# Asymmetric product of left braces and simplicity; new solutions of the Yang-Baxter equation

## TL;DR

This paper advances the understanding of finite simple left braces by using asymmetric products to construct new classes and interpret existing ones, significantly contributing to the classification of solutions to the Yang-Baxter equation.

## Contribution

It introduces a novel application of asymmetric products to construct and interpret finite simple left braces, expanding the known classes and providing new examples with solvable multiplicative groups.

## Key findings

- Constructed new classes of simple left braces.
- Interpreted all known constructions as asymmetric products.
- Produced finite simple left braces with solvable multiplicative groups of arbitrary derived length.

## Abstract

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the Yang-Baxter equation. In this paper we continue the study of finite simple left braces with the emphasis on the application of the asymmetric product of left braces in order to construct new classes of simple left braces. We do not only construct new classes but also we interpret all previously known constructions as asymmetric products. Moreover, a construction is given of finite simple left braces with a multiplicative group that is solvable of arbitrary derived length.