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

2025/04/14 by Fedor Vylegzhanin, Vylegzhanin, Fedor, Yakov Veryovkin +1
Mathematics · #20F12 #20F14 #20F55 #57S12 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2504.10305

openalex publication_date 2025/04/14 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

We study the graded Lie algebra L(RC_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(RC_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.

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