{
  "id": "1508.07203",
  "version": "v1",
  "published": "2015-08-28T13:33:15.000Z",
  "updated": "2015-08-28T13:33:15.000Z",
  "title": "The classification of the cyclic $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$--modules",
  "authors": [
    "Paolo Casati"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper we classify all the cyclic finite dimensional indecomposable\\\\ modules of the perfect Lie algebras $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$ with its standard representation. Furthermore, using the embeddings of the Lie algebras $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$ in $\\mathfrak{sl}(n+2)$, we show that any finite dimensional irreducible module of $\\mathfrak{sl}(n+2)$ restricted to $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$ is a cyclic module and that any cyclic $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$--modules can be constructed as quotient module of the restriction to $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$ of some finite dimensional irreducible $\\mathfrak{sl}(n+2)$--modules. This explicit realization of the cyclic $\\mathfrak{sl}(n+1)\\ltimes \\mathbbm{C}^{n+1}$--modules plays a role in their classification.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-28T13:33:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "22E70"
    ],
    "keywords": [
      "classification",
      "perfect lie algebras",
      "simple lie algebra",
      "finite dimensional irreducible module",
      "cyclic finite dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}