Vertex operators for imaginary gl₂ subalgebras of the Monster Lie algebra

The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V^♮⊗V_1,1, where V^♮ 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 II_1,1. 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.