Constructing skew bracoids via abelian maps, and solutions to the {Y}ang-{B}axter equation

TL;DR

The paper develops a framework to construct bracoids from abelian maps and identifies their strong left ideals, connecting these to brace quotients and to new, explicit YBE solutions. By analyzing two dual perspectives on bi-skew braces, it provides criteria under which a subgroup yields a bracoid or a brace, and shows how to generate families of bracoids via fixed point sets and ker/ images of $\psi$. It then builds right non-degenerate set-theoretic solutions to the Yang–Baxter equation from abelian-map bracoids, including a idempotent-map construction and a two-block product construction with explicit $\lambda$ and $\rho$ maps. The work also furnishes concrete examples and shows how to obtain braces embedded inside bracoids, enabling the application of established YBE reduction techniques. Overall, it broadens the toolkit for producing YBE solutions from algebraic structures related to braces, with potential implications for Hopf–Galois theory and related areas.

Abstract

We show how one can use the skew braces constructed using abelian maps to generate families of skew bracoids as defined by Martin-Lyons and Truman. Under certain circumstances, these bracoids give right non-degenerate solutions to the Yang-Baxter equation.

Theorems & Definitions (36)

• Theorem A
• Theorem B
• Theorem C
• Proposition 2.1
• proof
• Corollary 2.2
• Remark 2.3
• Remark 2.4
• Example 2.5
• Example 2.6