Lie dimension subrings

Abstract

We compare, for [Formula: see text] a Lie ring over the integers, its lower central series [Formula: see text] and its dimension series defined by [Formula: see text] in the universal enveloping algebra of [Formula: see text]. We show that [Formula: see text] for all [Formula: see text], but give an example showing that they may differ if [Formula: see text]. We introduce simplicial methods to describe these results, and to serve as a possible tool for further study of the dimension series.