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Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups

Chaithra P, Deniz Kus, R. Venkatesh

TL;DR

Addressing the problem of determining root multiplicities in Borcherds-Kac-Moody (BKM) superalgebras, the paper develops a combinatorial framework based on partially commutative Lie superalgebras (PCLSAs) and marked graphs, introducing marked multi-colorings and the heaps monoid to derive a denominator identity and compute root multiplicities via Möbius inversion. It provides an explicit description of the root set Δ^G in graph-theoretic terms and establishes a correspondence between the universal enveloping algebras of PCLSAs and right-angled Coxeter groups, enabling closed-form Hilbert series formulas. The work strengthens the bridge between Lie theory, graph theory, and algebraic combinatorics, yielding computable tools for root data and related Hilbert series. Overall, the results give a concrete, graph-theoretic handle on root multiplicities in BKM superalgebras through marked chromatic polynomials and associated algebraic structures.

Abstract

Infinite-dimensional Lie superalgebras, particularly Borcherds-Kac-Moody (BKM) superalgebras, play a fundamental role in mathematical physics, number theory, and representation theory. In this paper, we study the root multiplicities of BKM superalgebras via their denominator identities, deriving explicit combinatorial formulas in terms of graph invariants associated with marked (quasi) Dynkin diagrams. We introduce partially commutative Lie superalgebras (PCLSAs) and provide a direct combinatorial proof of their denominator identity, where the generating set runs over the super heaps monoid. A key notation in our approach is marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and offer a method for computing root multiplicities. As applications, we characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, leading to explicit formulas for their Hilbert series. These results further deepen the interplay between Lie superalgebras, graph theory, and algebraic combinatorics.

Marked multi-colorings, partially commutative Lie superalgebras and right-angled Coxeter groups

TL;DR

Addressing the problem of determining root multiplicities in Borcherds-Kac-Moody (BKM) superalgebras, the paper develops a combinatorial framework based on partially commutative Lie superalgebras (PCLSAs) and marked graphs, introducing marked multi-colorings and the heaps monoid to derive a denominator identity and compute root multiplicities via Möbius inversion. It provides an explicit description of the root set Δ^G in graph-theoretic terms and establishes a correspondence between the universal enveloping algebras of PCLSAs and right-angled Coxeter groups, enabling closed-form Hilbert series formulas. The work strengthens the bridge between Lie theory, graph theory, and algebraic combinatorics, yielding computable tools for root data and related Hilbert series. Overall, the results give a concrete, graph-theoretic handle on root multiplicities in BKM superalgebras through marked chromatic polynomials and associated algebraic structures.

Abstract

Infinite-dimensional Lie superalgebras, particularly Borcherds-Kac-Moody (BKM) superalgebras, play a fundamental role in mathematical physics, number theory, and representation theory. In this paper, we study the root multiplicities of BKM superalgebras via their denominator identities, deriving explicit combinatorial formulas in terms of graph invariants associated with marked (quasi) Dynkin diagrams. We introduce partially commutative Lie superalgebras (PCLSAs) and provide a direct combinatorial proof of their denominator identity, where the generating set runs over the super heaps monoid. A key notation in our approach is marked multi-colorings and their associated polynomials, which generalize chromatic polynomials and offer a method for computing root multiplicities. As applications, we characterize the roots of PCLSAs and establish connections between their universal enveloping algebras and right-angled Coxeter groups, leading to explicit formulas for their Hilbert series. These results further deepen the interplay between Lie superalgebras, graph theory, and algebraic combinatorics.

Paper Structure

This paper contains 6 sections, 13 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lie superalgebras: Inversion lemma and denominator identity
  3. Marked multi-colorings and marked chromatic polynomials
  4. Root multiplicities, BKM superalgebras and marked chromatic polynomials
  5. Description of the set of roots of $\mathcal{P}(\mathcal{G})$
  6. Right-angled Coxeter Groups and Partially commutative superalgebras

Key Result

Proposition 1

Let $\mathcal{G}$ be a marked graph. Then forms a basis for the universal enveloping algebra $\mathbf{U}_{\mathcal{G}}$ of $\mathcal{P}(\mathcal{G})$.

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Definition
  • Definition
  • Remark
  • Definition
  • Example
  • Definition
  • Proposition
  • proof
  • Remark
  • Lemma
  • ...and 24 more