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arXiv · 2506.12503

One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation

Marco Castelli
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Abstract

We study the class of one-generator solutions to the Yang-Baxter equation, extending some recent results concerning the classes of involutive and multipermutation solutions. Moreover we show the precise relationship between indecomposable solutions to the Yang-Baxter equation and finite one-generator skew braces, giving a positive answer to a question posed by Agata and Alicja Smoktunowicz. In the last part, we apply our results to the involutive case, and we present some numerical results involving solutions of small size.

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Marco Castelli. 2025-06-14. One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation. https://arxiv.org/abs/2506.12503

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