Classification of non-Kac compact quantum groups of SU(n) type

TL;DR

This paper classifies non-Kac compact quantum groups of SU(n) type with specific fusion rules, using categorical Poisson boundary techniques, and explores autoequivalences of twisted q-deformations of simple Lie groups.

Contribution

It provides the first classification of non-Kac compact quantum groups of SU(n) type with given fusion rules, and establishes a categorical framework relating quantum subgroups and cohomology.

Findings

Classification of non-Kac compact quantum groups of SU(n) type.

A categorical Poisson boundary approach to quantum subgroup structure.

Bijection between second cohomology groups of duals of G and K.

Abstract

We classify up to isomorphism all non-Kac compact quantum groups with the same fusion rules and dimension function as $SU(n)$. For this we first prove, using categorical Poisson boundary, the following general result. Let $G$ be a coamenable compact quantum group and $K$ be its maximal quantum subgroup of Kac type. Then any dimension-preserving unitary fiber functor $Rep\ G \to Hilb_f$ factors, uniquely up to isomorphism, through $Rep\ K$. Equivalently, we have a canonical bijection $H^2(\hat G; T) \cong H^2(\hat K; T)$. Next, we classify autoequivalences of the representation categories of twisted $q$-deformations of compact simple Lie groups.