Tannaka-Krein duality for compact groupoids I, representation theory

Tannaka-Krein reconstruction and ergodic actions of easy quantum groups

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory

An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module structure over the tensor category Rep(G) of finite dimensional representations of G. We show that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C*-categories over Rep(G) up to natural equivalence. This gives a global approach to the duality theory for ergodic actions…

Banach丛上群胚的几乎表示

Almost Representations of Groupoids on Banach Bundles

本研究将紧群几乎表示的经典结果推广到带连续Haar测度系统的真群胚，证明其在有限秩Hilbert丛上局部有足够多连续表示，进而得到纯拓扑下群胚的Tannaka对偶新推广。

Yetter-Drinfeld C*-algebras 的范畴双子结构：超越辫对易情况

Categorical dualtiy for Yetter-Drinfeld C*-algebras. Beyond the braided-commutative case

本文研究了Yetter-Drinfeld C*-代数在紧量子群下的范畴双子结构，提出了一种超越辫对易情况的双子理论。

Categorically Morita equivalent compact quantum groups

We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital C*-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galo…

Towards a classification of compact quantum groups of Lie type

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 $H$.…