Intermediate Growth in Lie Algebras and Their Enveloping Algebras

Abstract

In this paper a series of dimensions is suggested which includes as first terms dimension of a vector space, Gelfand–Kirillov dimension, and superdimension. In terms of these dimensions we describe the change of a growth in transition from a Lie algebra to its universal enveloping algebra. Also, we find the growth of free polynilpotent finitely generated Lie algebras; as an application we specify those algebras with rational Hilbert–Poincaré series. As a corollary we find an asymptotic growth of lower central series ranks for free polynilpotent finitely generated groups.