Graded identities of matrix algebras and the universal graded algebra

Abstract

We consider fine group gradings on the algebra Mn(ℂ) of n by n matrices over the complex numbers and the corresponding graded polynomial identities. Given a group G and a fine G-grading on Mn(ℂ), we show that the T-ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra U (depending on the group G and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of U is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of Mn(ℂ). Finally, we consider the ring of central quotients of U which is a central simple algebra over the field of quotients of the center of U. Using earlier results of the authors we show that this is a division algebra if and only if the group G is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example. © 2010 American Mathematical Society.