New simple solutions of the Yang--Baxter equation and their permutation groups
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.
Keywords
Cite
@article{arxiv.2401.12904,
title = {New simple solutions of the Yang--Baxter equation and their permutation groups},
author = {Ferran Cedo and Jan Okninski},
journal= {arXiv preprint arXiv:2401.12904},
year = {2024}
}
Comments
arXiv admin note: text overlap with arXiv:2112.07271 typos in the formulation of Theorem 3.1 corrected