Graded Polynomial Identities for Tensor Products by the Grassmann Algebra

Abstract

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ⊗ E, then deduce its G × ℤ2-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ℤ n × ℤ2-graded identities of the algebras M n (E). Moreover we give the graded cocharacter sequence of M 2(E), and show that M 2(E) is PI-equivalent to M 1,1(E) ⊗ E. This fact is a particular case of a more general result obtained by Kemer.