Vertex operators for imaginary $\mathfrak{gl}_2$ subalgebras in the Monster Lie Algebra

TL;DR

This paper constructs vertex algebra elements for specific subalgebras in the Monster Lie algebra related to imaginary roots, revealing non-trivial symmetry actions of the Monster group on these substructures.

Contribution

It introduces a method to explicitly construct vertex algebra elements for  subalgebras associated with imaginary roots in the Monster Lie algebra, leveraging primary vectors in the Moonshine module.

Findings

Constructed vertex algebra elements for  subalgebras in  Monster Lie algebra.

Proved the existence of pairs of primary vectors satisfying certain conditions.

Demonstrated non-trivial  group actions on these subalgebras for small indices.

Abstract

The Monster Lie algebra $\mathfrak m$ is a quotient of the physical space of the vertex algebra $V=V^\natural\otimes V_{1,1}$, where $V^\natural$ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and $V_{1,1}$ is the vertex algebra corresponding to the rank 2 even unimodular lattice $\textrm{II}_{1,1}$. We construct vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_{2}$, corresponding to each imaginary simple root, denoted $(1,j)$ for $j>0$. Our method requires the existence of pairs of primary vectors in $V^{\natural}$ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group $\mathbb{M}$ on the subspace of primary vectors in $V^\natural$ induces an $\mathbb{M}$-action on the set of $\mathfrak{gl}_2$ subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of $V^\natural$ to prove that this action is non-trivial for small values of $j$.