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

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

TL;DR

This work develops a comprehensive framework for central nilpotency in left skew braces, clarifying their role in set-theoretic Yang–Baxter equation solutions and addressing intricate ideal-theoretic questions. A central achievement is proving the equivalence of two universal-algebra–inspired commutator definitions, thereby unifying approaches to the commutator of ideals via absorbing polynomials and other polynomials. The paper then builds a robust theory of nilpotent ideals through the Fitting and Frattini concepts, together with a torsion theory and a well-defined index for subbraces, and extends these ideas to local and hypercentral settings. Through detailed worked examples, it exposes fundamental peculiarities of braces that differ from groups and rings, including nonexistence of idealisers and the failure of naive product-closure for central-nilpotent ideals, thereby providing precise tools for analyzing multipermutation YBE solutions and guiding future research in brace theory.

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.

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

TL;DR

This work develops a comprehensive framework for central nilpotency in left skew braces, clarifying their role in set-theoretic Yang–Baxter equation solutions and addressing intricate ideal-theoretic questions. A central achievement is proving the equivalence of two universal-algebra–inspired commutator definitions, thereby unifying approaches to the commutator of ideals via absorbing polynomials and other polynomials. The paper then builds a robust theory of nilpotent ideals through the Fitting and Frattini concepts, together with a torsion theory and a well-defined index for subbraces, and extends these ideas to local and hypercentral settings. Through detailed worked examples, it exposes fundamental peculiarities of braces that differ from groups and rings, including nonexistence of idealisers and the failure of naive product-closure for central-nilpotent ideals, thereby providing precise tools for analyzing multipermutation YBE solutions and guiding future research in brace theory.

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.
Paper Structure (9 sections, 37 theorems, 102 equations, 5 tables)

This paper contains 9 sections, 37 theorems, 102 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unifying Universal Algebra definitions of commutators of ideals in braces
  4. Central nilpotency of braces
  5. Torsion theory
  6. The index of a subbrace
  7. Central nilpotency for ideals
  8. Worked examples
  9. Acknowledgments

Key Result

Theorem 3.6

Let $I, J$ be ideals of a brace $B$. Then:

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Example 3.7
  • Definition 4.1
  • ...and 81 more