Representations of Virasoro-Heisenberg algebras and Virasoro-toroidal algebras

Virasoro-toroidal algebras, ((J) over tilde)([n]), are semi-direct products of toroidal algebras J([n]) and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let Gamma be an extension of a simply laced lattice Q by a hyperbolic lattice of rank two. There is a Fock space V(Gamma) corresponding to Gamma with a decomposition as a complex vector space: V(Gamma) = coproduct (m is an element of Z) K(m). Fabbri and Moody have shown that when m not equal 0, K(m) is an irreducible representation of (J) over tilde([2].) In this paper we produce a filtration of (J) over tilde([2])-submodules of K(0). When L is an arbitrary geometric lattice and n is a positive integer, we construct a Virasoro-Heisenberg algebra (H) over tilde(L, n). Let Q be an extension of Q by a degenerate rank one lattice. We determine the components of V(Gamma) that are irreducible (H) over tilde(Q, 1)-modules and we show that the reducible components have a filtration of (H) over tilde(Q, 1)-submodules with completely reducible quotients. Analogous results are obtained for (H) over tilde(Q, 2). These results complement and extend results of Fabbri and Moody.