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 RepH. This is further refined to a correspondence between the braided-commutative Yetter-Drinfeld H-algebras and the tensor functors from RepH. 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 ≠1. Together with earlier results on autoequivalences of the categories RepGq, 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.