{
  "abstract": "A charged particle bound by an isotropic harmonic potential to a Dirac magnetic monopole, with the centrifugal term that makes the orbits close, has degeneracies beyond rotational invariance: for integer $q$, the product of the electric and magnetic charges, the levels are those of the oscillator with the states of angular momentum $l<|q|$ removed. The conserved rank-2 tensor $T$ built from the ladder operators that accounts for them was identified thirty-five years ago, but the algebra it generates with the angular momentum $\\mathbf{J}$ was not obtained. We obtain its relations in covariant form and identify it. It is polynomial but not quadratic, and the shift $T'=T+2(H-1)J^{(2)}$ by the quadrupole of $\\mathbf{J}$, and only this one, makes all commutators polynomials in $\\mathbf{J}$. In each sector of fixed energy and $J_z$, $(\\mathbf{J}^2,T'_{zz})$ generates a cubic algebra of Heun--Racah type whose parameters are symmetric, up to a scalar shift, in the magnetic charge and the magnetic quantum number. At $q=0$, $T$ is the algebraic Heun operator of the pair (Casimir of $\\mathfrak{so}(3)$, quadrupole of $\\mathfrak{su}(3)$): the symmetry algebra is the $\\mathfrak{so}(3)$-covariant extension of a Heun--Racah algebra. We define it abstractly and classify its multiplicity-free finite-dimensional representations at the energies of the system: besides the levels, they include, for nonzero integer $q$, a second family carried by the states the monopole removes, the kernel of the Newman--Penrose operator. All relations are derived from the dynamical algebra of the ladder operators and hold for integer and half-integer charges alike.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25408",
  "author_total": 2,
  "authors": [
    {
      "name": "Ian Marquette"
    },
    {
      "name": "Luc Vinet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45c4pq8egtt05.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25408",
  "fulltext_url": null,
  "id": "45c4pq8egtt05",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25408",
  "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.MP"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Non-Hermitian Physics"
    ]
  ],
  "openalex": "W7223029637",
  "pdf_url": "https://arxiv.org/pdf/2610.25408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45c4pq8egtt05.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.25408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Symmetry algebra of a charged harmonic oscillator in the field of the Dirac magnetic monopole",
  "title_text": "Symmetry algebra of a charged harmonic oscillator in the field of the Dirac magnetic monopole",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/45c4pq8egtt05",
  "venue": "arXiv",
  "versions": []
}