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

We discuss tensor categories motivated by Conformal Field Theory, their unitarizability with applications to various models including the affine VOAs. We discuss classification of type A Verlinde fusion categories. We propose a direct proof of Kazhdan-Lusztig-Finkelberg theorem. This theorem gives a ribbon equivalence between the fusion category associated to a quantum group at a certain root of unity and that associated to a corresponding affine vertex operator algebra at a suitable positive integer level. Our proof develops ideas of a 1998 paper by Wenzl and relies on the use of the unitary structure. More precisely, Wenzl described a fusion tensor product in quantum group fusion categories, and related it to the unitary structure. Given two irreducible objects, the inner product of the fusion tensor product is non-trivial and induced by the braiding of Uq(g). Moreover, the paper suggests a suitable untwisting procedure by some square root construction to make the unitary structure trivial. Then it also describes a continuous path that intuitively connects objects of the quantum group fusion category to representations of the simple Lie group defining the affine Lie algebra. We study this untwisting procedure inspired by some analogy with the work of Wassermann via the idea of primary fields, which play a central role. Our results rely on the notion of weak-quasi-Hopf algebra of Drinfeld-Mack-Schomerus. We were also guided by Drinfeld original proof of Drinfeld-Kohno theorem, by the general scheme settled by Bakalov and Kirillov for Drinfeld-Kazhdan-Lusztig at generic parame- ters over C which reduces the theorem to Hopf and quasi-Hopf algebras, and by Neshveyev and Tuset Tannakian approach via discrete algebras for a positive parameter. Differences arise from the unitary structure, the problem that an analogue of Uq(g) and of Drinfeld category have to be constructed, in such a way that the Zhu algebra replaces U(g), and the associativity morphisms. One of our main results is the construction of a Hopf algebra in a weak sense associated to quantum group fusion category and of a twist giving a wqh structure on the Zhu algebra, and a unitary modular fusion category structure on the category of C*-representations of the affine Lie algebra, confirming an early view by Frenkel and Zhu. In particular, the braiding arising from our construction of the associated affine Lie algebra is of a very simple form similarly to the case of Drinfeld quasi-Hopf algebra, but differences arise that parallel the case of CFT. The associator is a 3-coboundary in a suitable weak sense. We show that our unitary modular tensor category structure can directly be compared with the ribbon tensor category structure by Wassermann, Toledano-Laredo, Huang and Lepowsky, CKLW, Gui for affine VOAs. This leads to a direct proof of Kazhdan-Lusztig- Finkelberg theorem. We next summarize our results in a more precise way. Our main tool is Tannaka-Krein duality for semisimple categories. After developing general algebraic theory of weak quasi-Hopf algebras and reviewing the Tannakian for- malism, we discuss a corresponding analytic theory, which is based on the notion of Ω-involution by Gould and Lekatsas. We introduce the notion of w-Hopf algebra as an analogue of the notion of Hopf algebra in a weak setting. We extend the theory of compact quantum groups in the work by Woronowicz’, and several authors. We notice that weak quasi-Hopf algebras may be associated to semisimple tensor categories under very mild assumptions, e.g. amenability, that allow to construct integral valued submultiplicative dimension functions (weak dimension functions), extending orig- inal results by Mack and Schomerus and Haring-Oldenburg. We use this idea to construct unitary tensor structures on linear C∗-categories that are tensor equivalent to unitary tensor categories. Applications include unitarization of affine VOAs, built on the known tensor equivalence by Kazhdan-Lusztig-Finkelberg-Huang- Lepowsky equivalence and unitarity of quantum group fusion categories by Kirillov-Wenzl- Xu. In particular, we apply our approach to solve a problem posed by Galindo on uniqueness of the unitary tensor structure. Motivated by the need of a better understanding of whether our approach to unitarizability of affine VOAs via weak quasi-Hopf algebras is a natural manifestation of structural aspects, in the second part of the paper we study unitary tensor structures of Verlinde fusion categories more in detail. We introduce specific Ω-involution on a general ribbon wqh algebra associated to the braiding, that we call unitary coboundary wqh by abstract- ing the case of Uq(g). We give a categorical characterization and turns out to extend symmetric tensor functors in Doplicher-Roberts theorem. In the case of Verlinde fusion categories of type A, we start with Kazhdan-Wenzl theory and use a number of ideas in the quantum group literature including a result by Bischoff for sl2 at integer level and Neshveyev and Yamashita for slN in the generic case and use the specific structure on a w-Hopf algebra previously constructed by two of us to classify the category completely. Then we approach the connection problem between affine VOAs and quantum group fusion categories. We follow a Tannakian scheme indicated by Neshveyev-Tuset-Yamashita for q generic based on the use of discrete quasi-Hopf algebras of Drinfeld, extending it to the weak generalization introduced by Mack and Schomerus, that is we work with discrete weak quasi-Hopf algebras. These weak versions still admit a notion of twist. We generalize the notion of 3-coboundary associator to the weak setting. We formulate an abstract converse of Drinfeld-Kohno theorem in an analytic setting for the unitary coboundary wqh providing an untwisted unitary coboundary wqh algebra in the subclass, that is with the mentioned very simple R-matrix similarly to Drinfeld case and also a trivial unitary structure on sufficiently many representations following Wenzl that determine the whole structure. We construct a semisimple unitary coboundary w-Hopf algebra structure on Wenzl algebra AW (a semisimple subquotient of Uq(g)) with representation category equivalent to the corresponding Verlinde quantum group fusion category. In this case ∆(I) is given by Wenzl idempotent P. Subclass membership follows from the w-Hopf property. We apply our Drinfeld-Kohno to the twist T = R1/2∆(I). In this way we construct a 3-coboundary Drinfeld associator. Finally, we transport an untwisted unitary coboundary wqh algebra structure to the Zhu algebra AZ of the associated affine VOA at integer level via the twist and Wenzl path and from this to the corresponding affine Lie algebra representation category that makes it into a unitary modular fusion category. We compare with early work by Kirillov and then by Wassermann, Toledano-Laredo, Gui, Huang and Lepowsky braiding and associativity structures and we show that they are the same.