GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS

摘要

Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ?, the algebra Mp,q(E) can be turned into a ?p+q × ?2-algebra by combining an elementary ?p+q-grading with the natural ?2-grading on E. The tensor product Mp,q(E) ? Mr,s(E) can be turned into a ?(p+q)(r+s) × ?2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ? Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemer's PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).