{
  "id": "2105.01086",
  "version": "v1",
  "published": "2021-05-03T18:00:06.000Z",
  "updated": "2021-05-03T18:00:06.000Z",
  "title": "Racah algebras, the centralizer $Z_n(\\mathfrak{sl}_2)$ and its Hilbert-Poincaré series",
  "authors": [
    "Nicolas Crampe",
    "Julien Gaboriaud",
    "Loïc Poulain d'Andecy",
    "Luc Vinet"
  ],
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra $sR(n)$, is then introduced. Using results from classical invariant theory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer $Z_{n}(\\mathfrak{sl}_2)$ of the diagonal embedding of $U(\\mathfrak{sl}_2)$ in $U(\\mathfrak{sl}_2)^{\\otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(\\mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincar\\'e series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-03T18:00:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "centralizer",
      "higher rank racah algebra",
      "special racah algebra",
      "special askey-wilson algebra",
      "higher rank generalizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}