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

Abstract

We develop Doplicher-Roberts quantum group duality program for the WZW model within the framework of vertex operator algebras. We establish that a weak quasi-fibre structure on a functor preserving a Drinfeld coboundary symmetry naturally extends a symmetric functor under permutation symmetry. Utilizing Wenzl's functor associated with the unitary quantum group fusion category, we construct a weak tensor structure, yielding a new class of unitary coboundary weak Hopf $C^*$-algebras for all Lie types and levels. Via a specialized Drinfeld twist and the Wenzl de-quantization curve, this structure is transported onto the Zhu algebra--which consequently becomes a unitary coboundary weak quasi-Hopf $C^*$-algebra with a 3-coboundary associator--providing a uniform, self-contained construction of unitary rigid braided tensor categories for categories of affine VOA modules at positive integer levels. Furthermore, we analyze the type A case via classification methods based on Kazhdan--Wenzl theory and our weak Hopf algebra framework, providing key insight into the determination of associativity from the braiding in the general case. We develop a cohomology theory for braided tensor categories with a generating object enabling a complete identification of our ribbon braided tensor structure with the constructions of Huang and Lepowsky for the classical Lie types and G_2, while bypassing their original reliance on the KZ equations and the Verlinde formula entirely. Our methods solve several long-standing problems: Galindo's question on the uniqueness of unitary tensor structures, Kirillov's conjecture on the positivity of a certain Hermitian form on the module category of an affine Lie algebra by Beilinson-Feigin-Mazur, the quantum group structure on the Zhu algebra sought by Frenkel and Zhu, and provide a direct proof of the Kazhdan-Lusztig-Finkelberg equivalence settling an open problem of Huang.