{
  "abstract": "We summarize a coordinate-free classification of five-dimensional real Lie algebras up to isomorphism. Rather than solving the overdetermined quadratic Jacobi system modulo $\\mathrm{GL}(5,\\mathbb{R})$, the method uses Levi--Malcev theory and nilradical stratification. In the non-solvable case, the Levi factor is three-dimensional, $\\mathfrak{so}(3)$ or $\\mathfrak{sl}(2,\\mathbb{R})$, acting on a two-dimensional radical; this yields exactly five classes, including the indecomposable $\\mathfrak{sl}(2,\\mathbb{R})\\ltimes\\mathbb{R}^2$. In the solvable case, Schur's bound on commutative subalgebras of $\\mathfrak{gl}(2,\\mathbb{R})$ implies $\\dim\\mathfrak{n}\\in\\{3,4,5\\}$, eliminating one- and two-dimensional nilradicals. The nilpotent case $\\dim\\mathfrak{n}=5$ has nine classes: three decomposable and six indecomposable, including $\\mathfrak{h}_5$ and two filiform algebras. For $\\dim\\mathfrak{n}=4$, classification reduces to normal forms of outer derivations for $\\mathbb{R}^4$, $\\mathfrak{h}_3\\oplus\\mathbb{R}$, and $\\mathfrak{n}_4$ under automorphism and scaling. For $\\dim\\mathfrak{n}=3$, it uses two-dimensional abelian subalgebras of $\\mathrm{OutDer}(\\mathfrak{n})$ containing no nonzero nilpotent operators. The work also computes minimal faithful representation dimensions $μ(\\mathfrak{g})$, gives a dictionary with classical tables, and designs a deterministic identification algorithm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06971",
  "author_total": 1,
  "authors": [
    {
      "name": "Mehdi Nadjafikhah"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5r1g674yp6wqm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06971",
  "fulltext_url": null,
  "id": "5r1g674yp6wqm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06971",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06971",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5r1g674yp6wqm.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.06971",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06971"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Classification of real Lie algebras of dimension five",
  "title_text": "Classification of real Lie algebras of dimension five",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/5r1g674yp6wqm",
  "venue": "arXiv",
  "versions": []
}