Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and on an approach to 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 an approach to 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. We develop ideas of a 1998 paper by Wenzl. Our results rely on the notion of weak-quasi-Hopf algebra of Drinfeld-Mack-Schomerus. We were also guided by Drinfeld first proof of Drinfeld-Kohno, by the general scheme settled by Bakalov and Kirillov and by Neshveyev and Tuset for a generic parameter but differences arise. 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 induced by the braiding of $U_q({mathfrak g})$. Moreover, in our interpretation 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. To approach this, we study this untwisting procedure. One of our main results is the construction of a Hopf algebra in a weak sense (w-Hopf algebra) associated to quantum group fusion category and of a twist of it giving a wqh structure on the Zhu algebra and thus a unitary modular fusion category structure on the category of C*-representations of the affine Lie algebra. In particular, the braiding associated to the affine Lie algebra is of a very simple form similarly to the case of Drinfeld quasi-Hopf algebra. The associator is a $3$-coboundary in a suitable weak sense. We conjecture that this modular tensor category structure is equivalent to that obtained via the tensor product theory of VOAs by Huang and Lepowsky. A proof of our conjecture leads to a proof of Kazhdan-Lusztig-Finkelberg theorem. We shall try to develop our conjecture in a different paper, or in a later update of this paper. 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 formalism, we discuss a corresponding analytic theory, which is based on the notion of $Omega$-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 many others. 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 original results by Mack and Schomerus and Haring-Oldenburg. We use this idea to construct unitary tensor structures on $C^*$-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. In the second part of the paper we study unitary tensor structures of Verlinde fusion categories more in detail, 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. We classify Verlinde fusion categories of type $A$, based on Kazhdan-Wenzl theory and on the w-Hopf algebra previously constructed by the first and last named authors, extending a result by Bischoff for ${mathfrak sl}_2$ at integer level and Nashveyev and Yamashita for ${mathfrak sl}_N$ in the generic case. Then we approach the connection problem between affine VOAs and quantum group fusion categories. We follow a 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 introduce the notion of unitary coboundary wqh. In this case the $Omega$-involution is induced by the braiding by abstracting the case of $U_q({mathfrak g})$. We give a categorical characterization and turns out to extends symmetric tensor functors in Doplicher-Roberts theorem. We formulate an abstract converse of Drinfeld-Kohno theorem in an analytic setting for a specific subclass 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. We construct a semisimple unitary coboundary w-Hopf algebra structure on Wenzl algebra $A_W$ (a semisimple subquotient of $U_q({mathfrak g})$) with representation category equivalent to the corresponding Verlinde quantum group fusion category. In this case $Delta(I)$ is given by Wenzl idempotent $P$. Subclass membership follows from the w-Hopf property. We apply our Drinfeld-Kohno to the twist $T=overline{R}^{1/2}Delta(I)$. In this way we construct a $3$-coboundary Drinfeld associator. Finally, we transport an untwisted unitary coboundary cocommutative wqh algebra structure to the Frenkel-Zhu algebra $A_Z$ via Wenzl path and from this to the corresponding affine Lie algebra representation category that makes it into a unitary modular fusion category. Possible future directions that we feel interested and we wish to complete in an updated version is to resume our approach to a direct proof of Kazhdan-Lusztig-Finkelberg equivalence theorem between UMFC categories from quantum groups and affine VOAs starting with the the tensor product theory by Huang and Lepowsky that is only briefly hinted in this version. Moreover, we would like to propose to interested people including ourselves to develop more connections between quantum groups and works in conformal net theory by Longo, Guido-Longo, A. Wassermann approach with the idea of primary fields, Toledano-Laredo work, or on their relation with VOAs by Carpi-Kawahigashi-Longo-Weiner, and Gui, or as an analogue of the idea of a compact gauge group by Doplicher and Roberts in high dimensional QFT theory. Any comment is welcome.