{
  "abstract": "Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\\dim H=2\\dim L$ is a normalized quadratic extension of $L$. For a fixed support $(g,h)$, such extensions are described by a coalgebra Hochschild $2$-cocycle together with Ore-type data $(σ,δ,u,v)$ satisfying explicit compatibility conditions. Their isomorphism classes are parameterized by the orbits of gauge transformations and Hopf automorphisms of $L$. As an application, we classify non-connected pointed Hopf algebras of dimension $16$ in characteristic $2$ with one-dimensional infinitesimal braiding whose diagram is not a Nichols algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11622",
  "author_total": 1,
  "authors": [
    {
      "name": "Rongchuan Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5evdkp9p0wf1j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11622",
  "fulltext_url": null,
  "id": "5evdkp9p0wf1j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11622",
  "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.QA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7212319245",
  "pdf_url": "https://arxiv.org/pdf/2609.11622",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5evdkp9p0wf1j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11622",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11622"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Normalized quadratic extensions of pointed Hopf algebras",
  "title_text": "Normalized quadratic extensions of pointed Hopf algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/5evdkp9p0wf1j",
  "venue": "arXiv",
  "versions": []
}