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.