Tannaka-Krein duality for compact groupoids II, Fourier transform

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…

等变 Tannaka-Krein 重构与离散结构的量子自同构群

Equivariant Tannaka-Krein reconstruction and quantum automorphism groups of discrete structures

本文通过推广 Tannaka-Krein 重构定理，为一大类离散结构构造量子自同构群，并给出从连通局部有限离散结构得到相应范畴的方法，以量子 Cayley 图为例加以说明。

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…

Equivariant covering spaces of quantum homogeneous spaces

We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras. In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences. As its application, we show that every Jones' value appears as the index of an equivariant conditional expectation. In the latter half of this paper, we give an imprimitivity theorem in some cases: for general compact quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo deformation $G_q$ of a simply-connected compact Lie group $G$. As an ap…

Partition C*-algebras II - Links to Compact Matrix Quantum Groups

In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up article, we show that often we can associate a Hopf algebra structure to partition C*-algebras, and also a compact matrix quantum group structure. This follows the lines of Banica and Speicher's approach to quantum groups; however, we access them in a more algebraic way circumventing Tannaka-Krein duality. We give criteria when these quantum groups are quantum subgroups of Wang's free orthogonal quantum gro…

Unitary Easy Quantum Groups: the free case and the group case

Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of ($C^*$-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group $O_n^+$. We now give a definition of unitary easy quantum groups using colored partitions to tack…

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 $H$.…