### Abstract

© 2019 World Scientific Publishing Company. The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation (YBE) 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 YBE. 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.

Original language | English |
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Article number | 1850042 |

Journal | Communications in Contemporary Mathematics |

Volume | 21 |

DOIs | |

Publication status | Published - 1 Dec 2019 |

### Keywords

- Yang-Baxter equation
- asymmetric product
- brace
- set-theoretic solution
- simple brace

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## Cite this

Bachiller, D., Cedó, F., Jespers, E., & Okniński, J. (2019). Asymmetric product of left braces and simplicity; New solutions of the Yang-Baxter equation.

*Communications in Contemporary Mathematics*,*21*, [1850042]. https://doi.org/10.1142/S0219199718500426