{
  "abstract": "Non-commutative friezes have been introduced by Berenstein and Retakh and studied further by the authors. In this paper we consider non-commutative friezes over normed division rings, like for instance Hamilton's quaternions or more general quaternion algebras. We address the fundamental question in the theory of friezes of whether over a certain subset there are finitely or infinitely many non-commutative friezes (with 1's on the boundary) for any height. As an application of a theorem bounding the norm of quiddity entries we deduce that for every norm-finite subset of a normed division ring there are only finitely many such non-commutative friezes for every height. In particular this result applies to the Lipschitz quaternions and the Hurwitz quaternions of Hamilton's quaternions. We then study more generally non-commutative friezes over Lipschitz subrings of non-split quaternion algebras $(a,b)_{\\mathbb{Q}}$. We determine the frieze subrings for all $a,b<0$, and as a consequence we see that all such non-commutative friezes are known if $a\\le -4$ and $b\\le -4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32257",
  "author_total": 3,
  "authors": [
    {
      "name": "Michael Cuntz"
    },
    {
      "name": "Thorsten Holm"
    },
    {
      "name": "Peter Jorgensen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3k7cbqc18tw6r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32257",
  "fulltext_url": null,
  "id": "3k7cbqc18tw6r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32257",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214744086",
  "pdf_url": "https://arxiv.org/pdf/2609.32257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3k7cbqc18tw6r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-commutative frieze patterns over quaternion algebras and other normed division rings",
  "title_text": "Non-commutative frieze patterns over quaternion algebras and other normed division rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/3k7cbqc18tw6r",
  "venue": "arXiv",
  "versions": []
}