Abstract

Representation theory of simple Lie groups has interesting deformations in the framework of tensor categories. What kind of categories can have the fusion rules of type A theory (corresponding to finite-dimensional representations of special linear groups) is well understood by the work of Kazhdan and Wenzl (1993). We look at the remaining classical series, namely the fusion rules of type BCD theories (corresponding to orthogonal and symplectic groups), and give a similar classification. The proof is based on Liu’s technique to find Kauffman bracket relations in planar algebras, and certain torsion freeness of the Lie type categories by Voigt and Goffeng for compact quantum groups, and by Schopieray and Gannon for modular categories. This, combined with our previous work on the classification of dimension-preserving fiber functors and categorical Poisson boundary, leads to a classification of non-Kac type compact quantum groups of the classical fusion type. Based on ongoing joint work with P. Grossman and S. Neshveyev.