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

TL;DR

This paper introduces di-skew braces, a new algebraic structure, demonstrating their role in systematically producing bijective solutions to the set-theoretic Yang-Baxter equation and exploring their structural properties.

Contribution

It defines di-skew braces and shows how generalized digroups lead to new solutions of the Yang-Baxter equation, expanding the class of known solutions.

Findings

Di-skew braces systematically generate bijective, non-degenerate solutions.

Solutions can be decomposed as hemi-semidirect products involving skew braces.

Concrete solutions are constructed using averaging operators on groups.

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.