Eswara Rao, S.; Moody, R. V. Vertex representations for \(n\)-toroidal Lie algebras and a generalization of the Virasoro algebra. (English) Zbl 0808.17018 Commun. Math. Phys. 159, No. 2, 239-264 (1994). 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 \(n=1\). Cited in 6 ReviewsCited in 67 Documents MSC: 17B69 Vertex operators; vertex operator algebras and related structures 17B68 Virasoro and related algebras Keywords:toroidal Lie algebra; vertex operator representations; universal central extension; Virasoro algebra PDF BibTeX XML Cite \textit{S. Eswara Rao} and \textit{R. V. Moody}, Commun. Math. Phys. 159, No. 2, 239--264 (1994; Zbl 0808.17018) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.