Lie Dimension Subrings

TL;DR

This paper investigates the relationship between the lower central series and the dimension series of Lie rings over integers, showing they coincide for small n but may differ at n=4, and introduces simplicial methods for analysis.

Contribution

It establishes the equality of the lower central and dimension series for n<4 and introduces simplicial techniques for studying these series in Lie rings.

Findings

ment series coincide for n<4

Counterexample at n=4 shows they can differ

Simplicial methods provide new tools for analysis

Abstract

We compare, for L a Lie ring over the integers, its lower central series (\gamma_n(L))_{n>0} and its dimension series defined by \delta_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that \gamma_n(L)=\delta_n(L) for all n<4, 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.