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Toroidal Lie algebras and vertex representations
Authors:Robert V Moody  Senapathi Eswara Rao  Takeo Yokonuma
Institution:(1) Department of Mathematics, University of Alberta, T6G 2G1 Edmonton, Canada;(2) School of Mathematics, Tata Institute for Fundamental Research, Bombay, India;(3) Department of Mathematics, Sophia University, Tokyo, Japan
Abstract:The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus Copf××Copf× 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.To Professor J. Tits for his sixtieth birthday
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