{
  "id": "1306.2582",
  "version": "v3",
  "published": "2013-06-11T17:02:34.000Z",
  "updated": "2015-04-15T22:25:26.000Z",
  "title": "On endotrivial modules for Lie superalgebras",
  "authors": [
    "Andrew J. Talian"
  ],
  "comment": "27 pages; updates to section 7",
  "doi": "10.1016/j.jalgebra.2015.02.022",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g} = \\mathfrak{g}_{\\overline{0}} \\oplus \\mathfrak{g}_{\\overline{1}}$ be a Lie superalgebra over an algebraically closed field, $k$, of characteristic 0. An endotrivial $\\mathfrak{g}$-module, $M$, is a $\\mathfrak{g}$-supermodule such that $\\operatorname{Hom}_k(M,M) \\cong k_{ev} \\oplus P$ as $\\mathfrak{g}$-supermodules, where $k_{ev}$ is the trivial module concentrated in degree $\\overline{0}$ and $P$ is a projective $\\mathfrak{g}$-supermodule. In the stable module category, these modules form a group under the operation of the tensor product. We show that for an endotrivial module $M$, the syzygies $\\Omega^n(M)$ are also endotrivial, and for certain Lie superalgebras of particular interest, we show that $\\Omega^1(k_{ev})$ and the parity change functor actually generate the group of endotrivials. Additionally, for a broader class of Lie superalgebras, for a fixed $n$, we show that there are finitely many endotrivial modules of dimension $n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-30T08:00:05.000Z",
      "abstract": "Let $\\mathfrak{g}$ be a Lie superalgebra over an algebraically closed field, $k$, of characteristic 0. An endotrivial $\\mathfrak{g}$-module, $M$, is a $\\mathfrak{g}$-supermodule such that $\\Hom_k(M,M) \\cong k \\oplus P$ as $\\mathfrak{g}$-supermodules, where $k$ is the trivial module concentrated in degree $\\bar{0}$ and $P$ is a projective $\\mathfrak{g}$-supermodule. In the stable module category, these modules form a group under the operation of the tensor product. We show that for an endotrivial module $M$, the syzygies $\\Omega^{n}(M)$ are also endotrivial and for certain Lie superalgebras of particular interest, we show that $\\Omega^{1}(k)$ and the parity change functor actually generate the group of endotrivials. Additionally, for a broader class of Lie superalgebras, for a fixed $n$, we show that there are finitely many endotrivial modules of dimension $n$.",
      "comment": "21 pages; updates to section 5",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-15T22:25:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie superalgebra",
      "endotrivial module",
      "supermodule",
      "parity change functor",
      "broader class"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.2582T"
    }
  }
}