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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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[1] [B] Borcherds, R.: Vertex algebras, Kac-Moody algebras and the Monster. Proc. Natl. Acad. Sci. USA83, 3068–3071 (1986) · Zbl 0613.17012 · doi:10.1073/pnas.83.10.3068
[2] [BM] Bermans, S., Moody, R.V.: Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent. Math.108, 323–347 (1992) · Zbl 0778.17018 · doi:10.1007/BF02100608
[3] [BC] Berman, S., Cox, B.: Enveloping algebras and representations of toroidal Lie algebras. Pacific J. Math., to appear · Zbl 0809.17022
[4] [EMY] Eswara Rao, S., Moody, R.V., Yokonuma, T.: Lie algebras and Weyl groups arising from vertex operator representations. Nova J. of Algebra and Geometry1, 15–58 (1992) · Zbl 0867.17022
[5] [FM] Fabbri, M., Moody, R.V.: Irreducible representations of Virasoro-toroidal Lie algebras. Commun. Math. Phys., to appear · Zbl 0796.17024
[6] [G] Garland, H.: The arithemetic theory of loop groups. Math. IHES52, 5–136 (1980) · Zbl 0475.17004
[7] [GO] Goddard, P., Olive, D.: Algebras, lattices and strings, Vertex operators in Mathematics and Physics. Publ. Math. Sci. Res. Inst. #3, Springer-Verlag, 1984 · Zbl 0556.17004
[8] [K] Kassel, C.: Kahler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. J. Pure and Appl. Algebra34, 256–275 (1985) · Zbl 0549.17009
[9] [KF] Frenkel, I., Kac, V.: Basic representation of affine Lie algebras and dual resonance models. Invent. Math.62, 23–66 (1980) · Zbl 0493.17010 · doi:10.1007/BF01391662
[10] [KMPS] Kass, S., Moody, R.V., Patera, J., Slansky, R.: Representations of Affine Algebras and Branching Rules. Berkeley: University of California Press 1990 · Zbl 0785.17028
[11] [MEY] Moody, R.V., Eswara Rao, S., Yokonuma, T.: Toroidal Lie algebras and vertex representations. Geom. Ded.35, 283–307 (1990) · Zbl 0704.17011 · doi:10.1007/BF00147350
[12] [MS] Moody, R.V., Shi, Z.: Toroidal Weyl groups. Nova J. Algebra and Geometry1, 317–337 (1992) · Zbl 0867.17015
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