{
  "id": "1311.0570",
  "version": "v3",
  "published": "2013-11-04T03:04:56.000Z",
  "updated": "2015-04-30T22:17:41.000Z",
  "title": "Coefficients of Šapovalov elements for simple Lie algebras and contragredient Lie superalgebras",
  "authors": [
    "Ian M. Musson"
  ],
  "comment": "revised version: There is a new section on powers of \\v Sapovalov elements, and the section on the type A case has been corrected and expanded",
  "categories": [
    "math.RT"
  ],
  "abstract": "We provide upper bounds on the degrees of the coefficients of \\v{S}apovalov elements for a simple Lie algebra. If $\\fg$ is a contragredient Lie superalgebra and $\\gc$ is a positive isotropic root of $\\fg,$ we prove the existence and uniqueness of the \\v{S}apovalov element for $\\gc$ and we obtain upper bounds on the degrees of their coefficients. For type A Lie superalgebras we give a closed formula for \\v{S}apovalov elements. Often the coefficients of \\v{S}apovalov elements are products of linear factors, and we provide some reasons for this coming from representation theory. We also explore the relationships between \\v{S}apovalov elements coming from different roots, and their behavior when the Borel subalgebra is changed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-27T23:22:36.000Z",
      "comment": "revised version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-30T22:17:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple lie algebra",
      "contragredient lie superalgebra",
      "coefficients",
      "upper bounds",
      "borel subalgebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0570M"
    }
  }
}