Central nilpotency of left skew braces and solutions of the Yang-Baxter equation
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| Format: | Preprint |
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2023
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| author | Ballester-Bolinches, Adolfo Esteban-Romero, Ramón Ferrara, Maria Pérez-Calabuig, Vicent Trombetti, Marco |
| author_facet | Ballester-Bolinches, Adolfo Esteban-Romero, Ramón Ferrara, Maria Pérez-Calabuig, Vicent Trombetti, Marco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_07474 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Central nilpotency of left skew braces and solutions of the Yang-Baxter equation Ballester-Bolinches, Adolfo Esteban-Romero, Ramón Ferrara, Maria Pérez-Calabuig, Vicent Trombetti, Marco Group Theory Rings and Algebras 16T25, 20F18, 81R50 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. |
| title | Central nilpotency of left skew braces and solutions of the Yang-Baxter equation |
| topic | Group Theory Rings and Algebras 16T25, 20F18, 81R50 |
| url | https://arxiv.org/abs/2310.07474 |