Weak skew left braces, Hopf-Galois theory, and the Yang-Baxter equation

弱斜左括号、Hopf-Galois 理论和 Yang-Baxter 方程

基本信息

批准号：
EP/W012154/1
负责人：
Paul Truman
金额：
$ 6.86万
依托单位：
Keele University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW012154%2F1
关键词：
Weak skew left braces Hopf

项目摘要

This proposal focusses on generalizing a recently-discovered connection between topics in abstract algebra and theoretical physics. The algebraic topic is Hopf-Galois theory; this is a generalization of Galois theory, a classical topic that arose from studying certain symmetries present amongst the roots of polynomial equations. The modern interpretation uses a field extension in place of a concrete equation, and studies this via a group, called the Galois group of the field extension. Hopf-Galois theory replaces the Galois group by a Hopf algebra; in fact, a given field extension may admit a number of so-called Hopf-Galois structures, each giving a different context in which we can study the field extension. Hopf-Galois theory is a fruitful area of research, with connections to number theory, group theory, and many other areas of abstract algebra. However, an unexpected connection has recently emerged between Hopf-Galois theory and methods for producing solutions to the Yang-Baxter equation in theoretical physics, which has applications in topics as diverse as integrable systems, knot theory, and quantum computing. The linchpin of this connection is a further algebraic object called a skew left brace; these are generalizations of braces, which were introduced by Rump in 2007 to generate and study solutions of the Yang-Baxter equation. It can be shown that there is a correspondence between Hopf-Galois structures on certain field extensions and skew left braces; these in turn yield solutions to the Yang-Baxter equation. It has subsequently been found that important properties of Hopf-Galois structures can be determined by studying the corresponding skew left braces. The overarching aim of this project is to formulate a more general object, a weak skew left brace, such that weak skew left braces correspond to Hopf-Galois structures on a much larger class of field extensions. The first objective of the project will be formulate the appropriate generalization of the definition of a skew left brace, and to establish fundamental consequences of this definition. Subsequent objectives will include enumerating and classifying weak skew left braces with specified properties, and investigating how properties of Hopf-Galois structures and weak skew left braces are related to one another. Since the original motivation for the introduction of skew left braces was the desire to generate and study solutions to the Yang-Baxter equation, it will be a most interesting to investigate what connection weak skew left braces might have with this question.

该提案的重点是概括最近发现的抽象代数和理论物理主题之间的联系。代数题目是Hopf-Galois理论；这是伽罗瓦理论的概括，伽罗瓦理论是一个经典主题，源于研究多项式方程根中存在的某些对称性。现代解释使用场扩展代替具体方程，并通过称为场扩展伽罗瓦群的群来研究这一点。 Hopf-Galois 理论用 Hopf 代数代替了 Galois 群；事实上，一个给定的域扩展可能允许许多所谓的 Hopf-Galois 结构，每个结构都给出了我们可以研究域扩展的不同上下文。霍普夫-伽罗瓦理论是一个卓有成效的研究领域，与数论、群论和抽象代数的许多其他领域都有联系。然而，最近，霍普夫-伽罗瓦理论与理论物理中杨-巴克斯特方程解的方法之间出现了意想不到的联系，该方程在可积系统、结理论和量子计算等多种主题中都有应用。这种连接的关键是另一个称为斜左大括号的代数对象；这些是大括号的推广，由 Rump 于 2007 年引入，用于生成和研究 Yang-Baxter 方程的解。可以证明，某些域扩展上的Hopf-Galois结构与斜左括号之间存在对应关系；这些反过来又产生了杨-巴克斯特方程的解。随后发现，Hopf-Galois 结构的重要性质可以通过研究相应的斜左括号来确定。该项目的总体目标是制定一个更通用的对象，即弱斜左括号，使得弱斜左括号对应于更大类别的域扩展上的 Hopf-Galois 结构。该项目的首要目标是对左斜括号的定义进行适当的概括，并确定该定义的基本结果。后续目标将包括对具有指定属性的弱斜左括号进行枚举和分类，并研究 Hopf-Galois 结构和弱斜左括号的属性如何相互关联。由于引入斜左括号的最初动机是希望生成和研究 Yang-Baxter 方程的解，因此研究弱斜左括号与这个问题可能有什么联系将是最有趣的。