{
  "abstract": "Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\\dim H/|\\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\\bigoplus_{x\\in X}\\K e_x$, $X'\\subsetneq X$ is a nonempty subquandle, $V'=\\bigoplus_{x\\in X'}\\K e_x$, and $s\\in X\\setminus X'$, then $\\mathcal B(V)\\simeq\\K[e_s]/(e_s^{m_s})\\otimes C_{s,X'}\\otimes\\mathcal B(V')$ for some graded vector space $C_{s,X'}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)_z\\,\\mathcal H_{\\mathcal B(V')}(z)\\mid\\mathcal H_{\\mathcal B(V)}(z)$. In the non-group case, this yields a $p^2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29853",
  "author_total": 1,
  "authors": [
    {
      "name": "Rongchuan Xiong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ybft65mj1ysa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29853",
  "fulltext_url": null,
  "id": "7ybft65mj1ysa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29853",
  "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": "W7214297617",
  "pdf_url": "https://arxiv.org/pdf/2609.29853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ybft65mj1ysa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension",
  "title_text": "A q-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7ybft65mj1ysa",
  "venue": "arXiv",
  "versions": []
}