Central nilpotency of left skew braces and solutions of the Yang–Baxter equation

Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Maria Ferrara, Vicent Pérez-Calabuig and Marco Trombetti

Vol. 335 (2025), No. 1, 1–32

DOI: 10.2140/pjm.2025.335.1

Abstract

This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory, we introduce an “index” for subbraces, but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a good Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient is the characterisation of the commutator of ideals in terms of star products, which solves a problem of Bonatto and Jedlička (J. Algebra Appl. 22:12 (2023), art. id. 2350255). Moreover, we provide an example showing that the idealiser of a subbrace does not exist in general.

Keywords

brace, Yang–Baxter equation, nilpotency, Fitting ideal, commutator of ideals, idealiser

Mathematical Subject Classification

Primary: 16T25, 20F18

Secondary: 81R50

Milestones

Received: 18 March 2024

Revised: 13 January 2025

Accepted: 25 February 2025

Published: 24 March 2025