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

Fedor Vylegzhanin, Yakov Veryovkin

Published 2025-04-14, updated 2025-05-13Version 3

We study the graded Lie algebra $L(RC_K)$ associated with the lower central series of a right-angled Coxeter group $RC_K$. We prove that its commutator subalgebra is a quotient of the polynomial ring over an auxiliary Lie subalgebra $N_K$ of the graph Lie algebra $L_K$, and conjecture that the quotient map is an isomorphism. The epimorphism is defined in terms of a new operation in the associated Lie algebra, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra $U(N_K)$ is the mod 2 loop homology algebra of the corresponding moment-angle complex $Z_K$. This allows us to give a presentation of the Lie algebra $N_K$ by generators and relations.