{
  "id": "2209.09749",
  "version": "v1",
  "published": "2022-09-20T14:29:20.000Z",
  "updated": "2022-09-20T14:29:20.000Z",
  "title": "Reachable elements in basic classical Lie superalgebras",
  "authors": [
    "Leyu Han"
  ],
  "comment": "24 pages, 9 tables, submitted to Journal of Algebra",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let \\mathfrak{g}=\\mathfrak{g}_{\\bar{0}}\\oplus\\mathfrak{g}_{\\bar{1}} be a basic classical Lie superalgebra over \\mathbb{C}, e\\in\\mathfrak{g}_{\\bar{0}} a nilpotent element and \\mathfrak{g}^{e} the centralizer of e in \\mathfrak{g}. We study various properties of nilpotent elements in \\mathfrak{g}, which have previously only been considered in the case of Lie algebras. In particular, we prove that e is reachable if and only if e satisfies the Panyushev property for \\mathfrak{g}=\\mathfrak{sl}(m|n), m\\neq n or \\mathfrak{psl}(n|n) and \\mathfrak{osp}(m|2n). For exceptional Lie superalgebras \\mathfrak{g}=D(2,1;\\alpha), G(3), F(4), we give the classification of e which are reachable, strongly reachable or satisfy the Panyushev property. In addition, we give bases for \\mathfrak{g}^{e} and its centre \\mathfrak{z}(\\mathfrak{g}^{e}) for \\mathfrak{g}=\\mathfrak{psl}(n|n), which completes results of Han on the relationship between \\dim\\mathfrak{g}^{e}, \\dim\\mathfrak{z}(\\mathfrak{g}^{e}) and the labelled Dynkin diagrams for all basic classical Lie superalgebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-20T14:29:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "basic classical lie superalgebra",
      "reachable elements",
      "nilpotent element",
      "panyushev property",
      "exceptional lie superalgebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}