The Gel'fand-Kirillov dimension of relatively free prime algebras of arbitrary signature

Abstract

The growth function VA(n) of an algebra A is the dimension of the space spanned by words of length at most n. The Gel’fand–Kirillov dimension of A is the limit GKdim(A) = limn→∞ lnVA(n)/ lnn if it exists. GKdim does not depend on the choice of generators. It has been calculated for the algebra of generic matrices, a free algebra in the variety generated by the Cayley–Dickson algebra O, and the simple exceptional Jordan algebra HC3 in the case when char(K) = 2 or 3 (see [2], [3]). It is calculated in [4], [5] for the algebra of generic matrices and for certain other relatively free algebras such as those in Var(O), along with similar asymptotics (the growth exponent).