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Vertex representations for \(n\)-toroidal Lie algebras and a generalization of the Virasoro algebra. (English) Zbl 0808.17018
We construct faithful vertex operator representations for the universal central extension \(\tau_ n\) of \({\mathfrak g}\otimes \mathbb{C}[ t_ 1^{\pm1}, t_ 2^{\pm1}, \dots, t_ n^{\pm1} ]\), where \({\mathfrak g}\) is a simple, simply-laced finite dimensional Lie algebra over \(\mathbb{C}\). We call \(\tau_ n\) the \(n\)-toroidal Lie algebra. These representations also afford representations for an abelian extension of the Lie algebra of derivations of \(\mathbb{C}[ t_ 1^{\pm1}, t_ 2^{\pm1}, \dots, t_ n^{\pm1} ]\). This latter Lie algebra is a generalization of the Virasoro algebra, and so this whole construction is a generalization of both the Frenkel-Kac and the Segal-Sugawara construction which are well known for the case \(n=1\).

17B69 Vertex operators; vertex operator algebras and related structures
17B68 Virasoro and related algebras
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