Graded tensor products

Abstract

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z 2 -graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274–296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M n ( F ) over the field F is regular, which is closely related to M n ( F ) being Z n -graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A ⊗ B theorem for Z 2 -graded PI algebras.