Abstract
The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ℂ××ℂ× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.
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References
Borcherds, R., ‘Vertex algebras, Kac-Moody algebras and the Monster’, Proc. Natl. Acad. Sci. USA 83 (1986), 3068–3071.
Frenkel, I. and Kac, V., ‘Basic representations of affine Lie algebras and dual resonance models’, Invent. Math. 62 (1980), 23–66.
Frenkel, I., Lepowsky, J., and Meurman, A., Vertex Operator Algebras and the Monster, Academic Press, Boston, 1989.
Garland, H., ‘The arithmetic theory of loop groups’, Publ. Math. IHES 52 (1980), 5–136.
Goddard, P. and Olive, D., ‘Algebras, lattices and strings’, Vertex Operators in Mathematics and Physics, Publ. Math. Sci. Res. Inst. #3, 51–96, Springer-Verlag, 1984.
Hilton, P. and Stammbach, U., A Course in Homological Algebra, Springer-Verlag, New York, 1971.
Jacobson, N., Structure of Rings, AMS, 1956.
Kassel, C., ‘Kähler differentials and converings of complex simple Lie algebras extended over a commutative algebra’, J. Pure Appl. Algebra 34 (1985), 265–275.
Kass, S., Moody, R., Patera, J., and Slansky, R., Representations of Affine Algebras and Branding Rules, Univ. of California Press (to appear).
Moody, R., ‘A new class of Lie algebras’, J. Algebra 10 (1968), 211–230.
Moody, R. and Pianzola, A., ‘Infinite dimensional Lie algebras (a unifying overview)’, Algebras, Groups, and Geometries 4 (1987), 165–213.
Moody, R. and Pianzola, A., Lie Algebras with Triangular Decomposition, Vol. 1, Wiley (to appear).
Morita, J. and Yoshii, Y., ‘Universal central extensions of Chevalley algebras over Laurent series polynomial rings and GIM Lie algebras’, Proc. Japan Acad. Ser. A 61 (1985), 179–181.
Santharoubane, L. J., ‘The second cohomology group for Kac-Moody Lie algebras and Kähler differentials’, J. Algebra 125 (1989), 13–26.
Wilson, R., ‘Euclidean Lie algebras are universal central extensions’, Lie Algebras and Related Topics, Lecture Notes in Math. 933, 210–213, Springer-Verlag, 1982.
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To Professor J. Tits for his sixtieth birthday
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Moody, R.V., Rao, S.E. & Yokonuma, T. Toroidal Lie algebras and vertex representations. Geom Dedicata 35, 283–307 (1990). https://doi.org/10.1007/BF00147350
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DOI: https://doi.org/10.1007/BF00147350