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

TL;DR

This paper explores the concept of nilpotency in skew braces and its implications for solutions to the Yang-Baxter equation, establishing analogies with nilpotent groups and extending classical theorems.

Contribution

It introduces annihilator nilpotent skew braces and generalizes key group theory results to the skew brace framework.

Findings

Annihilator nilpotent skew braces form a significant class analogous to nilpotent groups.

Several classical theorems in group theory are extended to skew braces.

Insights into the structure of solutions to the Yang-Baxter equation through nilpotency concepts.

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.