Skew left braces and 2-reductive solutions of the Yang-Baxter equation

2023/03/27 by Přemysl Jedlička, Jedlička, Přemysl, Agata Pilitowska +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Primary: 16T25 Secondary: 20B25 #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2303.15154

openalex publication_date 2023/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study 2-reductive non-involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. We give a combinatorial construction of any such solution of any (even infinite) size. We also prove that solutions associated to a skew left brace are 2-reductive if and only if the skew left brace is nilpotent of class 2. Moreover, all such skew left braces are actually bi-skew left braces. We focus on these structures and we give several equivalent properties characterizing solutions associated to bi-skew left braces.

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• Indecomposable non-degenerate 2-permutational solutions of the Yang-Baxter equation

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