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arXiv · 2401.12904

New simple solutions of the Yang--Baxter equation and their permutation groups

Ferran Cedo·Jan Okninski
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Abstract

A new class of indecomposable, irretractable, involutive, non-degenerate set-theoretic solutions of the Yang--Baxter equation is constructed. This class complements the class of such solutions constructed in \cite{CO22} and together they generalize the class of solutions described in \cite[Theorem 4.7{CO21}. Necessary and sufficient conditions are found in order that these new solutions are simple. For a rich subclass of these solutions the structure of their permutation groups, considered as left braces, is determined. In particular, these results answer a question stated in \cite{CO21}. In the finite case, all these solutions have square cardinality. A new class of finite simple solutions of non-square cardinality such that their permutation groups are simple left braces is also constructed.

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Ferran Cedo, Jan Okninski. 2024-01-23. New simple solutions of the Yang--Baxter equation and their permutation groups. https://arxiv.org/abs/2401.12904

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