Tensor structures arising from affine Lie algebras. II

Abstract

In Part I of this paper we studied a category &" of representations of an affine Lie algebra with central charge K -h (where K E C -Q>o) and we defined a tensor product functor which associates to several objects-of &K another object of &K' Here we will establish the commutativity isomorphism and the associativity isomorphism for this tensor product functor.The numbering of sections will continue that of Part L In §9 we consider certain modules of coinvariants and prove some finiteness results for them.(The idea to consider such spaces of coinvariants has been used in the case of &K in the work of Beilinson and Feigin cited in the introduction to Part L) In § §1O-12 we introduce integrable connections on these modules of coinvariants; these should be regarded as generalizations of the Knizhnik-Zamolodchikov equations.In § 14 we establish the commutativity isomorphisms, and in § §15-18 we establish the associativity isomorphisms.We are very indebted to A. Beilinson for many useful discussions.The idea to use formal schemes in the definition of the associativity isomorphism was suggested to us by him; the definition of the diagona,l modules (see 16.10), which plays a crucial role in the proof, is due to A. Beilinson.We also thank Lepowsky and Huang for useful discussions.We have benefited from reading Deligne's manuscript [De], which contains a discussion of the connection between tensor categories and local systems on the moduli space of stable marked curves of genus O. CONTENTS 9. Finiteness of coinvariants 10.Vector fields 11.Sugawara operators and coinvariants 12. Connections 13.Tensor product and coinvariants 14.Commutativity isomorphism 15.Degeneration of quadrics (I) 16.Degeneration of quadrics (II) 17.Degeneration of quadrics and connections