On the Yang-Baxter equation and left nilpotent left braces

DOI10.1016/J.JPAA.2016.07.014zbMATH Open1397.16033arXiv1601.07131OpenAlexW2962844998MaRDI QIDQ343571

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Publication date: 28 November 2016

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Abstract: We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show 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 also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into a finite braided group. For a left brace A, we explore the close relation between the multipermutation level of the solution associated with it and the radical chain $A^{(n+1)}=A^{(n)}*A$ introduced by Rump.

Full work available at URL: https://arxiv.org/abs/1601.07131

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