{
  "abstract": "Let $G$ be a reductive affine algebraic group over $\\mathbb C$ with toral identity component $T$ and finite component group $K$. We describe finite indecomposable semisimple module categories over $\\Rep(G)$ using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in $K$ and its intersection with $T$. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of $\\widehat T$ of index at most $|K|$. When $\\dim T=1$, every Hopf $2$-cocycle for $G$ has finite support of order at most $|K|^2$. We compute all gauge classes for the split and nonsplit extensions with component group $\\mathbb{Z}/2\\mathbb{Z}$ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group $G$, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces $G$ to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18599",
  "author_total": 1,
  "authors": [
    {
      "name": "Shlomo Gelaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dpr9vhhs2xkgt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18599",
  "fulltext_url": null,
  "id": "dpr9vhhs2xkgt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18599",
  "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.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7170046541",
  "pdf_url": "https://arxiv.org/pdf/2607.18599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dpr9vhhs2xkgt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2607.18599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hopf $2$-cocycles for certain affine algebraic reductive groups",
  "title_text": "Hopf $2$-cocycles for certain affine algebraic reductive groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/dpr9vhhs2xkgt",
  "venue": "arXiv",
  "versions": []
}