Polynomial identities of algebras in positive characteristic

Abstract

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between these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case.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 M 1 , 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: M a , b ( E ) ⊗ E and M a + b ( E ) ; M a , b ( E ) ⊗ E and M c , d ( E ) ⊗ E when a + b = c + d , a ⩾ b , c ⩾ d and a ≠ c ; and finally, M 1 , 1 ( E ) ⊗ M 1 , 1 ( E ) and M 2 , 2 ( E ) . Here E stands for the infinite-dimensional Grassmann algebra with 1, and M a , b ( E ) is the subalgebra of M a + b ( E ) of the block matrices with blocks a × a and b × b on the main diagonal with entries from E 0 , and off-diagonal entries from E 1 ; E = E 0 ⊕ E 1 is the natural grading on E.