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<p>We introduce a novel algebraic structure called <i>di-skew brace</i> by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang–Baxter equation. We study the structural properties of these solutions with a particular focus on their left derived shelves, which belong to the class of <i>conjugation racks</i>. Consistently, we show that these solutions belong to a broader class that includes skew brace solutions. In particular, we prove that each such solution can be decomposed as a <i>hemi-semidirect</i> product of a skew brace solution endowed with a certain compatible action on the idempotents of the associated di-skew brace structure. Finally, we provide concrete instances of these solutions through a suitable notion of <i>averaging operators</i> on groups.</p>

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Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang–Baxter equation

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  • 发表日期：

    2025年12月26日

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  • Andrea Albano,
  • Paola Stefanelli

摘要

We introduce a novel algebraic structure called di-skew brace by which we show that generalized digroups systematically yield bijective, non-degenerate solutions to the set-theoretic Yang–Baxter equation. We study the structural properties of these solutions with a particular focus on their left derived shelves, which belong to the class of conjugation racks. Consistently, we show that these solutions belong to a broader class that includes skew brace solutions. In particular, we prove that each such solution can be decomposed as a hemi-semidirect product of a skew brace solution endowed with a certain compatible action on the idempotents of the associated di-skew brace structure. Finally, we provide concrete instances of these solutions through a suitable notion of averaging operators on groups.