{
  "abstract": "We introduce a family of five-dimensional non-flexible quadratic real algebras $\\{\\mathcal A_γ\\}_{0\\leqγ\\leq1}$ with canonical basis $1,i,j,k,\\ell$, in which $γ$ changes the multiplication but not their norm, and study how this change affects geometry and dynamics. The algebras are pairwise non-isomorphic, $\\mathcal A_0$ contains Hamilton's quaternion algebra, and $\\mathcal A_1$ is the Pentarnion algebra. The alternating part of the trilinear form $\\langle u\\times_γv,w\\rangle$ induced by the multiplication of $\\mathcal{A}_γ$ defines a three-dimensional subspace $H_γ\\subset\\operatorname{span}_{\\mathbb R}\\{i,j,k,\\ell\\}$, tilted by the angle $\\arctan(γ/\\sqrt3)$, whose rotation group $G_γ\\cong\\mathrm{SO}(3)$ contains the automorphism group of $\\mathcal A_γ$. For $γ_1\\neγ_2$ the groups $G_{γ_1}$ and $G_{γ_2}$ generate $\\mathrm{SO}(V)\\cong\\mathrm{SO}(4)$. For the six ways of ordering and bracketing the product of $i,p,p$ in $\\mathcal A_γ$, two map the unit circle in ${\\rm span}\\{j,k\\}$ to a point and, for $γ>0$, four map it to ellipses. As an application, we derive a Hamiltonian model of a classical spin on $H_γ$ whose multiplication-induced quadratic energy is isotropic for $γ=0$ and anisotropic for $γ>0$. It conserves spin length and, for fixed $γ$, energy, and satisfies an exact work identity when $γ$ varies in time. Numerical experiments with periodically varying $γ$ indicate both regular and irregular spin dynamics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31485",
  "author_total": 2,
  "authors": [
    {
      "name": "Alireza Poshtkohi"
    },
    {
      "name": "Susanne Pumpluen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9p0j0014rsqm2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31485",
  "fulltext_url": null,
  "id": "9p0j0014rsqm2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31485",
  "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-ph"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7223398348",
  "pdf_url": "https://arxiv.org/pdf/2610.31485",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9p0j0014rsqm2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.31485",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31485"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "From Quaternions to Pentarnions I: Algebra, geometry, and a spin model",
  "title_text": "From Quaternions to Pentarnions I: Algebra, geometry, and a spin model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/9p0j0014rsqm2",
  "venue": "arXiv",
  "versions": []
}