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}}\\).