Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z2Z2-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 Mn(F)Mn(F) over the field FF is regular, which is closely related to Mn(F)Mn(F) being ZnZn-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⊗BA⊗B theorem for Z2Z2-graded PI algebras.