Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras

Abstract

AbstractVirasoro-toroidal algebras, , are semi-direct products of toroidal algebras and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let Γ be an extension of a simply laced lattice by a hyperbolic lattice of rank two. There is a Fock space V(Γ) corresponding to Γ with a decomposition as a complex vector space: V(Γ) = . Fabbri and Moody have shown that when m ≠ 0, K(m) is an irreducible representation of . In this paper we produce a filtration of -submodules of K(0). When L is an arbitrary geometric lattice and n is a positive integer, we construct a Virasoro-Heisenberg algebra . Let Q be an extension of by a degenerate rank one lattice. We determine the components of V(Γ) that are irreducible -modules and we show that the reducible components have a filtration of -submodules with completely reducible quotients. Analogous results are obtained for . These results complement and extend results of Fabbri and Moody.