if a (n-5)-filiform algebra is characteristically
nilpotent, then it must be 2... | math |
2,761 | Wedderburn Polynomials over Division Rings | math.RA | A Wedderburn polynomial over a division ring K is a minimal polynomial of an
algebraic subset of K. Special cases of such polynomials include, for instance,
the minimal polynomials (over the center F=Z(K)) of elements of K that are
algebraic over F. In this note, we give a survey on some of our ongoing work on
the stru... | math |
2,762 | Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras | math.RA | In this article we prove that for a basic classical Lie superalgebra the
annihilator of a strongly typical Verma module is a centrally generated ideal.
For a basic classical Lie superalgebra of type I we prove that the localization
of the enveloping algebra by a certain central element is free over its centre. | math |
2,763 | On certain families of naturally graded Lie algebras | math.RA | In this work large families of naturally graded nilpotent Lie algebras in
arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying
the centralizer property, are given. This condtion constitutes a
generalization, for a nilpotent Lie agebra, of the structural properties
charactrizing the Lie algebr... | math |
2,764 | On weight graphs for nilpotent Lie algebras I | math.RA | We introduce the concept of weight graph for the weight system $P\frak{g}(T)$
of a finite dimensional nilpotent Lie algebra $\frak{g}$ and analyze the
necessary conditions for a $(p,q)$-graph to be a weight graph for some
$\frak{g}$. | math |
2,765 | Some Properties of 3x3 Octonionic Hermitian Matrices with Non-Real Eigenvalues | math.RA | We discuss our preliminary attempts to extend previous work on 2x2 Hermitian
octonionic matrices with non-real eigenvalues to the 3x3 case. | math |
2,766 | Duality and Rational Modules in Hopf Algebras over Commutative Rings | math.RA | Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and
$A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules
and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can
also strengthen the Blattner-Montgomery duality theorem. Finally, we give
sufficient con... | math |
2,767 | Characteristically nilpotent Lie algebras : a survey | math.RA | We review the known results about characteristically nilpotent complex Lie
algebras, as well as we comment recent developements in the theory. | math |
2,768 | On the determination of 2-step solvable Lie algebra from its weight graph | math.RA | By using the concept of weight graph associated to certain nilpotent Lie
algebras $\frak{g}$, we find necessary and sufficient conditions for a
semidirect product $\frak{g}\oplus T_{i}$, where $T_{i}<T$ is a subalgebra of a
maximal torus of derivations $T$ of $\frak{g}$ which induces a decomposition of
$\frak{g}$ into ... | math |
2,769 | Integration on Lie supergroups. A Hopf superalgebra approach | math.RA | For a large class of finite-dimensional Lie superalgebras (including the
classical simple ones) a Lie supergroup associated to the algebra is defined by
fixing the Hopf superalgebra of functions on the supergroup. Then it is shown
that on this Hopf superalgebra there exists a non-zero left integral. According
to a rece... | math |
2,770 | On a generic inverse differential Galois problem for GL_n | math.RA | \newcommand{\GLn}{\operatorname{GL}_n} \newcommand{\GL}{\GLn(C)}
Let $F$ be a differential field with algebraically closed field of constants
$C$. We prove that $F< Y_{ij}>(X_{ij})\supset F< Y_{ij}>$ is a generic
Picard-Vessiot extension of $F$ for $\GL$. If $E\supset F$ is any
Picard-Vessiot extension with different... | math |
2,771 | Classification of (n-5)-filiform Lie algebras | math.RA | In this paper we consider the problem of classifying the $(n-5)$-filiform Lie
algebras. This is the first index for which infinite parametrized families
appear, as can be seen in dimension $7.$ Moreover we obtain large families of
characteristic nilpotent Lie algebras with nilpotence index 5 and show that at
least for ... | math |
2,772 | Exact interval solutions to the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra | math.RA | In this note we construct a solution of a matrix interval linear equation of
the form X=AX+B (the discrete stationary Bellman equation) over partially
ordered semirings, including the semiring of nonnegative real numbers and all
idempotent semirings. We discuss also the computational complexity of problems
in interval ... | math |
2,773 | Using noncommutative Groebner bases in solving partially prescribed matrix inverse completion problems | math.RA | We investigate the use of noncommutative Groebner bases in solving partially
prescribed matrix inverse completion problems. The types of problems considered
here are similar to those in [BLJW]. There the authors gave necessary and
sufficient conditions for the solution of a two by two block matrix completion
problem. O... | math |
2,774 | On the product by generators of characteristically nilpotent Lie S-algebras | math.RA | We introduce the product by generators of complex nilpotent Lie algebras,
which is a commutative product obtained from a central extension of the direct
sum of Lie algebras. We show that the product preserves also the characteristic
nilpotence provided that the multiplied algebras are $S$-algebras. In
particular, this ... | math |
2,775 | An approach to Hopf algebras via Frobenius coordinates II | math.RA | We study a Hopf algebra $H$, which is finitely generated and projective over
