{
  "id": "2312.16766",
  "version": "v1",
  "published": "2023-12-28T01:24:31.000Z",
  "updated": "2023-12-28T01:24:31.000Z",
  "title": "Jordan algebras and weight modules",
  "authors": [
    "Michael Lau",
    "Olivier Mathieu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We consider bounded weight modules for the universal central extension ${\\mathfrak{sl}}_2(J)$ of the Tits-Kantor-Koecher algebra of a unital Jordan algebra $J$. Universal objects called Weyl modules are introduced and studied, and a combinatorial dominance criterion is given for analogues of highest weights. Specializing $J$ to the free Jordan algebra $J(r)$ of rank $r$, the category $\\mathcal{C}^{fin}$ of finite-dimensional $\\mathbb{Z}$-graded ${\\mathfrak{sl}}_2(J)$-modules shares many properties with the representation theory of algebraic groups. Using a deep result of Zelmanov, we show that this subcategory admits Weyl modules. By analogy, we conjecture that $\\mathcal{C}^{fin}$ is a highest weight category. The resulting homological properties would then imply cohomological vanishing results previously conjectured as a way of determining graded dimensions of free Jordan algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-28T01:24:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B60",
      "17C05",
      "17C50"
    ],
    "keywords": [
      "free jordan algebra",
      "subcategory admits weyl modules",
      "universal central extension",
      "combinatorial dominance criterion",
      "unital jordan algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}