{
  "id": "math/0409055",
  "version": "v1",
  "published": "2004-09-03T21:16:16.000Z",
  "updated": "2004-09-03T21:16:16.000Z",
  "title": "An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "Let $\\g$ be a simple Lie algebra of type A or C. We show that the coadjoint representation of any seaweed subalgebra of $\\g$ has some properties similar to that of the adjoint representation of a reductive Lie algebra. Namely, a) the field of invariants is rational and b) there exists a generic stabiliser whose identity component is a torus. Our main tool for this is a result about coadjoint representations of some N-graded Lie algebras, which can be regarded as an extension of Rais' theorem for the index of semi-direct products. For all other simple types, we give a uniform description of a parabolic subalgebra such that its coadjoint representation has no generic stabiliser. The crucial property here is that if $\\g$ is not of type A or C, then the highest root is fundamental. We also show that, for any parabolic subgroup, the ring of regular invariants of the coadjoint representation is trivial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-03T21:16:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lie algebra",
      "seaweed subalgebra",
      "coadjoint representation",
      "generic stabiliser",
      "reductive lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9055P"
    }
  }
}