On a theorem of Stelzer for some classes of mixed groups

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $PSL_n(q)$ collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $PSp_{2n}(q)$, $PΩ^+_{4n}(q)$, $PΩ^-_{4n}(q)$, $^3D_4(q)$, $E_7(q)$, $E_8(q)$, $F_4(q)$, or $G_2(q)$ with q even is the group algebra.

关于局部酉变换类

On classes of local unitary transformations

给出了局部酉不变性下的密度矩阵类与酉群的双陪集之间的一一对应。我们证明了局域幺正等价多体混合态类之间的相互关系与本征值的实际值无关，只依赖于本征值的重数。本文还讨论了酉群的齐性空间的解释。

关于混合无单位群作用的本质自由性

On essential freeness of actions of mixed identity-free groups

本文针对文献[ACTT23]提出的问题，证明混合无单位群的忠实广义伯努利作用及紧阿贝尔群上的仿射作用均为本质自由，拓展了具有该性质的离散遍历作用类。

Suzuki-Ree groups as algebraic groups over $\mathbb{F}_{\sqrt{\smash[b]p}}$

Among the infinite classes of finite simple groups, the most exotic classes are probably the Suzuki groups and the Ree groups. They are "twisted versions" of groups of Lie type, but they cannot be directly obtained as groups of rational points of a suitable linear algebraic group. We provide a framework in which these groups do arise as groups of rational points of algebraic groups over a "twisted field"; in the finite case, such a twisted field can be interpreted as a "field with $\sqrt{\smash[b]p}$ elements". Our framework at once allows for other, perhaps less known, exotic families of grou…

Simply-laced mixed-sign Coxeter groups, with an associate graph is a line or a simple cycle

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing the standard geometric representation of the associated Coxeter group by concerning the labels of the vertices. The group which Hironaka get by that generalization is called mixed-sign Coxeter group. In this paper we classify the simply-laced mixed-sign Coxeter groups where the associated graph is either a line or a simple cycle. We show that all the defini…

扭曲和混合

Twisting and Mixing

我们提出了一个框架，连接三类有趣的群：扭曲群（也称为Suzuki-Ree群），混合群和奇异的伪约化群。对于一个给定的特征p，我们构造扭曲和混合计划的类别。普通计划是混合计划的一个满子范畴。混合格式是由基变化的扭曲格式产生的，尽管不是所有的混合格式都是这样产生的。这些类别中的组对象被称为扭曲和混合组方案。（1）扭Chevalley群${}^2\mathsf B_2$、${}^2\mathsf G_2$和${}^2\mathsf F_4$是扭群概型的有理点。(2)Tits意义下的混合群是作为混合域上混合群概型的有理点而产生的。(3)Conrad，Gabber和Prasad的奇异伪约化群是混合群格式的Weil限制。