Measurable Kac cohomology for bicrossed products

Examples of locally compact quantum groups through the bicrossed product construction

The main aim of this paper is to introduce some examples of non-compact locally compact quantum groups to a non-specialized audience. The major importance of these examples is their simplicity. Other examples as the quantum E(2) group of Woronowicz are much more difficult to construct. We will make use of the bicrossed product construction to obtain our examples. This construction dates back to Kac, Takeuchi and Majid. It was also considered by Baaj & Skandalis.

局部紧量子群在经典空间上的忠实作用

Faithful actions of locally compact quantum groups on classical spaces

该研究打破了非Kac紧量子群无法在经典紧空间忠实作用的认知，构造出可在经典非紧空间忠实遍历作用的双交叉积型局部紧量子群，并猜想经典流形无真正量子等距的结论在非紧情形也成立。

On the von Neumann algebra of class functions on a compact quantum group

We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups; in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we constr…

Property $T$ for general locally compact quantum groups

In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\wid…

Inner amenability and approximation properties of locally compact quantum groups

We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau--Paterson, resolve a recent conjecture of Ng--Viselter, and prove that, for inner amenable quantum groups $\mathbb{G}$, approximation properties of the dual operator algebras can be averaged to approximation properties $\mathbb{G}$. Similar homological techniques are …

与本质双射1-上循环相关的局部紧群的量子化

Quantization of locally compact groups associated with essentially bijective $1$-cocycles

该研究针对带阿贝尔核的局部紧群扩张与相容本质双射1-上循环，构造对偶酉2-上循环并证明对应形变为上循环双交叉积，推广了已有有限群等相关结果，还给出了从Yang–Baxter方程解等得到局部紧量子群的途径。