Towards a Classification of Compact Quantum Groups of Lie Type

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.