Simple solutions of the Yang-Baxter equation
Ilaria Colazzo, Eric Jespers, Łukasz Kubat, Arne Van Antwerpen
TL;DR
The paper develops a unified algebraic framework based on skew left braces to classify finite simple non-degenerate set-theoretic solutions of the Yang–Baxter equation. It proves that such solutions are either Lyubashenko type (prime-order, affine in structure) or arise from a finite brace $B$ additively generated by the solution set $X$, with the smallest non-zero ideal $V$ and $B/V$ trivial of cyclic type, complemented by a transitive action of $V$ on $X$; within this second class, three subcases governed by $B^{(2)}$ and $B^{(3)}$ are identified, with case (3) open. The authors provide complete descriptions in the Lyubashenko and abelian-$V$ or non-abelian-$V$ settings, give explicit formulae for the corresponding $r$, and construct new infinite families of simple solutions, including non-involutive and non-quandle examples, thereby extending and unifying prior work of Joyce, Castelli, and others. The framework also explains how simple skew left braces can yield additional simple solutions and connects with Byott’s brace families, opening avenues for further structural and constructive classifications. Overall, the work advances understanding of how simplicity is manifested in the associated brace structures and broadens the landscape of simple YBE solutions with concrete, algebraically tractable models.
Abstract
We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach includes and extends the work of Joyce for quandles and Castelli for involutive solutions, demonstrating that the simplicity of a solution can be understood through its associated permutation skew left brace. In particular, we show that this skew left brace must have the smallest non-zero ideal, and the quotient by this ideal gives a trivial skew left brace of cyclic type; clearly all simple skew left braces satisfy these assumptions. As an application of our approach we construct and characterise new infinite families of simple solutions that are neither involutive nor quandles. Additionally, we show that our method can be applied to simple skew left braces to generate further families of simple solutions.