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

TL;DR

This paper explores the structure of centrally nilpotent skew braces, their ideal theory, and their relation to solutions of the Yang-Baxter equation, introducing new concepts like the good Fitting ideal and characterizing ideal commutators.

Contribution

It introduces a new framework for understanding centrally nilpotent skew braces, including the concept of a good Fitting ideal and characterizes ideal commutators using absorbing polynomials.

Findings

Product of centrally nilpotent ideals need not be centrally nilpotent

Introduces the good Fitting ideal for skew braces

Characterizes the commutator of ideals in terms of absorbing polynomials

Abstract

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general.