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

点击翻译按钮获取中文摘要

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.