Abstract. We consider fine group gradings on the algebra M n (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 M n (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 M n (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.

No abstract

No abstract

We consider the algebra M k (C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by imbedding G in the symmetric group S k via the regular representation and imbedding S k in M k (C) in the usual way. This induces a natural G−grading on M k (C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g 1 , g 2 , . . . , g k ) of distinct elements g i ∈ G. We study the graded polynomial identities for M k (C) equipped with a crossed product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T -ideal of identities and the Amitsur-Levitsky Theorem.Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of M k (C).