关于整体域上有限交换Galois扩张的Dec群

关于拟局部域的Brauer群及其有限Galois扩张的范数群

On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions

本文证明了可分阿贝尔扭群可实现为拟局部域的Brauer群。它描述了主拟局部域的Brauer群的同构类，并解决了关于两种基本类型的Henselian值绝对稳定域的Brauer群的约化分支的类似问题。对于拟局部域E和有限可分扩张R/E，我们得到了范数群等式$N(R/E) = N(R_{0}/E)$成立的两个充分条件，其中R_{0}是E在R中的极大交换扩张.这是用于推导信息产生的具体关系之间的伽罗瓦群和有限伽罗瓦扩展E的范群。

作为可解扩张上Hopf-Galois结构的类型出现的非阿贝尔单群

Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension

本研究确定了可解扩张上作为Hopf-Galois结构类型的有限非阿贝尔单群，同时给出了乘法群可解的斜brace中作为加法群的这类单群的完整列表。

Graded-division algebras and Galois extensions

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras. On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field F is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of F and the isomorphism class of a G-Galois extension of F. This connection is used to classify the simple G-Galois extensions of F in terms of …

Bi-skew braces and Hopf Galois structures

We define a bi-skew brace to be a set $G$ with two group operations $\star$ and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group $(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace, then $G$ corresponds to a Hopf Galois structure of type $(G, \star)$ on any Galois extension of fields with Galois group isomorphic to $(G, \circ)$. If $G$ is a bi-skew brace, then $G$ also corresponds to a Hopf Galois structure of type $(G, \circ)$ on a Galois extension of fields with Galois group isomorphic to $(G, \star)$. Many non-trivial examples exist. One sou…

The commutative inverse semigroup of partial abelian extensions

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$ on the subalgebra of invariants $S^{α_H}$ of $S$ such that $S^{α_H}$ is an $α_{G/H}$-partial Galois extension of $S^{α_G}$. Second, assuming that $G$ is abelian, we construct a commutative inverse semigroup $T_{par}(G,R)$, whose elements are equivalence classes of $α_G$-part…

Torsion group schemes as iterative differential Galois groups

We are considering iterative derivations on the function field L of abelian schemes in positive characteristic p>0, and give conditions when the torsion group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are the ID-Galois groups of L over certain ID-subfields. For an explicit example, we even give a construction of (a family of) such iterative derivations.

p-幂零群块的辉煌Rickard等价的Galois下降

Galois descent of splendid Rickard equivalences for blocks of $p$-nilpotent groups

本文通过证明链复形的一般Galois下降准则，加强了p-幂零群块等价及辉煌Rickard等价的Galois下降结果，从而验证了Broué阿贝尔亏群猜想的一个精细化版本。

$p^{2} q$ 次扩张上的 Hopf-Galois 结构与 $p^{2} q$ 阶斜括号：初等阿贝尔 Sylow $p$-子群情形

Hopf-Galois structures on extensions of degree $p^{2} q$ and skew braces of order $p^{2} q$: the elementary abelian Sylow $p$-subgroup case

本文完成了当伽罗瓦群 Sylow $p$-子群为初等阿贝尔群时，$p^{2}q$ 次扩张的 Hopf-Galois 结构及相应斜括号的分类，通过伽马函数与全形正则子群的一一对应发展了更概念化的枚举方法。

非阿贝尔Hopf上同调

Non-Abelian Hopf Cohomology

引入了系数在Hopf模中的霍普夫代数的非交换上同调集.我们证明了这些集合推广了Serre的非阿贝尔群上同调理论.使用下降的技术，我们建立，我们的建设，使分类以及扭曲的形式，在霍普夫-伽罗瓦扩展的Torsors在霍普夫模块。

关于四元数与二面体brace的数量及Hopf--Galois结构

On the number of quaternion and dihedral braces and Hopf--Galois structures

本研究证明了乘群为广义四元数群的给定阶braces数量的相关猜想，给出了二面体群情形的类似结果，并枚举了对应Galois扩张上的阿贝尔型Hopf-Galois结构。

有理共形场论中的Galois群

Galois Groups in Rational Conformal Field Theory

在有理共形场论（RCFT）中的融合环可以被描述为多项式环，具有整数系数，模某些关系。我们使用这些关系的伽罗瓦群，以获得一个局部的一组方程的融合品种的点。这些方程足以将所有的RCFT、Galois群按Galois群分类。证明了Galois群等价于伪RCFT群。我们证明了在RCFT中遇到的伽罗瓦群都是阿贝尔群，这意味着模矩阵的根是可解的。