New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces

TL;DR

This paper explores simple solutions to the Yang-Baxter equation, establishing a link with simple left braces, and provides new methods to construct such solutions, addressing open problems in the field.

Contribution

It introduces new simple solutions to the Yang-Baxter equation and connects them to simple left braces, expanding the known classes of solutions and providing construction techniques.

Findings

Identified conditions for solutions to be simple

Established a link between simple solutions and simple left braces

Constructed new examples of simple solutions

Abstract

Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions are found in order that these solutions are simple. Then a link between simple solutions and simple left braces is established, that allows us to construct more examples of simple solutions. In particular, the results answer some problems stated in the recent paper [13].