Realizations of the Monster Lie algebra

TL;DR

This paper explores the structure of the Monster Lie algebra, constructing it from Kac-Moody subalgebras and modules, and uses this to simplify proofs related to moonshine series and their properties.

Contribution

It introduces a new construction of the Monster Lie algebra from Kac-Moody subalgebras and modules, providing a novel approach to understanding moonshine series.

Findings

Constructed the Monster Lie algebra from Kac-Moody subalgebras and modules.

Demonstrated that certain generalized Verma modules are standard and irreducible.

Provided new recursion relations for moonshine series coefficients.

Abstract

We study aspects of the theory of generalized Kac-Moody Lie algebras (or Borcherds algebras) and their standard modules. It is shown how such an algebra with no mutually orthogonal imaginary simple roots, including Borcherds' Monster Lie algebra $\frak m$, can be naturally constructed from a certain Kac-Moody subalgebra and a module for it. We observe that certain generalized Verma (induced) modules for generalized Kac-Moody algebras are standard modules and hence irreducible. In particular, starting from the moonshine module for the Monster group $M$, we construct a certain $\{frak gl}_2$- and $M$-module, the tensor algebra over which carries a natural structure of irreducible module for $\frak m$, which is realized as an explicitly prescribed $M$-covariant Lie algebra of operators on this tensor algebra. The existence of large free subalgebras of $\frak m$ is further exploited to provide a simplification of Borcherds' proof of the Conway-Norton conjectures for the McKay-Thompson series of the moonshine module. The coefficients of these series are shown to satisfy natural recursion relations (replication formulas) equivalent to, but different from, those obtained by Borcherds.