{
  "abstract": "We study Ore extensions $R[t;σ,δ]$ of rings graded by an arbitrary group $Γ$. The grading of $R$ extends with $t$ of degree $γ$ if and only if $σ$ is compatible with conjugation by $γ$ and $δ$ with left translation by $γ$. The extended grading is then unique. For nonabelian $Γ$ the degree of $t$ need not commute with the degrees of $R$, and this twist cannot always be removed. If $R$ is gr-simple and $σ$ is an automorphism, then $R[t;σ,δ]$ is gr-simple if and only if it has no homogeneous normal element that is monic of positive $t$-degree. The quantized Weyl algebra shows that the graded forms of the classical necessary conditions do not suffice. Over a group algebra $kG$, these extensions correspond to actions of $G$ on the affine line, and gr-simplicity means that the action has no finite orbit. In characteristic zero this is the nonvanishing of a class in $H^1(G,k_χ)$, and simplicity forces $G$ to be metabelian. We prove the analogous results for Poisson polynomial rings and relate the two settings by semiclassical limits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38131",
  "author_total": 1,
  "authors": [
    {
      "name": "Yassine Ait Mohamed"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/336dcnqfxd2s9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38131",
  "fulltext_url": null,
  "id": "336dcnqfxd2s9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38131",
  "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": "W7214912094",
  "pdf_url": "https://arxiv.org/pdf/2609.38131",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/336dcnqfxd2s9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2609.38131",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38131"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Group-graded Ore extensions and graded simplicity",
  "title_text": "Group-graded Ore extensions and graded simplicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/336dcnqfxd2s9",
  "venue": "arXiv",
  "versions": []
}