Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces

Abstract

Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right $p$-nilpotent skew braces. The theory of left $p$-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

arXiv comment23 pages, 3 tables. Final version, accepted for publication in International Journal of Algebra and Computation

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• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• Enumeration of set-theoretic solutions to the Yang-Baxter equation
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation
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