{
  "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 $\\dot f=\\{f,H_1,H_2\\}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27300",
  "author_total": 3,
  "authors": [
    {
      "name": "Hani Abdelwahab"
    },
    {
      "name": "Kobiljon Abdurasulov"
    },
    {
      "name": "Ivan Kaygorodov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3yhssz3fd45j2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27300",
  "fulltext_url": null,
  "id": "3yhssz3fd45j2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27300",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7223109395",
  "pdf_url": "https://arxiv.org/pdf/2610.27300",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3yhssz3fd45j2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.27300",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27300"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The algebraic and geometric classification of $3$-Poisson algebras",
  "title_text": "The algebraic and geometric classification of $3$-Poisson algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/3yhssz3fd45j2",
  "venue": "arXiv",
  "versions": []
}