Weak skew left braces, Hopf-Galois theory, and the Yang-Baxter equation
Lead Research Organisation:
Keele University
Department Name: Faculty of Natural Sciences
Abstract
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.
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.
Organisations
Publications
Martin-Lyons I
(2024)
Skew bracoids
in Journal of Algebra
Martin-Lyons I
(2023)
Skew bracoids
| Description | The context of this project begins with an abstract mathematical object called a skew brace. Initially introduced in order to create and study set theoretic solutions of the Yang-Baxter equation, instances of which occur throughout mathematics and theoretical physics, skew braces were found in 2017 to be connected with Hopf-Galois structures, which are used to study symmetries in equations of a very different kind arising in number theory and abstract algebra. The overarching aim of this project is to generalize this connection by defining a new mathematical object (a "weak skew brace") that corresponds to a larger class of Hopf-Galois structures, and to investigate possible applications of this object to solutions of the Yang-Baxter equation. Skew brace theory is an extremely active area of research, with many variations and generalizations of skew braces being formulated and studied. Other researchers have recently attached the name "weak skew brace" to a different object. We therefore propose to call our objects "near skew braces". The first objective of the project was to find the correct definition of a near skew brace, motivated by the existing connection with Hopf-Galois structures. This objective has been accomplished; in fact, a variety of approaches and techniques have been developed for characterizing and constructing near skew braces. As usual when defining a new algebraic object, various notions of substructures have been formulated: sub-near skew braces, left ideals, ideals, and "enhanced" ideals, along with a natural notion of a quotient near skew brace. Tools for comparing and relating near skew braces to one another (homomorphisms) have also been formulated, and have many of the properties that one would naturally demand. The question of determining when two near skew braces are "essentially the same" (isomorphisms) is rather more delicate; at present this is the only outstanding point from objective 1. The second objective of the project was to classify certain families of near skew braces. Results of this type cannot be completed until the correct notion of isomorphism has been formulated, but many families of examples can be readily generated. In addition: a family of "almost classical" near skew braces (named after an existing construction in Hopf-Galois theory) has emerged; these have many desirable properties and seem likely to occupy a distinguished place in the theory. The third objective was to investigate connections with the so-called "Hopf-Galois correspondence". Many satisfying and illuminating results have been obtained in this area: in some cases near skew braces provide a better framework than skew braces for studying these kinds of questions. In this context, the "almost classical" near skew braces discussed above have been found to be connected with the existing notion of "induced" Hopf-Galois structures, and may offer a route to generalize these objects. The final objective is to investigate potential connections with the Yang-Baxter equation. The existing technique for constructing a solution of the Yang-Baxter equation from a skew braces does not generalize smoothly to near skew braces. A new interpretation or perspective is required; this is current work. |
| Exploitation Route | The discovery of the connection between skew braces and Hopf-Galois theory brought together two previously unconnected communities of researchers, both of whom would be interested in using and building upon the results of this project. Researchers in Hopf-Galois theory could use near skew braces to create new families of Hopf-Galois structures with desirable or pathological properties. Research into solutions of the Yang-Baxter equation involves numerous algebraic constructions and objects (skew braces, heaps, trusses, racks, quandles etc); academic working in this area might be interested in relating near skew braces to these constructions, or in formulating corresponding generalizations. |
| Sectors | Other |
| URL | http://www.ilariacolazzo.info/gryb2023/ |