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Representations of VirasoroHeisenberg Algebras and VirasoroToroidal Algebras


Published:19990601
Printed: Jun 1999
Marc A. Fabbri
Frank Okoh
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Abstract
Virasorotoroidal algebras, $\tilde\mathcal{T}_{[n]}$, are
semidirect products of toroidal algebras $\mathcal{T}_{[n]}$ and
the Virasoro algebra. The toroidal algebras are, in turn,
multiloop versions of affine KacMoody 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 VirasoroHeisenberg 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.