Towards a classification of compact quantum groups of Lie type

TL;DR

This survey reviews recent advances in classifying compact quantum groups of Lie type, focusing on their representation categories, dualities, and fiber functors, especially for simple Lie groups like SU(n).

Contribution

It introduces a categorical duality framework for classifying quantum groups of Lie type, linking actions, module categories, and fiber functors, extending previous results.

Findings

Classification of dimension-preserving fiber functors via maximal Kac quantum subgroups.

Complete classification of quantum groups of G-type for compact simple Lie groups G.

Exhaustive classification of non-Kac quantum groups for G=SU(n).

Abstract

This is a survey of recent results on classification of compact quantum groups of Lie type, by which we mean quantum groups with the same fusion rules and dimensions of representations as for a compact connected Lie group $G$. The classification is based on a categorical duality for quantum group actions recently developed by De Commer and the authors in the spirit of Woronowicz's Tannaka--Krein duality theorem. The duality establishes a correspondence between the actions of a compact quantum group $H$ on unital C$^*$-algebras and the module categories over its representation category Rep $H$. This is further refined to a correspondence between the braided-commutative Yetter--Drinfeld $H$-algebras and the tensor functors from Rep $H$. Combined with the more analytical theory of Poisson boundaries, this leads to a classification of dimension-preserving fiber functors on the representation category of any coamenable compact quantum group in terms of its maximal Kac quantum subgroup, which is the maximal torus for the $q$-deformation of $G$ if $q\ne1$. Together with earlier results on autoequivalences of the categories Rep $G_q$, this allows us to classify up to isomorphism a large class of quantum groups of $G$-type for compact connected simple Lie groups $G$. In the case of $G=SU(n)$ this class exhausts all non-Kac quantum groups.