{
  "id": "math/0506579",
  "version": "v2",
  "published": "2005-06-28T19:21:58.000Z",
  "updated": "2007-10-17T20:57:34.000Z",
  "title": "Semi-direct products of Lie algebras and their invariants",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "49 pages, title changed, section 11 is shortened, numerous minor corrections; accepted version, to appear in Publ. RIMS 43(2007)",
  "categories": [
    "math.AG",
    "math.RT"
  ],
  "abstract": "The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive Lie algebras whose adjoint representation has a polynomial algebra of invariants. We extend results of Takiff, Geoffriau, Rais-Tauvel, and Levasseur-Stafford concerning Takiff Lie algebras to a wider class of semi-direct products. This includes $Z_2$-contractions of simple Lie algebras and generalised Takiff algebras.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-10-17T20:57:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "invariants",
      "non-reductive lie algebras",
      "levasseur-stafford concerning takiff lie algebras",
      "generic isotropy groups",
      "semi-direct product construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......6579P"
    }
  }
}