Graded identities of matrix algebras and the universal graded algebra

Abstract

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of Mn(C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on Mn(C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼= ⊕g∈GRg is a grading and dimk(Rg) ≤ 1. Thus any component is either 0 or isomorphic to k as a k–vector space. It is easy to show that Supp(R), the subset of elements of G for which Rg is not 0, is a subgroup of G. Moreover