Cocycle superrigidity for ergodic actions of non-semisimple Lie groups

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension $16$

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension $16$. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group

The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.

Actions of semisimple Lie groups on circle bundles

Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.

关于某些仿射代数约化群的霍普夫2-上循环

Hopf $2$-cocycles for certain affine algebraic reductive groups

本文受仿射代数约化群\(G\)的霍普夫2-上循环分类问题启发，利用相关命题对单位连通分支为环面\(T\)的\(G\)的（极小）霍普夫2-上循环分类，通过分类有限不可分解半单模范畴及秩为1的模范畴，证明纤维函子与霍普夫2-上循环一一对应。

半单张量范畴的扭曲德利涅积

Twisted Deligne products of semisimple tensor categories

探讨半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积分类，给出特定条件下唯一扭曲德利涅积是普通积的结论，借助相关工作实现群论分类及融合范畴精确分解，附录引入范畴$n$-上链概念并回答相关问题。

Non-existence of Hopf orders for a twist of the alternating and symmetric groups

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n \geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m \geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $\mathbb{Z}_2^m$. This provides more examples of com…