PI (non)equivalence and Gelfand-Kirillov dimension 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; we showed that the so-called Tensor Product Theorem cannot be extended for infinite fields of positive characteristic p > 2. Furthermore we studied the Gelfand-Kirillov dimension of the relatively free algebras of verbally prime and related algebras. In this paper we compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras Ma,a(E) ⊗ E and M2a(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) ⊗ Mc,d(E) and Mac+bd,ad+cb(E); and Ma,b(E) ⊗ Mc,d (E) and Me, f (E) ⊗ Mg,h(E) when a ≥ b, c ≥ d, e ≥ f, g ≥ h, ac + bd = eg+ f h, ad +bc = eh + fg and ac ≠ eg. 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.