{
  "id": "math/0407065",
  "version": "v2",
  "published": "2004-07-05T15:24:32.000Z",
  "updated": "2004-09-22T19:31:32.000Z",
  "title": "The centralisers of nilpotent elements in classical Lie algebras",
  "authors": [
    "O. S. Yakimova"
  ],
  "comment": "Replaced with english translation",
  "categories": [
    "math.RT"
  ],
  "abstract": "The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_\\alpha$ of elements $\\alpha\\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\\in g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $x\\in so_8$ such that the coadjoint action of $z(x)$ has no generic stabiliser.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-09-22T19:31:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nilpotent element",
      "classical lie algebras",
      "centraliser",
      "reductive lie algebra",
      "generic stabiliser"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7065Y"
    }
  }
}