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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 $$n=1$$.

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 17B68 Virasoro and related algebras
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##### References:
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