{
  "id": "math/0608694",
  "version": "v1",
  "published": "2006-08-28T16:53:26.000Z",
  "updated": "2006-08-28T16:53:26.000Z",
  "title": "Distance-regular graphs and the $q$-tetrahedron algebra",
  "authors": [
    "Tatsuro Ito",
    "Paul Terwilliger"
  ],
  "comment": "22 pages",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "Let $\\Gamma$ denote a distance-regular graph with classical parameters $(D,b,\\alpha,\\beta)$ and $b\\not=1$, $\\alpha=b-1$. The condition on $\\alpha$ implies that $\\Gamma$ is formally self-dual. For $b=q^2$ we use the adjacency matrix and dual adjacency matrix to obtain an action of the $q$-tetrahedron algebra $\\boxtimes_q$ on the standard module of $\\Gamma$. We describe four algebra homomorphisms into $\\boxtimes_q$ from the quantum affine algebra $U_q({\\hat{\\mathfrak{sl}}_2})$; using these we pull back the above $\\boxtimes_q$-action to obtain four actions of $U_q({\\hat{\\mathfrak{sl}}_2})$ on the standard module of $\\Gamma$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-28T16:53:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "distance-regular graph",
      "tetrahedron algebra",
      "standard module",
      "dual adjacency matrix",
      "quantum affine algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8694I"
    }
  }
}