Moscow Mathematical Journal

Volume 26, Issue 2, April–June 2026 pp. 255–276.

The Commutator Subalgebra of the Lie Algebra Associated with a Right-Angled Coxeter Group

Authors: Yakov Veryovkin (1) and Fedor Vylegzhanin (2)
Author institution:(1) National Research University Higher School of Economics, Moscow, Russia
(2) National Research University Higher School of Economics, Moscow, Russia;
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Summary:

We study the graded Lie algebra $L(RC_{\mathcal K})$ 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 $N_{\mathcal K}$ to the commutator subalgebra of $L(RC_{\mathcal K})$, 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 $U(N_{\mathcal K})$ is isomorphic to the mod $2$ loop homology algebra of the corresponding moment-angle complex $\mathcal{Z}_{\mathcal K}$. This allows us to give a presentation of the Lie algebra $N_{\mathcal K}$ by generators and relations.

2020 Math. Subj. Class. 20F14, 20F12, 20F55; 57S12.

Keywords: Right-angled Coxeter group, lower central series, associated Lie algebra, moment-angle complex, polyhedral product.

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