{
  "abstract": "In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra $\\mathfrak{sl}(2,\\mathbb{C})$. That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solvable Lie subalgebra is reductive, with semisimple component isomorphic to $\\mathfrak{sl}(2,\\mathbb{C})$ itself. Every non-nilpotent solvable Lie subalgebra is the direct sum of an abelian center and a member of the family of solvable Lie algebras $\\mathfrak{r}(\\boldsymbol{d},\\boldsymbol{m})$, parameterized by a vector $\\boldsymbol{d}$ whose entries are strictly increasing positive integers that have greatest common divisor one together with a positive integer vector $\\boldsymbol{m}$ of the same length recording their multiplicities. Moreover, we discuss the corresponding classification in a real skew-hermitian setting and its relation to the realizations in the single- and two-mode Weyl algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04001",
  "author_total": 1,
  "authors": [
    {
      "name": "Tim Heib"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/56yv9b3z79f1r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04001",
  "fulltext_url": null,
  "id": "56yv9b3z79f1r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04001",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RA"
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      "arxiv",
      "math.RT"
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      "openalex",
      "2602|Advanced Topics in Algebra"
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  "openalex": "W7220669013",
  "pdf_url": "https://arxiv.org/pdf/2610.04001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/56yv9b3z79f1r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.04001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\\mathfrak{sl}(2,\\mathbb{C})$",
  "title_text": "Finite-dimensional Lie subalgebras of the universal enveloping algebra of sl(2,ℂ)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/56yv9b3z79f1r",
  "venue": "arXiv",
  "versions": []
}