{
  "id": "2207.02837",
  "version": "v1",
  "published": "2022-07-06T17:51:44.000Z",
  "updated": "2022-07-06T17:51:44.000Z",
  "title": "Quantum cluster algebras associated to weighted projective lines",
  "authors": [
    "Fan Xu",
    "Fang Yang"
  ],
  "comment": "38 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathbb{X}_{\\boldsymbol{p},\\boldsymbol{\\lambda}}$ be a weighted projective line. We define the quantum cluster algebra of $\\mathbb{X}_{\\boldsymbol{p},\\boldsymbol{\\lambda}}$ and realize its specialized version as the subquotient of the Hall algebra of $\\mathbb{X}_{\\boldsymbol{p},\\boldsymbol{\\lambda}}$ via the quantum cluster character map. Inspired by \\cite{Chen2021}, we prove an analogue cluster multiplication formula between quantum cluster characters. As an application, we obtain the polynomial property of the cardinalities of Grassmannian varieties of exceptional coherent sheaves on $\\mathbb{X}_{\\boldsymbol{p},\\boldsymbol{\\lambda}}$ . In the end, we construct several bar-invariant $\\mathbb{Z}[\\nu^{\\pm}]$-bases for the quantum cluster algebra of the projective line $\\mathbb{P}^1$ and show how it coincides with the quantum cluster algebra of the Kronecker quiver.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-06T17:51:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "17B20",
      "18F20",
      "20G42"
    ],
    "keywords": [
      "quantum cluster algebra",
      "weighted projective line",
      "quantum cluster character map",
      "analogue cluster multiplication formula",
      "exceptional coherent sheaves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}