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The algebraic and geometric classification of $3$-Poisson algebras

  • Hani Abdelwahab
  • Kobiljon Abdurasulov
  • Ivan Kaygorodov
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Abstract

$3$-Poisson (ternary Nambu--Poisson) algebras are the algebras of observables of Nambu mechanics, the generalization of Hamiltonian mechanics in which the time evolution ḟ={f,H1,H2} is governed by two Hamiltonians and preserves the phase volume; their $3$-Lie part also gives the gauge algebra of the Bagger--Lambert--Gustavsson model of multiple M2-branes. We give the algebraic classification of complex $4$-dimensional $3$-Poisson algebras (seven algebras) and of complex $5$-dimensional nilpotent $3$-Poisson algebras ($38$ algebras and one one-parameter family) with both multiplications non-zero. We also describe the algebraic counterpart of the Inönü--Wigner contractions of physics: the variety of complex $4$-dimensional $3$-Poisson algebras has dimension $16$ and seven irreducible components, one of which is the orbit closure of the simple $3$-Lie algebra of the M2-brane model, and the variety of complex $5$-dimensional nilpotent $3$-Poisson algebras has dimension $20$ and five irreducible components.

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Cite

BibTeXRIS
BibTeX
@misc{abdelwahab2026algebraic,
  title = {{The algebraic and geometric classification of $3$-Poisson algebras}},
  author = {Hani Abdelwahab and Kobiljon Abdurasulov and Ivan Kaygorodov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27300},
  eprint = {2610.27300},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27300},
}
RIS
TY  - UNPB
TI  - The algebraic and geometric classification of $3$-Poisson algebras
AU  - Hani Abdelwahab
AU  - Kobiljon Abdurasulov
AU  - Ivan Kaygorodov
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.27300
UR  - https://arxiv.org/abs/2610.27300
AB  - $3$-Poisson (ternary Nambu--Poisson) algebras are the algebras of observables of Nambu mechanics, the generalization of Hamiltonian mechanics in which the time evolution ḟ={f,H_1,H_2} is governed by two Hamiltonians and preserves the phase volume; their $3$-Lie part also gives the gauge algebra of the Bagger--Lambert--Gustavsson model of multiple M2-branes. We give the algebraic classification of complex $4$-dimensional $3$-Poisson algebras (seven algebras) and of complex $5$-dimensional nilpotent $3$-Poisson algebras ($38$ algebras and one one-parameter family) with both multiplications non-zero. We also describe the algebraic counterpart of the Inönü--Wigner contractions of physics: the variety of complex $4$-dimensional $3$-Poisson algebras has dimension $16$ and seven irreducible components, one of which is the orbit closure of the simple $3$-Lie algebra of the M2-brane model, and the variety of complex $5$-dimensional nilpotent $3$-Poisson algebras has dimension $20$ and five irreducible components.
ER  -