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 differencesbetween these two cases for the characteristic, and we showed that the so-called Tensor Product Theorem is in part no longer valid in the second case.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 $E\otimes E$ are not PI equivalent in characteristic $p>2$. Furthermore we show that that the following algebras are not PI equivalent in positive characteristic: $M_{a,b}(E)\otimes E$ and $M_{a+b}(E)$;$M_{a,b}(E)\otimes E$ and $M_{c,d}(E)\otimes E$ when $a+b=c+d$, $a\ge b$, $c\ge d$ and $a\ne c$; and finally, $M_{1,1}(E)\otimes M_{1,1}(E)$ 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 $a\times a$ and $b\times b$ on the main diagonal with entries from $E_0$, andoff-diagonal entries from $E_1$; $E=E_0\oplus E_1$ is the natural grading on $E$.