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arXiv · 2505.15387

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

Andrea Albano·Paola Stefanelli
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Abstract

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.

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Andrea Albano, Paola Stefanelli. 2025-05-21. Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation. https://doi.org/10.1007/s00233-025-10606-2

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