Poisson $n$-Lie algebras: constructions and the structure of solvable algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914556530393088 |
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| author | Cao, Xinru Normatov, Zafar Omirov, Bakhrom |
| author_facet | Cao, Xinru Normatov, Zafar Omirov, Bakhrom |
| contents | In this paper, we develop a construction of Poisson $n$-Lie algebras arising from $n$-Lie algebras of Jacobians and establish conditions under which this construction yields a Poisson $n$-Lie algebra. We also formulate a general conjecture in the unital case. In addition, we show that tensor products of Poisson algebras admit natural Poisson $n$-Lie structures via suitable quotient constructions. Conversely, we construct a Poisson algebra from a given Poisson $n$-Lie algebra, thereby establishing a correspondence between these classes of algebras.
Furthermore, we obtain analogues of Engel's and Lie's theorems and provide a characterization of solvable and nilpotent Poisson $n$-Lie algebras in terms of the underlying algebraic structures. We also introduce the notion of hypo-nilpotent ideals and prove results concerning maximal hypo-nilpotent ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we show that generalized eigenspaces of multiplication operators form ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01785 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Poisson $n$-Lie algebras: constructions and the structure of solvable algebras Cao, Xinru Normatov, Zafar Omirov, Bakhrom Rings and Algebras 17A42, 17B30, 17B63 In this paper, we develop a construction of Poisson $n$-Lie algebras arising from $n$-Lie algebras of Jacobians and establish conditions under which this construction yields a Poisson $n$-Lie algebra. We also formulate a general conjecture in the unital case. In addition, we show that tensor products of Poisson algebras admit natural Poisson $n$-Lie structures via suitable quotient constructions. Conversely, we construct a Poisson algebra from a given Poisson $n$-Lie algebra, thereby establishing a correspondence between these classes of algebras. Furthermore, we obtain analogues of Engel's and Lie's theorems and provide a characterization of solvable and nilpotent Poisson $n$-Lie algebras in terms of the underlying algebraic structures. We also introduce the notion of hypo-nilpotent ideals and prove results concerning maximal hypo-nilpotent ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we show that generalized eigenspaces of multiplication operators form ideals. |
| title | Poisson $n$-Lie algebras: constructions and the structure of solvable algebras |
| topic | Rings and Algebras 17A42, 17B30, 17B63 |
| url | https://arxiv.org/abs/2605.01785 |