Lie algebra cohomology and the fusion rules |
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Authors: | Constantin Teleman |
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Affiliation: | (1) Department of Mathematics, Harvard University, 02138 Cambridge, MA, USA |
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Abstract: | We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointszk in , with a finite-dimensional irreducible representationVk of assigned to each, the Lie algebra of-valued polynomials acts on eachVk, via evaluation atzk. Then, the relative Lie algebra cohomologyH* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott. |
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