## Project Details

### Description

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.

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.

Acronym | FWOTM921 |
---|---|

Status | Finished |

Effective start/end date | 1/10/18 → 30/09/20 |

### Flemish discipline codes

- Applied mathematics in specific fields not elsewhere classified

### Keywords

- Yang-Baxter