Cocharacters and gradedGelfand-Kirillov dimension for PI-algebras

Abstract

One among the most interesting problems in the area of algebra is the Specht problem. In particular, in the case of PI-algebras, the Specht problem has the following form: let A be a PI-algebra over F such that it is finitely generated, then is it true that the T-ideal of A has a finite number of generators as a T-ideal? In a famous work, Kemer solved into affirmative the previous question in the case F is a field of characteristic 0. Now we may ask if there is a general method in order to obtain the generators of the T-ideal of any PI-algebra. Then the answer is merely far to be provided. In fact we just have a few list of PI-algebras with a well known set of generators of their T-ideals. That is why we are going to look for other paths. After a result of Regev, it seems more useful to study the S_n-module of multilinear polynomials that are not polynomial identities for the algebra A, to say, V_n(A). The best path to study the S_n-module V_n(A) is to study the characters of V_n(A) or the cocharacters of A. A similar thing can be said toward the homogeneous polynomials that are not polynomial identities for A and will be a good idea to study the growth of the latter vector space. The responsible researcher already worked and published papers on this topics. He also started collaborations with high level researchers such as Vesselin Drensky (BAS-Bulgary), Onofrio Mario Di Vincenzo (Universit\'adella Basilicata-Italy) and Eli Aljadeff (Technion-Israel). The goal is to obtain a general algorithm in order to compute the cocharacters of one of the most important algebras in the theory, the upper triangular matrices with entries from the infinite dimensional Grassmann algebra. In the same way, we want to develop a theory for the graded Gelfand-Kirillov dimension for graded-semisimple PI-algebras. (AU)