Skew bracoids containing a skew brace
Ilaria Colazzo, Alan Koch, Isabel Martin-Lyons, Paul J. Truman
TL;DR
The paper addresses the problem of relating skew bracoids to left semibraces and using this link to generate set theoretic Yang-Baxter equation solutions. It proves a bijection between a large family of bracoids containing a brace and left semibraces, enabling YBE constructions to be obtained by translating bracoids into semibraces. Key contributions include detailed equivalences characterizing bracoids containing a brace, a rigorous bijection with semibraces via the lambda and rho maps, and explicit formulas for YBE solutions derived from bracoids. This work strengthens the connections between bracoids, braces, semibraces, and Hopf-Galois theory, and provides systematic methods and examples for generating and studying YBE solutions within this algebraic framework.
Abstract
Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A consequence of this correspondence is that skew bracoids in this family can be used to obtain and study solutions of the set-theoretic Yang--Baxter equation; we study this process and the resulting solutions. We give numerous examples of skew bracoids satisfying our hypothesis, drawing upon a variety of constructions in the literature.