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

Huang posed the problem of finding a direct proof of the combination of the Kazhdan-Lusztig and Finkelberg theorems establishing equivalence between two braided fusion categories: that of a quantum group at root of unity and that of an affine Lie algebra at positive integer level. We are motivated by the problem of extending Doplicher- Roberts theory for compact groups and reconstruction of fields to theories admitting a braided symmetry. We are also inspired by the Drinfeld-Kohno equivalence theorem and realize a fibre functor on these categories. We give a direct proof by constructing the structure of a unitary ribbon braided weak quasi-Hopf algebra (wqh) on the Zhu algebra associated to the affine vertex operator algebra at positive integer level, which induces a unitary rigid ribbon tensor category structure on its module category. We derive all the structure on the Zhu algebra from a unitary ribbon-braided weak Hopf algebra (wh) in a new sense, a quantum analogue of the compact group in Doplicher- Roberts theory, and a Drinfeld twist. This wh algebra is naturally associated with the unitary rigid ribbon-braided fusion category of the quantum group at the root of unity studied by Wenzl. We compare our ribbon-braided tensor structure with that of Huang and Lepowsky. In the type A case we obtain another proof based on our wh and classification methods that gives light to the role of the braided symmetry for the associator in the general case.