Poisson n-Lie algebras: constructions and the structure of solvable algebras

@misc{cao2026poisson,
  title = {{Poisson $n$-Lie algebras: constructions and the structure of solvable algebras}},
  author = {Xinru Cao and Zafar Normatov and Bakhrom Omirov},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.01785},
  eprint = {2605.01785},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.01785},
}

TY  - UNPB
TI  - Poisson n-Lie algebras: constructions and the structure of solvable algebras
AU  - Xinru Cao
AU  - Zafar Normatov
AU  - Bakhrom Omirov
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2605.01785
UR  - https://arxiv.org/abs/2605.01785
AB  - In this paper, we develop a construction of Poisson n-Lie algebras that generalizes the Jacobian n-Lie construction. Using the Grassmann--Plücker relations, we derive necessary and sufficient conditions under which the resulting bracket defines a Poisson n-Lie algebra. We also prove that suitable quotients of tensor products of Poisson algebras carry natural Poisson n-Lie structures. Conversely, we give a tensor-type procedure that associates a Poisson algebra to a given Poisson n-Lie algebra. The quotient and converse constructions thus provide two systematic methods for relating Poisson algebras to Poisson n-Lie algebras. We further establish analogues of Engel's theorem and Lie's theorem and characterize solvability and nilpotency of Poisson n-Lie algebras in terms of their underlying associative and n-Lie structures. We introduce hypo-nilpotent ideals and investigate maximal such ideals in finite-dimensional solvable Poisson n-Lie algebras. Finally, we prove that the generalized eigenspaces of multiplication operators are ideals.
ER  -