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

• DOI: 10.48550/arxiv.2101.10016
• Publication Date: 2021-01-01
• Authors: Sebastiano Carpi; Sergio Ciamprone; Marco Valerio Gia...; Claudia Pinzari
• Keywords: Fusion rules; Quantum affine algebra; Vertex operator algebra

We discuss tensor categories motivated by CFT, their unitarizability and applications to various models including the affine VOAs. classification of type A Verlinde fusion categories. propose an approach Kazhdan-Lusztig-Finkelberg theorem. This theorem gives a ribbon equivalence between category associated quantum group at certain root unity that corresponding vertex operator algebra suitable positive integer level. develop ideas Wenzl. Our results rely on notion weak-quasi-Hopf Drinfeld-Mack-Schomerus. were also guided Drinfeld original proof, Bakalov Kirillov Neshveyev Tuset work for generic parameter. Wenzl described product in categories, related it unitary structure. Given two irreducible objects, inner is induced braiding U_q(g), with q 1. Moreover, paper suggests untwisting procedure make structure trivial. Then describes continuous path intuitively connects objects representations simple Lie defining algebra. study this procedure. One our main construction Hopf weak sense twist giving wqh Zhu modular representation algebra, confirming early view Frenkel Zhu. show equivalent obtained via theory VOAs Huang Lepowsky. direct proof FKL