Localization of u-modules. III. Tensor categories arising from configuration spaces

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This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number ζ an abelian artinian category. We call its objects finite factorizable sheaves. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility ("factorization") and finiteness conditions. In Chapter 2 the tensor structure on is defined using functors of nearby cycles. It makes a braided tensor category. In Chapter 3 we define, using vanishing cycles functors, an exact tensor functor Φ: to the category connected with the corresponding quantum group. In Chapter 4 we show that Φ is an equivalence. Some proofs are only sketched.