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arXiv · 1904.11657

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

E. Acri·R. Lutowski·L. Vendramin
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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.

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E. Acri, R. Lutowski, L. Vendramin. 2019-04-26. Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces. https://doi.org/10.1142/s0218196719500656

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