Associative algebras with group action

Abstract

During her visit in IME/USP the Professor of the University of Brasília Irina Sviridova shall work on the research project in the area of non-commutative algebra, ring theory, in collaboration with professors and researchers of IME/USP, particularly with professor Ivan Shestakov.The research project concerns with the study of associative algebras with an action of some groups of automorphisms and anti-automorphisms.The study of group actions on various algebraic and topological structures is one of the principaland most interesting questions of modern mathematics (connected with the study of symmetries,with representation theory, invariant theory, etc.). Associative algebra is a classical objectof mathematics. It is very natural to consider automorphisms and anti-automorphisms of associative algebras or to study algebras with actions of some groups. Although this subject is very important, it has many open problems which are very interesting and have good application prospects. Among these problems is the study of behavior of free algebras and simple algebras with action.A major focus of the project is an effective and constructive description of simple finite dimensional algebras with an action of a group of automorphisms and anti-automorphisms (particularly, for case of an abelian and/or finite group). As a basis of the study can be considered some recent results of several authors (Yu.A.Bakhturin, S.K.Sehgal, M.V.Zaicev, I.P.Shestakov) on descriptions of simple algebras with actions in some particular cases of a group.It is also planned to apply the obtained results for the study of G-identities of associative algebras with actions. Particularly, they can be applied for characterization of G-identities of simple algebras with action by a group G, and to obtain the solution of the problem of PI-representability of finitely generated algebras with action, and of the Specht problem of finite basis for G-identities. (AU)