{
  "id": "0905.3714",
  "version": "v3",
  "published": "2009-05-22T15:58:42.000Z",
  "updated": "2010-03-19T13:10:50.000Z",
  "title": "On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type",
  "authors": [
    "Simon M. Goodwin",
    "Gerhard Roehrle",
    "Glenn Ubly"
  ],
  "comment": "14 pages, minor changes.",
  "doi": "10.1112/S1461157009000205",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We consider the finite $W$-algebra $U(\\g,e)$ associated to a nilpotent element $e \\in \\g$ in a simple complex Lie algebra $\\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a conjecture of Premet, that $U(\\g,e)$ always has a 1-dimensional representation, when $\\g$ is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in $U(\\g)$ whose associated variety is the coadjoint orbit corresponding to $e$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-03-19T13:10:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "17B10",
      "81R05"
    ],
    "keywords": [
      "simple lie algebras",
      "exceptional type",
      "finite w-algebras",
      "simple complex lie algebra",
      "modular lie algebras"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.3714G"
    }
  }
}