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

Abstract. This paper deals with some related problems posed since the late 80s by Doplicher-Roberts, Seiberg-Moore, Frenkel-Zhu, and Huang, specifically recalled in Sect. 1, regarding relations between quantum groups at roots of unity and categories arising from conformal field theory. Our main results are summarized in Sect. 3 and include the construction of unitary rigid ribbon braided tensor category structures on module categories of affine vertex operator algebras Vgk at positive integer levels k for all Lie types directly derived from fusion categories of quantum groups. The construction concerns the construction of certain semisimple unitary weak Hopf algebras AW (g, q, l) having roots in the work by Mack and Schomerus, regarded as quantum gauge groups in the sense of Doplicher-Roberts. We construct a Drinfeld twist from AW (g, q, l) to the Zhu algebra A(Vgk ) providing a weak quasi-Hopf algebra structure related to Frenkel-Zhu question. The twist is an analogue of Drinfeld twist for the quantized universal enveloping algebra Uh(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 G2. This is our solution to Huang’s problem. We apply this result to unitarize ribbon braided module categories of vertex operator algebras in a uniform and natural way. We also 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 the mentioned naturally associated semisimple weak Hopf algebras. A corollary of our general Tannakian methods gives a solution to a problem by Galindo on uniqueness of unitary tensor C∗-structures for tensor C ∗ -categories.