Polynomial identities for graded tensor products of algebras

Abstract

Let K be a field, char K = 0 . We study the polynomial identities satisfied by Z 2 -graded tensor products of T-prime algebras. Regev and Seeman proved that in a series of cases such tensor products are PI equivalent to T-prime algebras; they conjectured that this is always the case. We deal here with the remaining cases and thus confirm Regev and Seeman's conjecture. For some “small” algebras we can remove the restriction on the characteristic of the base field, and we show that the behaviour of the corresponding graded tensor products is quite similar to that for the usual (ungraded) tensor products. Finally we consider β-graded tensor products (also called commutation factors) and their identities. We show that Regev's A ⊗ B theorem holds for β-graded tensor products whenever the gradings are by finite abelian groups. Furthermore we study the PI equivalence of β-graded tensor products of T-prime algebras.