{
  "abstract": "We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field $\\mathbb{F}$ of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras $M_n(\\mathbb{F})^+$, the symmetric matrix algebras $H_n(\\mathbb{F})$, the symplectic Hermitian algebras $H_{2n}(\\mathbb{F},\\mathrm{sp})$, and the split Albert algebra. We determine exactly which elements are representable: except in $H_2(\\mathbb{F})$, these are all nonscalar elements together with $0$ and $1$, whereas in $H_2(\\mathbb{F})$ the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For $M_n(\\mathbb{F})^+$ over an arbitrary field, the maximal word length grows logarithmically in $n$, and for $n=2$ we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for $H_{2n}(\\mathbb{F},\\mathrm{sp})$. The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23581",
  "author_total": 2,
  "authors": [
    {
      "name": "Ilja Gogić"
    },
    {
      "name": "Mateo Tomašević"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dve331phh3as2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23581",
  "fulltext_url": null,
  "id": "dve331phh3as2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23581",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Advanced Topics in Algebra"
    ]
  ],
  "openalex": "W7213999856",
  "pdf_url": "https://arxiv.org/pdf/2609.23581",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dve331phh3as2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23581",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23581"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Idempotent words in finite-dimensional simple Jordan algebras",
  "title_text": "Idempotent words in finite-dimensional simple Jordan algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/dve331phh3as2",
  "venue": "arXiv",
  "versions": []
}