{
  "id": "1205.2144",
  "version": "v1",
  "published": "2012-05-10T02:43:10.000Z",
  "updated": "2012-05-10T02:43:10.000Z",
  "title": "Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type",
  "authors": [
    "Chalermpong Worawannotai"
  ],
  "comment": "66 pages. arXiv admin note: text overlap with arXiv:0708.1992, arXiv:math/0608694, arXiv:0710.4383, arXiv:1108.2484, arXiv:0705.0167, arXiv:math/0608623, arXiv:math/0602416, arXiv:0705.3918, arXiv:0911.0098, arXiv:1108.0458",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra U_q(sl_2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two U_q(sl_2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let \\Gamma denote a dual polar graph. Fix a vertex x of \\Gamma and let T = T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of \\Gamma and a certain diagonal matrix A* = A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A* to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two U_q(sl_2)-module structures on the standard module of \\Gamma. We describe how these two U_q(sl_2)-module structures are related. Each of these U_q(sl_2)-module structures induces a $\\mathbb{C}$-algebra homomorphism U_q(sl_2) \\rightarrow T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of \\Gamma we obtain a generating set L, F, R, K of T along with some attractive relations satisfied by these generators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-10T02:43:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "dual q-krawtchouk type",
      "dual polar graph",
      "leonard system",
      "quantum algebra",
      "structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2144W"
    }
  }
}