{
  "abstract": "We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on $SU(2)$ acting on two noncommuting variables. For $SU(2)$, we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree $2n$ the invariants are exactly the span of the $S_{2n}$-orbit of $(XY-YX)^n$. We also construct an explicit basis recursively from the operation $(A,B)\\mapsto XAYB-YAXB$, and show that the space of degree-$2n$ invariants has dimension equal to the $n$-th Catalan number. All proofs proceed by direct computation with the Haar integral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37185",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiachen Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5whvav7eq1x0p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37185",
  "fulltext_url": null,
  "id": "5whvav7eq1x0p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37185",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7224127994",
  "pdf_url": "https://arxiv.org/pdf/2610.37185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5whvav7eq1x0p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Some Lie Group Invariant noncommutative functions",
  "title_text": "On Some Lie Group Invariant noncommutative functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/5whvav7eq1x0p",
  "venue": "arXiv",
  "versions": []
}