{
  "id": "1210.8032",
  "version": "v4",
  "published": "2012-10-30T14:50:49.000Z",
  "updated": "2016-07-26T02:36:44.000Z",
  "title": "Centers of universal enveloping algebras of Lie superalgebras in prime characteristic",
  "authors": [
    "Junyan Wei",
    "Lisun Zheng",
    "Bin Shu"
  ],
  "comment": "This paper has been withdrawn by the author because the last two chapters are being essentially corrected",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\ggg=\\ggg_\\bz+\\ggg_\\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\\Lie(G)=\\ggg$, with the purely even subgroup $G_\\ev$ which is a reductive group. The center $\\cz:=\\cz(\\ggg)$ of the universal enveloping algebra of $\\ggg$ easily turns out to be a domain. In this paper, we prove that the quotient field of $\\cz$ coincides with that of the subalgebra generated by the $G_{\\ev}$-invariant ring $\\cz^{G_\\ev}$ of $\\cz$ and the $p$-center $\\cz_0$ of $U(\\ggg_\\bz)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-12T09:58:17.000Z",
      "abstract": "Let $\\ggg=\\ggg_\\bz+\\ggg_\\bo$ be a basic classical Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$, and $G$ be an algebraic supergroup satisfying $\\Lie(G)=\\ggg$, with the purely even subgroup $G_\\ev$ which is a reductive group. In this paper, we prove that the center $\\cz:=\\cz(\\ggg)$ of the universal enveloping algebra of $\\ggg$ is a domain, and $U(\\ggg)\\otimes_{\\cz}\\Frac(\\cz)$ is a simple superalgebra over $\\Frac(\\cz)$. And then we prove that the quotient field of $\\cz$ coincides with that of the subalgebra generated by the $G_{\\ev}$-invariant ring $\\cz^{G_\\ev}$ of $\\cz$ and the $p$-center $\\cz_0$ of $U(\\ggg_\\bz)$. In the case when $\\ggg=\\osp(1|2n)$, $\\cz$ is just generated by $\\cz^{G_\\ev}$ and $\\cz_0$. Furthermore, we will demonstrate the precise relation between the smooth points of the maximal spectrum ${Maxspec}(\\cz)$ and the corresponding irreducible modules for $\\osp(1|2)$.",
      "comment": "28 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2016-07-26T02:36:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "universal enveloping algebra",
      "prime characteristic",
      "basic classical lie superalgebra",
      "simple superalgebra",
      "smooth points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.8032W"
    }
  }
}