Set-theoretic Solutions to the Yang-Baxter and Reflection Equations from Braces

Robert Weston, Heriot-Watt University

Set theoretic solutions to the Yang-Baxter equation have been studied by various authors following a suggestion of Drinfeld. They have applications in a number of areas including soliton cellular automata and Yang-Baxter maps. In 2007, a new algebraic object, a brace, was defined by Rump. A brace is an Abelian group (A,+) that has an additional group multiplication * satisfying a*(b+c)=a*b+a*c-a. A brace can also be understood as a generalisation of a radical ring. Remarkably, there is a theorem due to Rump that states that all set-theoretic solutions to the Yang-Baxter equation can be understood in terms of braces. Recently, Smoktunowicz, Vendramin and I have used braces in order to construct solutions of the reflection equation. These solutions can then be used to construct transfer matrices of open systems. In this seminar I will summarise the definition and properties of braces and present some simple examples. I will then describe the use of braces in constructing solutions of both the Yang-Baxter and reflection equations.