{
  "id": "1008.2433",
  "version": "v1",
  "published": "2010-08-14T11:45:20.000Z",
  "updated": "2010-08-14T11:45:20.000Z",
  "title": "On matrix realizations of the Lie superalgebra D(2, 1 ; α)",
  "authors": [
    "Elena Poletaeva"
  ],
  "comment": "15 pages, to be published in Journal of Geometry and Physics 60 (2010), 1656-1664",
  "doi": "10.1016/j.geomphys.2010.06.008",
  "categories": [
    "math.RT"
  ],
  "abstract": "We obtain a realization of the Lie superalgebra $D(2, 1 ; \\alpha)$ in differential operators on the supercircle $S^{1|2}$ and in $4\\times 4$ matrices over a Weyl algebra. A contraction of $D(2, 1 ; \\alpha)$ is isomorphic to the universal central extension $\\hat{\\p\\s\\l}(2|2)$ of $\\p\\s\\l(2|2)$. We realize it in $4\\times 4$ matrices over the associative algebra of pseudodifferential operators on $S^1$. Correspondingly, there exists a three-parameter family of irreducible representations of $\\hat{\\p\\s\\l}(2|2)$ in a $(2|2)$--dimensional complex superspace.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-14T11:45:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B25",
      "17B05",
      "81R05"
    ],
    "keywords": [
      "lie superalgebra",
      "matrix realizations",
      "dimensional complex superspace",
      "universal central extension",
      "weyl algebra"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Geometry and Physics",
      "year": 2010,
      "month": "Nov",
      "volume": 60,
      "number": 11,
      "pages": 1656
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JGP....60.1656P"
    }
  }
}