a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions
in this setting, and provide a complete proof of Radford's formula for the
fourth power of the antipode, using Frobenius algebraic techniques. As further
applicati... | math |
2,776 | Modulization and the enveloping ringoid | math.RA | Let A be an algebra in a variety V. We study the modulization of a pointed
A-overalgebra P, show that it is totally in any variety that P is totally in,
and apply this theory to the construction of the enveloping ringoid Z[A,V]. | math |
2,777 | Are biseparable extensions Frobenius? | math.RA | In Secion~1 we describe what is known of the extent to which a separable
extension of unital associative rings is a Frobenius extension. A problem of
this kind is suggested by asking if three algebraic axioms for finite Jones
index subfactors are dependent. In Section~2 the problem in the title is
formulated in terms o... | math |
2,778 | The Structure of the Inverse to the Sylvester Resultant Matrix | math.RA | Given polynomials a(z) of degree m and b(z) of degree n, we represent the
inverse to the Sylvester resultant matrix of a(z) and b(z), if this inverse
exists, as a canonical sum of m+n dyadic matrices each of which is a rational
function of zeros of a(z) and b(z). As a result, we obtain the polynomial
solutions X(z) of ... | math |
2,779 | On an invariant related to a linear inequality | math.RA | Let A be an m-dimensional vector with positive real entries. Let A_{i,j} be
the vector obtained from A on deleting the entries A_i and A_j. We investigate
some invariant and near invariants related to the solutions E (m-2 dimensional
vectors with entries either +1 or -1) of the linear inequality |A_i-A_j| <
<E,A_{i,J}>... | math |
2,780 | Generalized Projection Operators in Geometric Algebra | math.RA | Given an automorphism and an anti-automorphism of a semigroup of a Geometric
Algebra, then for each element of the semigroup a (generalized) projection
operator exists that is defined on the entire Geometric Algebra. A single
fundamental theorem holds for all (generalized) projection operators. This
theorem makes previ... | math |
2,781 | On deformations of the filiform Lie superalgebra $L_{n,m}$ | math.RA | In this work, we recall that every filiform Lie superalgebra is a deformation
of the superalgebra $L_{n,m}$. We study the even cocycles which give this
nilpotent deformations. A family of independent bilinear maps will help us to
describe this cocycles. At the end an evaluation of the dimension of the space
$Z_0^2(L_{n... | math |
2,782 | Some Explicit Solutions of the Additive Deligne-Simpson Problem and Their Applications (Preprint) | math.RA | In this paper we construct three infinite series and two extra triples of
complex matrices B, C, and A=B+C of special spectral types associated to C.
Simpson's classification in his paper ``Products of Matrices'' and a
classification of multiple flag varieties with finitely many orbits of the
diagonal action of the gen... | math |
2,783 | Orthonormal Eigenbases over the Octonions | math.RA | We previously showed that the real eigenvalues of 3x3 octonionic Hermitian
matrices form two separate families, each containing 3 eigenvalues, and each
leading to an orthonormal decomposition of the identity matrix, which would
normally correspond to an orthonormal basis. We show here that it nevertheless
takes both fa... | math |
2,784 | Counting equivalence classes of irreducible representations | math.RA | Let $n$ be a positive integer, and let $R$ be a (possibly infinite
dimensional) finitely presented algebra over a computable field of
characteristic zero. We describe an algorithm for deciding (in principle)
whether $R$ has at most finitely many equivalence classes of $n$-dimensional
irreducible representations. When $... | math |
2,785 | Frobenius Functors of the second kind | math.RA | A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second
type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences
$\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide
examples of Frobenius pairs of the second kind. We study Frobenius pairs of the
second ki... | math |
2,786 | A note on the classification of naturally graded Lie algebras with linear characteristic sequence | math.RA | For sufficiently high dimensions, the naturally graded nonsplit nilpotent Lie
algebras with linear characteristic sequence are classified. | math |
2,787 | Simple roots of deformed preprojective algebras | math.RA | W. Crawley-Boevey has given a description of the set of dimension vectors of
simple representations of the deformed preprojective algebras. In this note we
give alternative descriptions. Note however that our descriptions depend on
irreducibility of the quotient varieties, so they do NOT give a shorter proof
of Crawley... | math |
2,788 | Parametrization by polytopes of intersections of orbits by conjugation | math.RA | Let S be an nXn real symmetric matrix with spectral decomposition S=Q^T
Lambda Q, where Q is an orthogonal matrix and Lambda is diagonal with simple
spectrum {lambda_1,..., lambda_n}. Also let O_S e R_S be the orbits by
conjugation of S by, respectively, orthogonal matrices and upper triangular
matrices with positive d... | math |
2,789 | On gradings of matrix algebras and descent theory | math.RA | We classify gradings on matrix algebras by a finite abelian group. A grading
is called good if all elementary matrices are homogeneous. For cyclic groups,
all gradings on a matrix algebra over an algebraically closed field are good.
