The Associated Lie Algebra of a Right-Angled Coxeter Group

Abstract

We study the lower central series of a right-angled Coxeter group \({\rm{R}}{{\rm{C}}_{\cal K}}\) and the associated graded Lie algebra \(L\left( {{\rm{R}}{{\rm{C}}_{\cal K}}} \right)\). The latter is related to the graph Lie algebra \({L_{\cal K}}\). We give an explicit combinatorial description of the first three consecutive factors of the lower central series of the group \({\rm{R}}{{\rm{C}}_{\cal K}}\).