{
  "id": "2005.13444",
  "version": "v1",
  "published": "2020-05-27T15:55:00.000Z",
  "updated": "2020-05-27T15:55:00.000Z",
  "title": "A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$",
  "authors": [
    "N. Crampe",
    "L. Poulain d'Andecy",
    "L. Vinet"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-27T15:55:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "clebsch-gordan series",
      "calabi-yau algebra",
      "central element",
      "fundamental invariant polynomials",
      "full weyl group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}