Some algebraic and categorical properties of skew braces

Details

Supervisors

Gran, Marino

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité approfondie

Abstract

enA skew brace is an algebraic structure, introduced by Guarnieri and Vendramin in 2017, to provide non-involutive solutions of the Yang-Baxter equation, which is a statistical mechanics equation. The skew brace structure is formed by a set equipped with two group operations that satisfy a specific distributive identity. In this thesis, we study some of its main properties and we explain how skew braces provide a wider range of solutions than other notions developed earlier, as the left braces. On the other hand, skew braces and their morphisms form a category, denoted by SKB. We first notice that this category has many common properties with the category of groups. Indeed, SKB is a semi-abelian category, which is a relevant property to develop, for instance, isomorphism theorems and homological theory. Moreover, the category is strongly protomodular, which gives a suitable framework for commutator theory. The aim of this thesis is to consider two different approaches of skew braces and to observe the interactions between these two theories.