Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras

http://dx.doi.org/10.4153/CJM-1999-024-x
Canad. J. Math. 51(1999), 523-545

  Published:1999-06-01
 Printed: Jun 1999

Virasoro-toroidal algebras, $\tilde\mathcal{T}_{[n]}$, are semi-direct products of toroidal algebras $\mathcal{T}_{[n]}$ 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 $\dot{Q}$ 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: $V(\Gamma) = \coprod_{m \in \mathbf{Z}}K(m)$. Fabbri and Moody have shown that when $m \neq 0$, $K(m)$ is an irreducible representation of $\tilde\mathcal{T}_{[2]}$. In this paper we produce a filtration of $\tilde\mathcal{T}_{[2]}$-submodules of $K(0)$. When $L$ is an arbitrary geometric lattice and $n$ is a positive integer, we construct a Virasoro-Heisenberg algebra $\tilde\mathcal{H}(L,n)$. Let $Q$ be an extension of $\dot{Q}$ by a degenerate rank one lattice. We determine the components of $V(\Gamma)$ that are irreducible $\tilde\mathcal{H}(Q,1)$-modules and we show that the reducible components have a filtration of $\tilde\mathcal{H}(Q,1)$-submodules with completely reducible quotients. Analogous results are obtained for $\tilde\mathcal{H} (\dot{Q},2)$. These results complement and extend results of Fabbri and Moody.