The centre of generic algebras of small PI algebras

Abstract

Verbally prime algebras are important in PI theory. They are well known over a field K of characteristic zero: 0 and K〈T〉 (the trivial ones), Mn(K), Mn(E), Mab(E). Here K〈T〉 is the free associative algebra with free generators T, E is the infinite dimensional Grassmann algebra over K, Mn(K) and Mn(E) are the n×n matrices over K and over E, respectively. Moreover Mab(E) are certain subalgebras of Ma+b(E), defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of Mn(K). The situation is rather unclear for the remaining nontrivial verbally prime algebras.In this paper we study the centre of the generic algebra of M11(E) in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.