Abstract
Toroidal Lie algebras and their vertex operator representations were introduced in [MEY] and a class of indecomposable modules were investigated. In this work, we extend the toroidal algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the toroidal algebra as an ideal and the Virasoro algebra as a subalgebra. With the use of vertex operators and certain oscillator representations of the Virasoro algebra it is proved that the corresponding Fock space gives rise to a class of irreducible modules for the Virasoro-toroidal algebra.
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[EM] Eswara, Rao, S., Moody, R.V.: Vertex representations for the universal central ex tensions for then-toroidal Lie algebras and a generalization of the Virasoro algebra. Commun. Math. Phys., to appear
[EMY] Eswara, Rao S., Moody, R.V., Yokonuma T.: Lie algebras and Weyl groups arising from vertex operator representations. Nova J. Algebra and Geometry1, 15–57 (1992)
[F1] Fabbri, M.A.: The structure of a new class of modules over the Virasoro-toroidal algebra. Preprint
[F2] Fabbri, M.A.: Virasoro-toroidal algebras and vertex representations. C.R. Math. Rep. Acad. Sci. Canada14, 77–82 (1992)
[FF] Feigin, B.L., Fuks, D.B.: Invariant skew-symmetric differential operators on the line and Verma modules over the Virasoro algebra. Funct. Anal. Appl. (English Translation)16, 114–126 (1982)
[FK] Frenkel, I., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math.63, 23–66 (1980)
[FLK] Frenkel, I., Lepowsky, J., Meurman, A.: “Vertex Operators and the Monster.” New York, London: Academic Press, Inc. 1988
[GO] Goddard, P., Olive, D.: “Algebras, lattices and strings.” In: Vertex Operators in Mathematics and Physics, Edited by Lepowsky, J., Mandelstam, S., Singer, I.M., Mathematical Sciences Research Institute Publications3 (1985)
[Kac] Kac, V.G.: Infinite dimensional Lie algebras. Boston, Basel: Birkhäuser 1983
[K] Kaplansky, I.: The Virasoro algebra. Commun. Math. Phys.86, 49–54 (1982)
[Ka] Kassel, C.: Kahler differentials and coverings of complex simple Lie algebras extended over a commutative algebra. J. Pure Appl. Algebra34, 265–275 (1985)
[KR] Kac, V.G., Raina, A.K.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. Singapore: World Scientific 1987
[MEY] Moody, R.V., Eswara, Rao S., Yokonuma, T.: Toroidal Lie algebras and vertex representations. Geometricae Dedicata35, 283–307 (1990)
[MP] Moody, R.V., Pianzola, A.: Lie Algebras with Triangular Decomposition. J. Wiley, to appear
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Communicated by A. Jaffe
To A. John Coleman on the occasion of his 75th birthday
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Fabbri, M.A., Moody, R.V. Irreducible representations of Virasoro-toroidal Lie algebras. Commun.Math. Phys. 159, 1–13 (1994). https://doi.org/10.1007/BF02100482
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DOI: https://doi.org/10.1007/BF02100482