2023/12/15 by Ilaria Colazzo, Eric Jespers, Colazzo, Ilaria +5 · 1 citation
Mathematics · #16T25 20N99 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2312.09687
openalex publication_date 2023/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach includes and extends the work of Joyce for quandles and Castelli for involutive solutions, demonstrating that the simplicity of a solution can be understood through its associated permutation skew left brace. In particular, we show that this skew left brace must have the smallest non-zero ideal, and the quotient by this ideal gives a trivial skew left brace of cyclic type; clearly all simple skew left braces satisfy these assumptions. As an application of our approach we construct and characterise new infinite families of simple solutions that are neither involutive nor quandles. Additionally, we show that our method can be applied to simple skew left braces to generate further families of simple solutions.