{
  "abstract": "Let $\\Bbbk$ be an algebraically closed field of characteristic zero, and let $\\mathcal A_q=\\Bbbk_q[x,y]$ be the quantum plane, with $q\\in\\Bbbk^\\ast\\setminus\\{1\\}$. For a fixed automorphism $σ$ of $\\mathcal A_q$, we study $σ$-derivations and their isotropy groups under the conjugation action of the centralizer of $σ$ in $\\operatorname{Aut}(\\mathcal A_q)$. For $q\\neq\\pm1$, we use Jordan's classification of skew derivations for toric automorphisms, which generalizes the description of Almulhem and Brzeziński for the quantum plane, to describe these groups by character equations on the torus $(\\Bbbk^\\ast)^2$, obtaining an arithmetic criterion for finiteness. In this case, we also prove that, for every fixed $σ$, every finite subgroup of the torus occurs, up to isomorphism, as the isotropy group of a $σ$-derivation, including when $q$ is a root of unity. For $q=-1$, we classify the $σ$-derivations for both toric and flip automorphisms $σ$ and describe their isotropy groups. In particular, we explicitly determine the isotropy groups of ordinary derivations of $\\mathcal A_{-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.20516",
  "author_total": 2,
  "authors": [
    {
      "name": "R. Baltazar"
    },
    {
      "name": "R. Cavalheiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1p08mre8jh5hb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.20516",
  "fulltext_url": null,
  "id": "1p08mre8jh5hb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.20516",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7161936523",
  "pdf_url": "https://arxiv.org/pdf/2605.20516",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1p08mre8jh5hb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2605.20516",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.20516"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Isotropy Groups of $σ$-Derivations on the Quantum Plane",
  "title_text": "Isotropy Groups of σ-Derivations on the Quantum Plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/1p08mre8jh5hb",
  "venue": "arXiv",
  "versions": []
}