Graded components of the Lie algebra associated with the lower central series of a right-angled Coxeter group

TL;DR

This paper investigates the structure of the graded Lie algebra derived from the lower central series of right-angled Coxeter groups, providing relations and a basis for specific cases with up to four generators.

Contribution

It introduces explicit relations in the graded components and describes a basis for the fourth component for groups with up to four generators.

Findings

Relations in graded components are explicitly obtained.

A basis for the fourth graded component is described for groups with ≤4 generators.

Structural insights into the Lie algebra associated with right-angled Coxeter groups.

Abstract

The lower central series of the rgiht-angled Coxeter group $RC_\mathcal K$ and the corresponding graded Lie algebra $L(RC_\mathcal K)$ associated with the lower central series of a right-angled Coxeter group are studied. Relations are obtained in the graded components of the Lie algebra $L(RC_\mathcal K)$. A basis of the fourth graded component $L(RC_\mathcal K)$ for groups with at most $4$ generators was described.