Skip to content
DraconiticResearch
Search Sources API
184,584 studies31 sourcesUpdated 16 min ago
PreprintCC BY 4.0arXiv

Poisson closures and Poisson A-algebras

  • Amir Fernández Ouaridi
  • Salvatore Siciliano
  • David A. Towers
Open at sourcePDFDOICite

Abstract

We study the interaction between the Lie and Poisson subalgebras of a Poisson algebra. Given a Lie subalgebra U of the underlying Lie algebra đť’«L, we consider the Poisson subalgebra generated by U, called its {Poisson closure}, and investigate when Lie-theoretic properties such as nilpotency and solvability are preserved under this construction. We also study the stability under Poisson closure of several distinguished Lie subalgebras, including maximal nilpotent and solvable subalgebras, Engel and Cartan subalgebras and the derived algebra. As an application, we investigate Poisson A-algebras, defined by the condition that every nilpotent subalgebra is a zero algebra. In the finite-dimensional case, we obtain structural results relating Poisson A-algebras to Lie A-algebras and to the associative structure of the algebra.

Related

  1. Extending structures for the linear framework of Universal Algebra with applications to transposed Poisson algebras
    Changhuan Liu, Tingfeng Wan, Guodong Zhou
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Affinization of algebraic structures: Poisson algebras
    Tomasz Brzeziński, Krzysztof Radziszewski, Brais Ramos Pérez
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Classification of Novikov-Poisson Algebras and Their Applications
    Ziyi Zhang, Zeyu Hao, Yining Sun +1
    arXivPreprintCC0Full textAlgebra & number theoryarXiv
  4. Dedekind Poisson Algebras over Arbitrary Fields
    A. I. Plakosh, O. O. Pypka
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. The geometric classification of Novikov-Poisson and Gelfand-Dorfman algebras
    Zeyu Hao, Ivan Kaygorodov, Azamat Saydaliyev +1
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{ouaridi2026poisson,
  title = {{Poisson closures and Poisson $A$-algebras}},
  author = {Amir Fernández Ouaridi and Salvatore Siciliano and David A. Towers},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35463},
  eprint = {2610.35463},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35463},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Poisson closures and Poisson A-algebras
AU  - Amir Fernández Ouaridi
AU  - Salvatore Siciliano
AU  - David A. Towers
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35463
UR  - https://arxiv.org/abs/2610.35463
AB  - We study the interaction between the Lie and Poisson subalgebras of a Poisson algebra. Given a Lie subalgebra U of the underlying Lie algebra đť’«_L, we consider the Poisson subalgebra generated by U, called its {Poisson closure}, and investigate when Lie-theoretic properties such as nilpotency and solvability are preserved under this construction. We also study the stability under Poisson closure of several distinguished Lie subalgebras, including maximal nilpotent and solvable subalgebras, Engel and Cartan subalgebras and the derived algebra. As an application, we investigate Poisson A-algebras, defined by the condition that every nilpotent subalgebra is a zero algebra. In the finite-dimensional case, we obtain structural results relating Poisson A-algebras to Lie A-algebras and to the associative structure of the algebra.
ER  -