{
  "id": "2212.14761",
  "version": "v1",
  "published": "2022-12-30T15:13:02.000Z",
  "updated": "2022-12-30T15:13:02.000Z",
  "title": "Split Casimir operator for simple Lie algebras in the cube of $\\mathsf{ad}$-representation and Vogel parameters",
  "authors": [
    "A. P. Isaev",
    "S. O. Krivonos",
    "A. A. Provorov"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations $\\mathsf{ad}$ and deduced a certain class of subrepresentations in $\\mathsf{ad}^{\\otimes 3}$. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in $\\mathsf{ad}^{\\otimes 3}$ were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in $\\mathsf{ad}^{\\otimes 3}$ for simple Lie algebras in terms of the Vogel parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-30T15:13:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lie algebras",
      "split casimir operator",
      "vogel parameters",
      "characteristic identities",
      "subrepresentations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}