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.

该建议的重点是概括抽象代数与理论物理学中的主题之间最近发现的联系。代数主题是Hopf-Galois理论。这是Galois理论的概括，Galois理论是一个经典的主题，它是由于研究多项式方程的根源中存在的某些对称性而引起的。现代解释使用场扩展代替混凝土方程，并通过一个称为场扩展的Galois组进行研究。 Hopf-Galois理论用Hopf代数取代了Galois集团。实际上，给定的场扩展可以接收许多所谓的Hopf-Galois结构，每个结构都提供了不同的环境，我们可以研究场扩展。 Hopf-Galois理论是一个富有成果的研究领域，与数字理论，群体理论以及抽象代数的许多其他领域的联系。然而，最近在Hopf-Galois理论和为Yang-Baxter方程生成理论物理学的解决方案的方法之间出现了意外的联系，该物理学在类似于集成系统，结理论和量子计算的主题中应用。该连接的linchpin是一个代数的另一个代数对象，称为偏斜的左撑杆。这些是牙套的概括，Rump在2007年引入了牙式，以生成和研究Yang-Baxter方程的解决方案。可以证明，在某些场扩展上的Hopf-Galois结构与左括号偏斜之间存在对应关系。这些反过来将解决方案产生到Yang-Baxter方程。随后发现，可以通过研究相应的偏斜左括号来确定Hopf-Galois结构的重要特性。该项目的总体目的是制定一个更通用的对象，一个弱偏斜的左支架，以使弱偏斜的左括号对应于更大类场扩展类别的Hopf-Galois结构。该项目的第一个目标是制定偏斜左支架定义的适当概括，并确定该定义的基本后果。随后的目标将包括具有指定特性的弱偏斜左括号和分类，并研究Hopf-Galois结构的特性和弱偏斜左括号如何相互关联。由于引入偏斜的左括号的最初动机是渴望生成和研究Yang-Baxter方程的解决方案，因此研究该问题可能会有什么联系偏斜的左括号可能是什么。