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Non-commutative frieze patterns over quaternion algebras and other normed division rings

  • Michael Cuntz
  • Thorsten Holm
  • Peter Jorgensen
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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)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≤ −4 and b≤ −4.

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Cite

BibTeXRIS
BibTeX
@misc{cuntz2026noncommutative,
  title = {{Non-commutative frieze patterns over quaternion algebras and other normed division rings}},
  author = {Michael Cuntz and Thorsten Holm and Peter Jorgensen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32257},
  eprint = {2609.32257},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32257},
}
RIS
TY  - UNPB
TI  - Non-commutative frieze patterns over quaternion algebras and other normed division rings
AU  - Michael Cuntz
AU  - Thorsten Holm
AU  - Peter Jorgensen
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32257
UR  - https://arxiv.org/abs/2609.32257
AB  - 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)_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≤ −4 and b≤ −4.
ER  -