Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem

Abstract: This paper deals with some related problems posed by Doplicher-Roberts, Seiberg-Moore, Frenkel-Zhu, Huang regarding relations between quantum groups at roots of unity and categories arising from conformal field theory. We construct unitary rigid ribbon braided tensor category structures on module categories of affine vertex operator algebras at positive integer levels for all Lie types, derived from fusion categories of quantum groups via the construction of certain semisimple weak Hopf algebras having roots in the work by Mack and Schomerus, and a Drinfeld twist to the Zhu algebra analogous to Drinfeld twist for the quantized universal enveloping algebra U_h(g) and Drinfeld quasi-Hopf algebra structure over U(g). We use our construction to give a direct proof of an equivalence theorem (going back to Kazhdan-Lusztig and Finkelberg for affine Lie algebras) between fusion categories of quantum groups and affine vertex operator algebras at positive integer levels, with respect to Huang-Lepowsky ribbon braided tensor structure, for the classical Lie types and G_2. We apply this result to unitarize ribbon braided module categories of vertex operator algebras. We classify fusion categories with type A fusion rules by their ribbon structure. In our work, the unitary structure of quantum group fusion categories obtained by Wenzl and an associated fibre functor to the Hilbert spaces play a main role to construct naturally associated semisimple weak Hopf algebras.