Generalized Moonshine IV: Monstrous Lie algebras

2012/08/30 by Scott Carnahan, Carnahan, Scott · 1 citation
Mathematics · Physics and Astronomy · #11F22 (Primary) 81R10 #17B65 #17B69 #81T40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1208.6254

openalex publication_date 2012/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each element of the Fischer-Griess Monster sporadic simple group, we construct an infinite dimensional Lie algebra equipped with a projective action of the centralizer of that element. Our construction is given by a string-theoretic "add a spacetime torus and quantize" functor applied to an abelian intertwining algebra that is formed from a family of twisted modules of the Monster vertex operator algebra. We prove that for all Fricke elements in the Monster, the characters of centralizers acting on the corresponding irreducible twisted modules are Hauptmoduln. From these results, we resolve Norton's Generalized Moonshine Conjecture.

Citations

• A resolution for standard modules of Borcherds Lie algebras
• Symmetric invariant bilinear forms on vertex operator algebras
• Modular functions arising from some finite groups

Cited by

• The structure of non-Fricke Monstrous Lie algebras

Related

• A short introduction to Monstrous Moonshine
• Moonshine for Rudvalis's sporadic group I
• Some twisted sectors for the Moonshine Module
• Monstrous Moonshine and the uniqueness of the Moonshine module
• Realizations of the Monster Lie algebra