{
  "id": "2205.07138",
  "version": "v1",
  "published": "2022-05-14T21:42:54.000Z",
  "updated": "2022-05-14T21:42:54.000Z",
  "title": "Bounded weight modules for basic classical Lie superalgebras at infinity",
  "authors": [
    "Dimitar Grantcharov",
    "Ivan Penkov",
    "Vera Serganova"
  ],
  "comment": "37 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We classify simple bounded weight modules over the complex simple Lie superalgebras $\\mathfrak{sl}(\\infty |\\infty)$ and $\\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\\infty$. For $\\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\\mathfrak{o} (m)$-modules and oscillator-type $\\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\\mathfrak{osp} (m | 2n)$ (under the assumption $\\dim \\, \\mathfrak{osp} (m | 2n) = \\infty$) by reducing its study to already known categories of representations of $\\mathfrak{sp} (2n)$, where $n$ possibly equals $\\infty$. When classifying simple bounded weight $\\mathfrak{sl}(\\infty |\\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\\mathfrak{sl}(\\infty |\\infty)_{\\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\\mathfrak{sl} (\\infty |\\infty)$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-14T21:42:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B65",
      "17B10"
    ],
    "keywords": [
      "basic classical lie superalgebras",
      "complex simple lie superalgebras",
      "classify simple bounded weight modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}