We can count the number of good gradings by a cyclic group. Using descent
theory, we cl... | math |
2,790 | Simple completable contractions of nilpotent Lie algebras | math.RA | We study a certain class of non-maximal rank contractions of the nilpotent
Lie algebra $\frak{g}_{m}$ and show that these contractions are completable Lie
algebras. As a consequence a family of solvable complete Lie algebras of
non-maximal rank is given in arbitrary dimension. | math |
2,791 | Strongly graded hereditary orders | math.RA | Let R be a Dedekind domain with global quotient field K. The purpose of this
note is to provide a characterization of when a strongly graded R-order with
semiprime 1-component is hereditary. This generalizes earlier work by the first
author and G. Janusz (Trans. Amer. Math. Soc. 352 (2000), 3381-3410). | math |
2,792 | Applications of Perron-Frobenius Theory to Population Dynamics | math.RA | By the use of Perron-Frobenius theory, simple proofs are given of the
Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the
net reproductive rate occurring in matrix models of population dynamics. The
latter result is further refined with some additional nonnegative matrix
theory. When the fer... | math |
2,793 | Dualizing Complexes and Tilting Complexes over Simple Rings | math.RA | We prove that two-sided tilting complexes, and dualizing complexes, over
simple Goldie rings (with some technical conditions) are always shifts of
invertible bimodules. This allows us to describe the derived Picard groups of
such rings, and to deduce these are Gorenstein (and sometimes even
Auslander-Gorenstein Cohen-M... | math |
2,794 | Automorphisms of tiled orders | math.RA | Let Lambda be a tiled R-order. We give a description of Aut_R(Lambda) as the
semidirect product of Inn(Lambda) and a certain subgroup of Aut(Q(Lambda)),
where Q(Lambda) is the link graph of Lambda. Additionally, we give criteria for
determining when an element of Aut(Q(Lambda)) belongs to this subgroup in terms
of the ... | math |
2,795 | Fields of definition for division algebras | math.RA | Let $A$ be a finite-dimensional division algebra containing a base field $k$
in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if
$A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that:
(1) In many cases $A$ can be defined over a rational extension of $k$. (2) If
$A$ has ... | math |
2,796 | $K_0$ of purely infinite simple regular rings | math.RA | We extend the notion of a purely infinite simple C*-algebra to the context of
unital rings, and we study its basic properties, specially those related to
K-Theory. For instance, if $R$ is a purely infinite simple ring, then
$K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated
projective $R$-modul... | math |
2,797 | Contractions and generalized Casimir invariants | math.RA | We prove that if $\frak{g}^{\prime}$ is a contraction of a Lie algebra
$\frak{g}$ then the number of functionally independent invariants of
$\frak{g}^{\prime}$ is at least that of $\frak{g}$. This allows to determine
explicitly the number of invariants of Lie algebras carrying a supplementary
structure, such as linear ... | math |
2,798 | Balanced d-lattices are complemented | math.RA | We show that all balanced d-lattices must be complemented, answering a
question of Chajda and Eigenthaler.
(A bounded lattice is balanced if any two congruences agree on their
1-classes iff they agree on their 0-classes.)
Our main tool is the characterization of d-lattices (a class of bounded
lattices including the... | math |
2,799 | Maschke functors, semisimple functors and separable functors of the second kind. Applications | math.RA | We introduce separable functors of the second kind (or $H$-separable
functors) and $H$-Maschke functors. $H$-separable functors are generalizations
of separable functors. Various necessary and sufficient conditions for a
functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir
elements (integrals... | math |
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