Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation

Abstract

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang-Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.

arXiv comment18 pages. In this version, Definition 5.5 and Proposition 5.6 were modified following arXiv:2310.07474. Postprint version

References in corpus (8)

• Set-theoretical solutions to the quantum Yang-Baxter equation
• On set-theoretical solutions of the quantum Yang-Baxter equation
• Skew braces and the Yang-Baxter equation
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Skew left braces of nilpotent type
• Factorizations of skew braces
• Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces

Cited by in corpus (6)

• Central nilpotency of left skew braces and solutions of the Yang-Baxter equation
• Schur covers of skew braces
• On two-sided skew braces
• The structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented
• Solutions of the Yang-Baxter equation and strong semilattices of skew braces
• On Grün's lemma for perfect skew braces