Towards a Classification of Compact Quantum Groups of Lie Type

Abstract

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 = 1. Together with earlier results on autoequivalences of the categories Rep Gq, 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.

Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem

“…If a weak quasi bialgebra A ′ is obtained from another such bialgebra A by replacing the associator of the latter with a new one but leaving the rest of the structure unchanged, then Rep(A) and Rep(A ′ ) have isomorphic Grothendieck rings. The following proposition, inspired by a similar statement in [101] for Hopf algebras, shows that at an abstract level an isomorphism of Grothendieck rings of fusion categories can always be visualized in this way.…”

“…This follows from the fact that rk(F ) is bounded above by the quantum dimension [131]. In this setting, it is important to recall the remarkable work by Neshveyev and Yamashita on the classification of compact quantum groups that beyond the fusion rules, share the integral dimensions with a given compact simple simply connected Lie group G, see [101] and references therein. 16.…”

Maximal torus theory for compact quantum groups

“…There are many other situations where these two technical ingredients are available, at least to some extent. Without getting into details here, let us just mention that: (1) the product operations ×, * can be investigated by using [34], (2) the free complexification operation can be investigated by using [29], (3) for deformations, evidence comes from [13], [27], (4) for free wreath products, evidence comes from [23], [24], (5) the twoparametric free quantum groups can be studied by using [4], and ( 6) for the various growth conjectures, substantial evidence comes from the computations in [7], [18].…”