Lie dimension subrings

Abstract

We compare, for L a Lie ring over the integers, its lower central series (γn(L))n≥1 and its dimension series defined by δn(L):=L∩ϖn(L) in the universal enveloping algebra of L. We show that γn(L)=δn(L) for all n≤3, but give an example showing that they may differ if n=4. We introduce simplicial methods to describe these results, and to serve as a possible tool for further study of the dimension series.