The commutator subalgebra of the Lie algebra associated with a right-angled Coxeter group

Abstract

We study the graded Lie algebra $L(R{C}_K)$ associated with the lower central series of a right-angled Coxeter group. We construct a surjective homomorphism from the polynomial ring over an explicit Lie algebra $N_K$ to the commutator subalgebra of $L(R{C}_K)$, and conjecture that it is an isomorphism. The homomorphism is defined in terms of a new operation in Lie algebras associated with groups generated by involutions, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra $U(N_K)$ is isomorphic to the mod 2 loop homology algebra of the corresponding moment-angle complex $ZK$. This allows us to give a presentation of the Lie algebra $N_K$ by generators and relations.

19 pages, accepted to Moscow Mathematical Journal