Polynomial identities and their numerical invariants

Abstract

We will study some topics in the Theory of algebras satisfying polynomial identities in this proposal for a Master Degree. We will approach the basics of the theory, namely polynomial identities, PI algebras, varieties of algebras, T-ideals, the linearization process, the homogeneous and the multilinear structures of relatively free algebras. To this end we shall need some facts from Ring theory such that the theorems due to Wedderburn and Artin, Wedderburn and Malcev, and also the theory of representations of the symmetric and of the general linear group. We shall cover the topics in the depth needed for our project. Moreover we shall study the notions of a codimension, codimension growth. We shall introduce the notion of the PI exponent of an algebra. We shall see the celebrated theorem due to A. Regev about the exponential growth of the codimensions, and we shall see how to apply it in order to prove another theorem due to Regev: that the tensor product of two PI algebras is once again a PI algebra. Afterwards we shall cover the theorem due to Giambruno and Zaicev that shows the PI exponent of an associative PI algebra in characteristic 0 always exists and is a nonnegative integer. Finally we shall describe the PI algebras whose codimensions are of polynomial growth.