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

Doplicher and Roberts originally posed the problem of extending their duality theory for compact groups and field reconstruction to theories admitting braided symmetry. In this paper, we address this problem for the Wess-Zumino-Witten model within the framework of vertex operator algebras. We construct a weak Hopf $C^{*}$-algebra in a new sense and endow it with coboundary symmetry acting as a quantum gauge group. We give a direct proof of the Kazhdan-Lusztig and Finkelberg equivalence between the braided fusion categories of quantum groups at roots of unity and affine Lie algebras at positive integer levels, settling an open problem by Huang. Furthermore, we resolve a problem posed by Frenkel and Zhu regarding a quantum group structure of the Zhu algebra, and solve Galindo's question on the uniqueness of unitary structures in tensor categories. We present a uniform, self-contained construction of unitary rigid braided tensor structures for categories of modules over affine VOAs at positive integer levels. This structure is explicitly equivalent to the quantum group fusion category and is independent of the Knizhnik-Zamolodchikov equations. Our weak Hopf algebra is canonically associated with the unitary rigid ribbon-braided fusion category of the quantum group at roots of unity, as studied by Wenzl. By applying a Drinfeld twist, we derive the complete structure of the Zhu algebra from this weak Hopf algebra. Finally, we identify our ribbon-braided tensor structure with the framework of Huang and Lepowsky for all Lie types and many objects. For type $A$ cases and pointed categories, we provide an alternative proof using classification methods and our weak Hopf algebra which provides key insights into the role of braided symmetry for the uniqueness of the associator in the general setting.