Beilinson Conjecture for Finite-dimensional Associative Algebras

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.

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Cayley Symmetries in Associative Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1963-031-8 ◽

1963 ◽

Vol 15 ◽

pp. 285-290 ◽

Cited By ~ 2

Author(s):

Earl J. Taft

Keyword(s):

Associative Algebra ◽

Associative Algebras ◽

Wedderburn Decomposition ◽

Finite Dimensional ◽

Principal Theorem ◽

Algebra Over A Field

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

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ON INNER DERIVATIONS OF FINITE DIMENSIONAL ASSOCIATIVE ALGEBRAS

Far East Journal of Mathematical Sciences (FJMS) ◽

10.17654/ms102102177 ◽

2017 ◽

Vol 102 (10) ◽

pp. 2177-2188

Author(s):

M. A. Fiidow ◽

N. F. Mohammed ◽

I. S. Rakhimov ◽

Sh. K. Said Husain

Keyword(s):

Associative Algebras ◽

Finite Dimensional

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A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2007/18915 ◽

2007 ◽

Vol 2007 ◽

pp. 1-24 ◽

Cited By ~ 5

Author(s):

Abdenacer Makhlouf

Keyword(s):

Lie Algebras ◽

Associative Algebras ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Noncommutative Version ◽

The Common ◽

Global Deformation ◽

Formal Power ◽

Deformation Approach ◽

Geometric Study

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an “extension” of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.

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On H-simple not necessarily associative algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501627 ◽

2019 ◽

Vol 18 (09) ◽

pp. 1950162

Author(s):

A. S. Gordienko

Keyword(s):

Associative Algebra ◽

Hopf Algebras ◽

Simple Algebra ◽

Polynomial Identities ◽

Associative Algebras ◽

Graded Algebras ◽

Finite Dimensional ◽

Graded Polynomial Identities ◽

Group Gradings ◽

Polynomial Formula

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

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