### Abstract

© 2016 Elsevier B.V. We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang–Baxter equation, we call them solutions. We prove that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We find an explicit relation between the multipermutation level of a left brace and the length of the radical chain A (n+1) =A (n) ⁎A introduced by Rump. We also show that a finite solution of the Yang–Baxter equation can be embedded in a convenient way into a finite left brace, or equivalently into a finite involutive braided group.

Original language | English |
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Pages (from-to) | 751-756 |

Journal | Journal of Pure and Applied Algebra |

Volume | 221 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Apr 2017 |

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## Cite this

Cedó, F., Gateva-Ivanova, T., & Smoktunowicz, A. (2017). On the Yang–Baxter equation and left nilpotent left braces.

*Journal of Pure and Applied Algebra*,*221*(4), 751-756. https://doi.org/10.1016/j.jpaa.2016.07.014