{
  "id": "1206.1436",
  "version": "v2",
  "published": "2012-06-07T10:08:30.000Z",
  "updated": "2014-05-27T14:14:16.000Z",
  "title": "$χ$-admissible subalgebras of $\\sl_{pn}(\\C)$ and finite $W$-algebras",
  "authors": [
    "Guilnard Sadaka"
  ],
  "comment": "This paper has been withdrawn and replaced by arXiv:1405.6390 where one can find further generalizations of the results",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading for e, we recover the finite W-algebra associated to e and it is well-known that gr(H) is isomorphic to \\C[S] as a graded Poisson algebra where S is the Slodowy slice of e and gr(H) is the graded algebra associated to the Kazhdan filtration. In this paper, we consider the case where g =\\sl_{pn}(\\C) and e consists of p Jordan blocks all of the same size n. Here, the only good grading for e is the Dynkin grading and we construct admissible subalgebras non isomorphic to the one derived from this good grading. For these algebras m, we prove that gr(H) is isomorphic to \\C[S], generalizing Premet and Gan-Ginzburg's result in this particular case, where S denotes an analogue to the Slodowy slice.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-27T14:14:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B20",
      "17B35"
    ],
    "keywords": [
      "complex simple lie algebra",
      "slodowy slice",
      "construct admissible subalgebras non isomorphic",
      "endomorphism algebra",
      "finite w-algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.1436S"
    }
  }
}