{
  "abstract": "Let $\\Bbbk$ be an algebraically closed field of characteristic zero. We classify the locally finite derivations of arbitrary Ore extensions of $\\Bbbk[x]$, thus extending van den Essen's \\cite{V92} classification for the polynomial algebra $\\Bbbk[x,y]$ to this noncommutative setting. More precisely, we consider the three families arising in the classification of Ore extensions of $\\Bbbk[x]$: the quantum plane, the first quantum Weyl algebra, and the differential Ore extensions \\[ A_h=\\Bbbk[x][t;h(x)\\partial_x]. \\] For both the quantum plane and the first quantum Weyl algebra, we determine the locally finite derivations and explain how the resulting classifications are related to the work of Suárez-Alvarez and Vivas \\cite{SuarezVivas} on generalized Weyl algebras. For the algebras $A_h$, with $h$ nonconstant, we obtain a complete classification in both the square-free and non-square-free cases. As a consequence, we show that $\\LFD(A_h)$ is a solvable and weakly locally finite Lie subalgebra of $\\Der(A_h)$, although it is not locally finite as a set of derivations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28257",
  "author_total": 4,
  "authors": [
    {
      "name": "R. Baltazar"
    },
    {
      "name": "A. Bianchi"
    },
    {
      "name": "M. Veloso"
    },
    {
      "name": "J. Schwarz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c3zxwe2bt8w33.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28257",
  "fulltext_url": null,
  "id": "c3zxwe2bt8w33",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28257",
  "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": "W7204835464",
  "pdf_url": "https://arxiv.org/pdf/2608.28257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c3zxwe2bt8w33.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Locally Finite Derivations in Ore Extensions",
  "title_text": "On Locally Finite Derivations in Ore Extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/c3zxwe2bt8w33",
  "venue": "arXiv",
  "versions": []
}