Hopf-Galois Theory and Skew Braces

该项目将探索和发展两个相当不同的代数领域之间的联系，即Hopf-Galois理论和（斜）括号。Galois理论是代数领域，它研究使用群的数字系统（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，它固定所有的真实的数并生成一个2阶群，复数的伽罗瓦群作为真实的数的扩展。除此之外，伽罗瓦理论解释了为什么5次或更高次的多项式方程通常不能用公式求解。霍普夫-伽罗瓦理论通过用一个霍普夫代数代替伽罗瓦群来推广这种经典情况。一个给定的域扩张可以有许多霍普夫-伽罗瓦结构，根据Greither和Pareigis（1987）的结果，找到它们就相当于一个群论中的组合问题。有许多结果列举或限制霍普夫伽罗瓦结构的扩展与各种伽罗瓦集团，和首席研究员一直是一个关键的贡献者，这一努力。（斜）括号是给出杨-巴克斯特方程的解的代数对象。他们是相当大的兴趣，因为杨巴克斯特方程起着基础作用，在许多领域的理论物理和数学。Rump（2007）介绍了支具，此后许多作者对其进行了研究。Guarnieri和Vendramin（2017）将其推广到斜撑。Bachiller（2016）首先指出，Hopf-Galois结构和支撑之间存在联系。这种连接实际上延伸到斜撑。这种联系是因为霍普夫-伽罗瓦结构和斜括号都对应于另一个群的全形中的正则子群。这意味着在霍普夫-伽罗瓦结构和斜括号之间存在对应关系，虽然不是一对一的，但允许将霍普夫-伽罗瓦结构上的结果重新解释为斜括号上的结果，反之亦然。这个项目将研究几个重要的开放问题的括号和斜括号，以及他们的类似物的霍普夫伽罗瓦结构。这些问题都涉及到括号或Hopf-Galois结构的扩展概念，研究的初始阶段将是从各种角度理解这些扩展。从这样做获得的见解，然后将被用来枚举四元数和二面角括号，研究斜括号与不溶性乘法组和可溶性添加剂组，寻找新的例子，可溶性群体，这不是对合杨巴克斯特集团，并分类一些类的简单括号和简单斜括号。

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.