Skew Braces as Remnants of Co-quasitriangular Hopf Algebras in $\mathrm{SupLat}$

TL;DR

This paper explores the connection between skew braces, Hopf algebras in the category of complete lattices, and solutions to the Yang-Baxter equation, providing a new algebraic framework for understanding these structures.

Contribution

It introduces a novel approach linking Hopf algebras in $ ext{SupLat}$ to skew braces and YBE solutions, unifying these concepts through the notion of remnants.

Findings

Hopf algebras in $ ext{SupLat}$ have associated groups called remnants.

Co-quasitriangular structures induce YBE solutions compatible with group structures.

Universal skew braces can be recovered from FRT-type Hopf algebras in $ ext{SupLat}$.

Abstract

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, $\mathrm{SupLat}$, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, $\mathcal{H}$ in $\mathrm{SupLat}$, has a corresponding group, $R(\mathcal{H})$, which we call its remnant and a co-quasitriangular structure on $\mathcal{H}$ induces a YBE solution on $R(\mathcal{H})$, which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, $\mathcal{H}$, this secondary group structure appears as the projection of the transmutation of $\mathcal{H}$. Finally, for any YBE solution, we obtain a FRT-type Hopf algebra in $\mathrm{SupLat}$, whose remnant recovers the universal skew brace of the solution.