Graded identities for tensor products of matrix (super)algebras over the Grassmann algebra

In this paper we study the graded identities satisfied by the superalgebras Ma,b over the Grassmann algebra and by their tensor products. These algebras play a crucial role in the theory developed by A. Kemer that led to the solution of the long standing Specht problem. It is well known that over a field of characteristic 0, the algebras Mpr+qs,ps+qr and Mp,q⊗Mr,s satisfy the same ordinary polynomial identities. By means of describing the corresponding graded identities we prove that the T-ideal of the former algebra is contained in the T-ideal of the latter. Furthermore the inclusion is proper at least in case (r,s)=(1,1). Finally we deal with the graded identities satisfied by algebras of type M2n-1,2n-1 and relate these graded identities to the ones of tensor powers of the Grassmann algebra. Our proofs are combinatorial and rely on the relationship between graded and ordinary identities as well as on appropriate models for the corresponding relatively free graded algebras.