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)花括号。Galois理论是使用群研究数系统(域)的对称性(自同构)的代数领域。一个熟悉的例子是复数共轭，它是复数的自同构，它固定所有实数，并生成一个2阶的群，作为实数的扩展的复数的伽罗瓦群。在其他方面，伽罗瓦理论解释了为什么5次或更多次的多项式方程通常不能用公式求解。Hopf-Galois理论通过用Hopf代数代替Galois群来推广这一经典情况。一个给定的域扩张可能有许多Hopf-Galois结构，根据Greither和Pareigis(1987)的结果，找到它们都相当于群论中的一个组合问题。有许多结果列举或限制了Hopf-Galois结构用于不同Galois群的扩张，而首席研究者一直是这一努力的关键贡献者。(斜)花括号是给出杨-巴克斯特方程的解的代数对象。由于杨-巴克斯特方程在理论物理和数学的许多领域中扮演着基本的角色，它们引起了人们的极大兴趣。花括号是由Rump(2007)引入的，此后许多作者对其进行了研究。Guarnieri和Vendramin(2017)将它们推广到斜撑。Bachiller(2016)首先指出，Hopf-Galois结构和支撑之间存在联系。这种联系实际上延伸到了斜撑。这种联系的产生是因为Hopf-Galois结构和斜花括号都对应于另一个群的全息中的正规子群。这意味着在Hopf-Galois结构和斜括号之间存在对应关系，这虽然不是一对一的，但允许将Hopf-Galois结构的结果重新解释为斜括号的结果，反之亦然。本项目将研究关于Hopf-Galois结构的支撑和斜支撑以及它们的类似物的几个重要的公开问题。这些问题都涉及到支撑或Hopf-Galois结构的外延的概念，研究的初始阶段将从不同的角度理解这些外延。这样做所得到的见解将被用于列举四元数和二面体花括号，研究具有不可解乘法群和可解加群的斜花括号，寻找不是对合杨-巴克斯特群的可解群的新例子，并对一些类的简单花括号和单斜花括号进行分类。

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.