{
  "id": "2201.06148",
  "version": "v2",
  "published": "2022-01-16T22:59:29.000Z",
  "updated": "2022-01-24T01:42:32.000Z",
  "title": "The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters",
  "authors": [
    "A. P. Isaev",
    "A. A. Provorov"
  ],
  "comment": "Submitted to Theoretical and Mathematical Physics; references corrected",
  "categories": [
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\\otimes 2}$ of the $osp(M|N)$ and $s\\ell(M|N)$ Lie superalgebras in the cases when $T$ is the defining and adjoint representations. For defining representations, the $osp(M|N)$- and $s\\ell(M|N)$-invariant solutions of the Yang-Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint representation, the characteristic identities and invariant projectors obtained are considered from the viewpoint of a universal description of Lie superalgebras by means of the Vogel parametrization. We also construct a universal generating function for higher Casimir operators of the $osp(M|N)$ and $s\\ell(M|N)$ Lie superalgebras in the adjoint representation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-24T01:42:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "split casimir operator",
      "lie superalgebras",
      "higher casimir operators",
      "yang-baxter equation",
      "adjoint representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}