Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras

Fabbri, Marc A. ; Okoh, Frank

Canadian journal of mathematics, Tome 51 (1999) no. 3, pp. 523-545

Résumé

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.

Détail

DOI : 10.4153/CJM-1999-024-x

Mots-clés : 17B65, 17B68

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