{
  "id": "1601.06253",
  "version": "v1",
  "published": "2016-01-23T09:03:30.000Z",
  "updated": "2016-01-23T09:03:30.000Z",
  "title": "The higher order $q$-Dolan-Grady relations and quantum integrable systems",
  "authors": [
    "Thi-Thao Vu"
  ],
  "comment": "PhD thesis, November 2014; 136 pages; Some basic material of Chapter 1,2 taken from other works (Terwilliger and coauthors, arXiv:math/0406555, ...; Baseilhac and co-authors arXiv:0906.1482, ...). Main results described in Chapter 3, published in arXiv:1312.3433, arXiv:1312.5897",
  "categories": [
    "math-ph",
    "math.MP",
    "math.QA"
  ],
  "abstract": "In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the $XXZ$ open spin chain at roots of unity) is considered. The main results of the thesis are: (i) for the class of $q-$Onsager algebras associated with $\\widehat{sl_2}$ and ADE type simply-laced affine Lie algebras, higher order analogs of Lusztig's relations are conjectured; (ii) for the open $XXZ$ spin chain at roots of unity, new elements (that are divided polynomials of $q-$Onsager generators) are introduced and some of their properties studied. These two elements together with the two basic elements of the $q-$Onsager algebra generate a new algebra, which can be understood as an analog of Lusztig's quantum group for the $q-$Onsager algebra.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-23T09:03:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum integrable systems",
      "higher order",
      "onsager algebra",
      "dolan-grady relations",
      "type simply-laced affine lie algebras"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 136,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106253V"
    }
  }
}