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.

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

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• Solutions of the Yang-Baxter equation and strong semilattices of skew braces
• On Grün's lemma for perfect skew braces