{
  "abstract": "Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-$1/2$ particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of $\\mathfrak{su}_{\\mathbb{R}}(2)$, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-$n/2$ representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto $\\mathfrak{su}_{\\mathbb{R}}(n+1)$ and $\\mathfrak{u}_{\\mathbb{R}}(n+1)$, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15302",
  "author_total": 2,
  "authors": [
    {
      "name": "Tim Heib"
    },
    {
      "name": "Pérola Milman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fchx7p33cd2dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15302",
  "fulltext_url": null,
  "id": "fchx7p33cd2dd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15302",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum many-body systems"
    ]
  ],
  "openalex": "W7213352517",
  "pdf_url": "https://arxiv.org/pdf/2609.15302",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fchx7p33cd2dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2609.15302",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15302"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction",
  "title_text": "Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/fchx7p33cd2dd",
  "venue": "arXiv",
  "versions": []
}