Polynomial identities of algebras in positive characteristic

The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p>2 little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case. In this paper we study the Gelfand–Kirillov dimension of the relatively free algebras of verbally prime and related algebras. We compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras M1,1(E) and E⊗E are not PI equivalent in characteristic p>2. Furthermore we show that the following algebras are not PI equivalent in positive characteristic: Ma,b(E)⊗E and Ma+b(E); Ma,b(E)⊗E and Mc,d(E)⊗E when a+b=c+d, a⩾b, c⩾d and a≠c; and finally, M1,1(E)⊗M1,1(E) and M2,2(E). Here E stands for the infinite-dimensional Grassmann algebra with 1, and Ma,b(E) is the subalgebra of Ma+b(E) of the block matrices with blocks a×a and b×b on the main diagonal with entries from E0, and off-diagonal entries from E1; E=E0⊕E1 is the natural grading on E.