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Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras

Published online by Cambridge University Press:  20 November 2018

Marc A. Fabbri  and
Frank Okoh
Marc A. Fabbri
Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287, U.S.A.
Frank Okoh
Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.
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Abstract

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Virasoro-toroidal algebras, ${{\tilde{J}}_{[n]}}$, 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 $\dot{Q}$ by a hyperbolic lattice of rank two. There is a Fock space $V\left( \Gamma \right)$ corresponding to $\Gamma $ with a decomposition as a complex vector space: $V\left( \Gamma \right)=\coprod{_{m\in z}K\left( m \right)}$ . Fabbri and Moody have shown that when $m\ne 0,\,K\left( m \right)$ is an irreducible representation of ${{\tilde{J}}_{[2]}}$ . In this paper we produce a filtration of ${{\tilde{J}}_{[2]}}$-submodules of $K\left( 0 \right)$. When $L$ is an arbitrary geometric lattice and $n$ is a positive integer, we construct a Virasoro-Heisenberg algebra $\tilde{H}\left( L,n \right)$ . Let $Q$ be an extension of $\dot{Q}$ by a degenerate rank one lattice. We determine the components of $V\left( \Gamma \right)$ that are irreducible $\tilde{H}\left( Q,1 \right)$ -modules and we show that the reducible components have a filtration of $\tilde{H}\left( Q,1 \right)$-submodules with completely reducible quotients. Analogous results are obtained for $\tilde{H}\left( \dot{Q},2 \right)$ . These results complement and extend results of Fabbri and Moody.

Keywords

17B6517B68
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 3 , 01 June 1999 , pp. 523 - 545
Copyright
Copyright © Canadian Mathematical Society 1999

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