{
  "id": "1412.6805",
  "version": "v1",
  "published": "2014-12-21T16:11:07.000Z",
  "updated": "2014-12-21T16:11:07.000Z",
  "title": "Finite $W$-superalgebras and the dimensional lower bounds for the representations of basic Lie superalgebras",
  "authors": [
    "Yang Zeng",
    "Bin Shu"
  ],
  "comment": "47 pages. This version is revised from the last 3 chapters of the manuscript \"Finite W-superalgebras for basic classical Lie superalgebras\" (arXiv:1404.1150 [math.RT]). arXiv admin note: text overlap with arXiv:0809.0663 by other authors",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\\mathfrak{g}_\\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type $A$. Under the assumption of this conjecture, we show that the lower bounds of dimensions in the modular representations of basic Lie superalgebras are attainable. Such lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated by Wang-Zhao in \\cite{WZ} for the modular representations of a basic Lie superalgebra ${\\ggg}_{{\\bbk}}$ over an algebraically closed field $\\bbk$ of positive characteristic $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-21T16:11:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "basic lie superalgebra",
      "dimensional lower bounds",
      "modular representations",
      "minimal dimensional representations",
      "complex numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1335872
    }
  }
}