Infinite dimensional analogues of nilpotent and solvable Lie algebras
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914072387125248 |
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| author | Haydarov, F. H. Omirov, B. A. Solijanova, G. O. |
| author_facet | Haydarov, F. H. Omirov, B. A. Solijanova, G. O. |
| contents | We study infinite-dimensional analogues of nilpotent and solvable Lie algebras, focusing on the classes of pro-nilpotent, residually nilpotent, pro-solvable and residually solvable Lie algebras. We extend classical triangularization results (Engel's and Lie's theorems) to the pro-setting and establish existence results for the pro-nilpotent radical in pro-solvable algebras and in certain residually solvable algebras. We adapt finite-dimensional construction methods to produce residually solvable extensions with a given pro-nilpotent radical under natural finiteness conditions. By analyzing derivations and maximal tori of pro-nilpotent algebras, we extend the notion of rank and show that, for pro-nilpotent algebras of maximal rank, every derivation of a maximal residually solvable extension is inner. Finally, we describe standard constructions (tensor and direct sum products, central extensions) that preserve pro-nilpotency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite dimensional analogues of nilpotent and solvable Lie algebras Haydarov, F. H. Omirov, B. A. Solijanova, G. O. Rings and Algebras We study infinite-dimensional analogues of nilpotent and solvable Lie algebras, focusing on the classes of pro-nilpotent, residually nilpotent, pro-solvable and residually solvable Lie algebras. We extend classical triangularization results (Engel's and Lie's theorems) to the pro-setting and establish existence results for the pro-nilpotent radical in pro-solvable algebras and in certain residually solvable algebras. We adapt finite-dimensional construction methods to produce residually solvable extensions with a given pro-nilpotent radical under natural finiteness conditions. By analyzing derivations and maximal tori of pro-nilpotent algebras, we extend the notion of rank and show that, for pro-nilpotent algebras of maximal rank, every derivation of a maximal residually solvable extension is inner. Finally, we describe standard constructions (tensor and direct sum products, central extensions) that preserve pro-nilpotency. |
| title | Infinite dimensional analogues of nilpotent and solvable Lie algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2510.02488 |