{
  "id": "2010.06852",
  "version": "v1",
  "published": "2020-10-14T07:36:24.000Z",
  "updated": "2020-10-14T07:36:24.000Z",
  "title": "Some homological properties of category $\\mathcal O$ for Lie superalgebras",
  "authors": [
    "Chih-Whi Chen",
    "Volodymyr Mazorchuk"
  ],
  "comment": "25 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\\Delta(\\lambda)$ to be such that every non-zero homomorphism from another Verma supermodule to $\\Delta(\\lambda)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\\mathfrak{pe}(n)$ and, furthermore, to reduce the problem of description of $\\mathrm{Ext}^1_{\\mathcal O}(L(\\mu),\\Delta(\\lambda))$ for $\\mathfrak{pe}(n)$ to the similar problem for the Lie algebra $\\mathfrak{gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\\mathcal O^{\\mathfrak p}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\\mathfrak{pe}(n)$ and the ortho-symplectic Lie superalgebra $\\mathfrak{osp}(2|2n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-14T07:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B55"
    ],
    "keywords": [
      "homological properties",
      "verma supermodule",
      "periplectic lie superalgebra",
      "classical lie superalgebras",
      "structural supermodules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}