{
  "id": "0909.4464",
  "version": "v6",
  "published": "2009-09-24T14:53:39.000Z",
  "updated": "2013-02-04T18:07:33.000Z",
  "title": "Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers",
  "authors": [
    "Vladimir V. Kisil"
  ],
  "comment": "LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two Figures; v2: minor improvements and corrections; v3: a section on raising/lowering operators is added; v4: typos are fixed; v5: Introduction is added, open problems are expanded.v6: Russian translation is added, references areupdated, NoWeb and C++ source codes are added at Data Conservation Project. arXiv admin note: substantial text overlap with arXiv:0707.4024",
  "categories": [
    "math.RT",
    "math.CV",
    "math.RA"
  ],
  "abstract": "In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operators",
  "revisions": [
    {
      "version": "v6",
      "updated": "2013-02-04T18:07:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22D30",
      "08A99",
      "13A50",
      "15A04",
      "20H05",
      "51M10"
    ],
    "keywords": [
      "induced representation",
      "hypercomplex numbers",
      "erlangen program",
      "subgroup prompt hypercomplex coefficients",
      "semisimple lie group"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.4464K"
    }
  }
}