A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries

Yetter-Drinfel'd algebras and coideals of Weak Hopf $C^*$-Algebras

We characterize braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras in categorical terms. Using this, we then study quotient type coideal subalgebras of a given weak Hopf $C^*$-algebra $\mathcal G$ and coideal subalgebras invariant with respect to the adjoint action of $\mathcal G$. Finally, as an example, we explicitly describe quotient type coideal subalgebras of the weak Hopf $C^*$-algebras associated with Tambara-Yamagami categories.

Compact Quantum Ergodic Systems

We develop theory of multiplicity maps for compact quantum groups, as an application, we obtain a complete classification of right coideal $C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are labeled with Dynkin diagrams, but classification results for positive and negative cases of $q$ are different. Many of the coideals are quantum spheres or quotient spaces by quantum subgroups, but we do have other ones in our classification list.

Relative Fourier transforms and expectations on coideal subalgebras

For an algebraic compact quantum group $H$ we establish a bijection between the set of right coideal $*$-subalgebras $A\to H$ and that of left module quotient $*$-coalgebras $H\to C$. It turns out that the inclusion $A\to H$ always splits as a map of right $A$-modules and right $H$-comodules, and the resulting expectation $E:H\to A$ is positive (and lifts to a positive map on the full $C^*$ completion on $H$) if and only if $A$ is invariant under the squared antipode of $H$. The proof proceeds by Tannaka-reconstructing the coalgebra $C$ corresponding to $A\to H$ by means of a fiber functor fro…

关于顺从与余顺从的余理想

On Amenable and Coamenable Coideals

基于类群投影研究离散量子群右余理想的顺从性与余顺从性，通过引入不变态刻画相对顺从性，并建立余顺从紧拟子群与幂等态的一一对应，推进了余理想对偶性理论。

表示的拓扑边界与余理想

Topological Boundaries of Representations and Coideals

该研究针对离散与紧量子群的余理想，推广顺从性与余顺从性概念，引入表示的Furstenberg-Hamana边界，刻画余理想余顺从性与唯一迹性质，是经典群论结果的量子类比，解答了领域内公开问题。

Drinfeld double的非交换泊松边界与Furstenberg-Hamana边界

Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles

该研究阐明量子群的非交换泊松边界与Drinfeld double的Furstenberg-Hamana边界的关系，证明其在多种量子群情形下的重合或商集关系，还将该边界构造推广至刚性C$^*$-张量范畴。