Vertex operators for imaginary [formula omitted] subalgebras of the Monster Lie algebra

Abstract

The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V♮⊗V1,1, where V♮ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and V1,1 is the vertex algebra corresponding to the rank 2 even unimodular lattice II1,1. We construct vertex algebra elements that project to bases for subalgebras of m isomorphic to gl2, corresponding to each imaginary simple root, denoted (1,j) for j>0. Our method requires the existence of pairs of primary vectors in V♮ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group M on the subspace of primary vectors in V♮ induces an M-action on the set of gl2 subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of V♮ to prove that this action is non-trivial for small values of j.