Solutions of the Yang-Baxter equation and associated algebraic structures

One of the fundamental equations in mathematical physics is the Yang-Baxter equation. Up until today, not all solutions of this equation are known. Therefore, in 1992, Drinfeld posed the question of describing all so called set-theoretic solutions of the Yang-Baxter equation.

This project is motivated by this open problem. In particular, we are very interested in the group and ring theoretic aspects that arise. More precisely, with every set-theoretic solution of the Yang- Baxter equation we can associate a group, called the "structure group", and consequent algebras. These structures will be studied intensively.

To do so, we will use a tool, recently introduced by Rump, that opened an entirely new area of research. This tool, called a left brace, is a set with two operations, such that the set forms an abelian group with one operation and a group with the other operation. Moreover, there is a correspondence between the two operations. Soon this notion was generalized into the notion of a skew left brace, and later into that of a left semi-brace. All these structures will be studied and we relate them with solutions of the Yang-Baxter equation and consequently with groups and algebras.