{
  "id": "1904.12468",
  "version": "v1",
  "published": "2019-04-29T06:52:09.000Z",
  "updated": "2019-04-29T06:52:09.000Z",
  "title": "BGG category for the quantum Schrödinger algebra",
  "authors": [
    "Genqiang Liu",
    "Yang Li"
  ],
  "comment": "18 pages",
  "categories": [
    "math.RT",
    "math.QA",
    "math.RA"
  ],
  "abstract": "In this paper, we study the BGG category $\\mathcal{O}$ for the quantum Schr{\\\"o}dinger algebra $U_q(\\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\\dot z\\neq 0$, using the module $B_{\\dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $\\mathcal{O}[\\dot z]$ consisting of modules with the central charge $\\dot z$ and the BGG category $\\mathcal{O}^{(\\mathfrak{sl}_2)}$ for the quantum group $U_q(\\mathfrak{sl}_2)$. In the case that $\\dot z=0$, we study the subcategory $\\mathcal{A}$ consisting of finite dimensional $U_q(\\mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in \\cite{DLMZ, Mak}, we directly construct an equivalent functor from $\\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\\mathfrak{s})$-modules is wild.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-29T06:52:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bgg category",
      "quantum schrödinger algebra",
      "central charge",
      "quantum weyl algebra",
      "finite dimensional representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}