Hopf-Galois Theory and Skew Braces

该项目将探索和发展代数的两个相当不同的领域之间的联系，即Hopf-Galois理论和（斜）括号。伽罗瓦理论是代数的一个领域，它研究使用群的数系统（域）的对称性（自同构）。一个熟悉的例子是复共轭，它是复数的自同构，固定所有实数并生成一个2阶的群，即作为实数扩展的复数的伽罗瓦群。其中，伽罗瓦理论解释了为什么5次或5次以上的多项式方程一般不能用公式求解。Hopf-Galois理论通过用Hopf代数代替Galois群来推广这一经典情形。给定的场扩展可能有许多Hopf-Galois结构，并且，根据Greither和Pareigis（1987）的结果，发现它们都相当于群论中的组合问题。有许多结果列举或限制Hopf-Galois结构与各种Galois群的扩展，首席研究员一直是这一努力的关键贡献者。斜括号是给出Yang-Baxter方程解的代数对象。由于杨-巴克斯特方程在理论物理和数学的许多领域中起着重要作用，因此它们引起了人们的极大兴趣。牙套是由Rump（2007）提出的，此后被许多作者研究。Guarnieri和Vendramin（2017）将其推广为倾斜牙套。Bachiller（2016）首先指出Hopf-Galois结构与括号之间存在联系。这种连接实际上延伸到斜撑。这种联系的产生是因为Hopf-Galois结构和斜撑都对应于另一个群的全纯态中的正则子群。这意味着Hopf-Galois结构和斜括号之间存在对应关系，虽然不是一对一的，但允许Hopf-Galois结构上的结果被重新解释为斜括号上的结果，反之亦然。本项目将研究Hopf-Galois结构中关于支撑和斜支撑及其类似问题的几个重要开放问题。这些问题都涉及到支撑或Hopf-Galois结构扩展的概念，研究的初始阶段将是从各种角度理解这些扩展。由此获得的见解将用于枚举四元数和二面体支撑，研究具有不可溶乘法群和可溶加性群的斜支撑，寻找非对合Yang-Baxter群的可溶群的新例子，并分类一些简单支撑和简单斜支撑的类别。

The project will explore and develop the connection between two rather different areas of algebra, namely Hopf-Galois theory and (skew) braces.Galois theory is the area of algebra which studies the symmetries (automorphisms) of number systems (fields) using groups. A familiar example is complex conjugation, which is an automorphism of the complex numbers that fixes all real numbers and generates a group of order 2, the Galois group of the complex numbers as an extension of the real numbers. Among other things, Galois theory explains why a polynomial equation of degree 5 or more cannot in general be solved by a formula. Hopf-Galois theory generalises this classical situation by replacing the Galois group with a Hopf algebra. A given field extension may have many Hopf-Galois structures, and, by a result of Greither and Pareigis (1987), finding them all amounts to a combinatorial problem in group theory. There are many results enumerating or restricting Hopf-Galois structures for extensions with various Galois groups, and the Principal Investigator has been a key contributor to this endeavour. (Skew) braces are algebraic objects which give solutions of the Yang-Baxter equation. They are of considerable interest since the Yang-Baxter equation plays a fundamental role in many areas of theoretical physics and of mathematics. Braces were introduced by Rump (2007) and have since been studied by many authors. They were generalised to skew braces by Guarnieri and Vendramin (2017). It was first noted by Bachiller (2016) that there is a connection between Hopf-Galois structures and braces. This connection in fact extends to skew braces. The connection comes about because both Hopf-Galois structures and skew braces correspond to regular subgroups in the holomorph of another group. This means that there is a correspondence between Hopf-Galois structures and skew braces which, while not one-to-one, allows results on Hopf-Galois structures to be reinterpreted as results on skew braces, and vice versa. This project will investigate several important open problems on braces and skew braces, and their analogues for Hopf-Galois structures. These problems all involve the notion of extensions of braces or Hopf-Galois structures, and the initial phase of the research will be to understand these extensions from a variety of perspectives. The insights gained from doing so will be then be used to enumerate quaternionic and dihedral braces, to study skew braces with insoluble multiplicative group and soluble additive group, to look for new examples of soluble groups which are not involutive Yang-Baxter groups, and to classify some classes of simple braces and simple skew braces.