Wilson conjecture for omega-categorical Lie algebras, the case 4-Engel characteristic 3
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915015037026304 |
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| author | d'Elbée, Christian |
| author_facet | d'Elbée, Christian |
| contents | We continue our study of the Wilson conjecture for $ω$-categorical Lie algebras and prove that $ω$-categorical $4$-Engel Lie algebras of characteristic $3$ are nilpotent. We develop a set of tools to adapt in the definable context some classical methods for studying Engel Lie algebras (Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others). We solve the case at hand by starting a systematic study of Lie algebras for which there is a $k$ such that the principal ideal generated by any element is nilpotent of class $<k$ (which we call $k$-strong Lie algebras). We use computer algebra to check basic cases of a conjectural arithmetical property of those, namely that $x^{k-1}y^{k-1} = (-1)^{k-1}y^{k-1}x^{k-1}$ is an identity for Lie elements of the enveloping algebra. The solution is given by reducing the problem to $k$-strong Lie algebras generated by particularly well behaved sandwiches in the sense of Kostrikin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wilson conjecture for omega-categorical Lie algebras, the case 4-Engel characteristic 3 d'Elbée, Christian Rings and Algebras Logic 03C60, 17B30, 20F18 We continue our study of the Wilson conjecture for $ω$-categorical Lie algebras and prove that $ω$-categorical $4$-Engel Lie algebras of characteristic $3$ are nilpotent. We develop a set of tools to adapt in the definable context some classical methods for studying Engel Lie algebras (Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others). We solve the case at hand by starting a systematic study of Lie algebras for which there is a $k$ such that the principal ideal generated by any element is nilpotent of class $<k$ (which we call $k$-strong Lie algebras). We use computer algebra to check basic cases of a conjectural arithmetical property of those, namely that $x^{k-1}y^{k-1} = (-1)^{k-1}y^{k-1}x^{k-1}$ is an identity for Lie elements of the enveloping algebra. The solution is given by reducing the problem to $k$-strong Lie algebras generated by particularly well behaved sandwiches in the sense of Kostrikin. |
| title | Wilson conjecture for omega-categorical Lie algebras, the case 4-Engel characteristic 3 |
| topic | Rings and Algebras Logic 03C60, 17B30, 20F18 |
| url | https://arxiv.org/abs/2411.00667 |