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

Fedor Vylegzhanin, Yakov Veryovkin

TL;DR

This work analyzes the commutator structure of the Lie algebra associated with a right-angled Coxeter group by embedding $L'(RC_K)$ into a graded polynomial extension over a distinguished subalgebra $N_K$ of the graph Lie algebra $L_K$. A squaring-inspired operation $h$ yields an epimorphism $\psi:N_K[t]\to L'(RC_K)$, and the authors conjecture that this map is an isomorphism; they prove surjectivity and, in key cases, establish isomorphism, linking the algebra to the topology of moment-angle complexes via $U(N_K)\cong H_*(\Omega Z_K;\mathbb{Z}_2)$. This topological connection enables a presentation of $N_K$ by GPTW generators and relations, and yields a Poincaré-series identity that encodes the first three graded components. They further relate loop-homology algebras of polyhedral products to partially commutative Hopf algebras, providing a framework in which the combinatorial data of $\mathcal{K}$ (notably chordality of $\mathcal{K}^1$) governs freeness of $N_K$ and the structure of $L'(RC_K)$. The results bridge Lie-theoretic properties of RC_K with toric topology, offering a conjectural canonical presentation for the commutator subalgebra and illuminating the role of GPTW generators in both algebraic and topological contexts.

Abstract

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.

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

TL;DR

This work analyzes the commutator structure of the Lie algebra associated with a right-angled Coxeter group by embedding into a graded polynomial extension over a distinguished subalgebra of the graph Lie algebra . A squaring-inspired operation yields an epimorphism , and the authors conjecture that this map is an isomorphism; they prove surjectivity and, in key cases, establish isomorphism, linking the algebra to the topology of moment-angle complexes via . This topological connection enables a presentation of by GPTW generators and relations, and yields a Poincaré-series identity that encodes the first three graded components. They further relate loop-homology algebras of polyhedral products to partially commutative Hopf algebras, providing a framework in which the combinatorial data of (notably chordality of ) governs freeness of and the structure of . The results bridge Lie-theoretic properties of RC_K with toric topology, offering a conjectural canonical presentation for the commutator subalgebra and illuminating the role of GPTW generators in both algebraic and topological contexts.

Abstract

We study the graded Lie algebra associated with the lower central series of a right-angled Coxeter group . We prove that its commutator subalgebra is a quotient of the polynomial ring over an auxiliary Lie subalgebra of the graph Lie algebra , 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 is the mod 2 loop homology algebra of the corresponding moment-angle complex . This allows us to give a presentation of the Lie algebra by generators and relations.

Paper Structure

This paper contains 18 sections, 28 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Simplicial complexes and graphs
  5. The Lie algebra associated with a group
  6. Right-angled Coxeter groups
  7. Nested commutators
  8. Properties of nested commutators
  9. General properties
  10. The squaring operation
  11. Computations in the associated Lie algebra
  12. The algebra NK and the surjection of NK[t] onto the commutator subalgebra
  13. Examples and computations
  14. First three graded components
  15. Polyhedral products and loop homology
  16. ...and 3 more sections

Key Result

Proposition 2.1

The square of any element of $\gamma_k(\hbox{\it RC}_\mathcal{K})$ is contained in $\gamma_{k+1}(\hbox{\it RC}_\mathcal{K})$. Hence $L(\hbox{\it RC}_\mathcal{K})$ is a Lie algebra over $\mathbb{Z}_2$.∎

Theorems & Definitions (66)

  • Proposition 2.1: veryovkin
  • Proposition 2.2: veryovkin
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: cf. vylegzhanin
  • Definition 2.6: vylegzhanin
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 56 more