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

这项建议侧重于概括最近发现的抽象代数和理论物理主题之间的联系。代数主题是Hopf-Galois理论；这是Galois理论的推广，Galois理论是研究多项式方程根之间的某些对称性而产生的经典主题。现代解释用域扩张来代替具体的方程，并通过一个群来研究这个问题，这个群称为域扩张的伽罗瓦群。Hopf-Galois理论用Hopf代数代替了Galois群；事实上，一个给定的域扩张可以允许许多所谓的Hopf-Galois结构，每个结构都给出了一个不同的上下文，我们可以在其中研究域扩张。Hopf-Galois理论是一个卓有成效的研究领域，它与数论、群论和许多其他抽象代数领域有联系。然而，最近在Hopf-Galois理论和理论物理中产生杨-Baxter方程解的方法之间出现了一种意想不到的联系，该方法在可积系统、纽结理论和量子计算等各种主题中都有应用。这种联系的关键是另一个称为左大括号的代数对象；这些是大括号的推广，由Rump在2007年引入，以生成和研究Yang-Baxter方程的解。可以证明，在某些域扩张上的Hopf-Galois结构和左斜大括号之间存在对应关系；这反过来又给出了Yang-Baxter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左斜撑来确定。这个项目的主要目标是建立一个更一般的对象，一个弱斜左括号，使得弱斜左括号对应于更大类域扩张上的Hopf-Galois结构。该项目的第一个目标将是对左斜撑的定义作出适当的概括，并确定这一定义的基本后果。随后的目标将包括列举和分类具有特定性质的弱斜左大括号，并研究Hopf-Galois结构和弱斜左大括号的性质是如何相互关联的。由于引入左斜花括号的最初动机是希望生成和研究杨-巴克斯特方程的解，因此研究弱左花括号与这个问题可能有什么联系将是非常有趣的。

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.