{
  "id": "2201.07723",
  "version": "v1",
  "published": "2022-01-19T17:02:44.000Z",
  "updated": "2022-01-19T17:02:44.000Z",
  "title": "Realization of Weyl groups by Ringel-Hall Lie algebras of preprojective algebras",
  "authors": [
    "Fan Xu",
    "Fang Yang"
  ],
  "categories": [
    "math.RT",
    "math.CT",
    "math.QA"
  ],
  "abstract": "Let $\\Lambda_Q$ be the preprojective algebra of a finite acyclic quiver $Q$ of non-Dynkin type and $D^b(rep^n \\Lambda_Q)$ be the bounded derived category of finite dimensional nilpotent $\\Lambda_Q$-modules. We define spherical twist functors over the root category $\\mathcal{R}_{\\Lambda_Q}$ of $D^b(rep^n \\Lambda_Q)$ and then realize the Weyl group associated to $Q$ as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra $\\mathfrak{g}(\\mathcal{R}_{\\Lambda_Q})$ of $\\mathcal{R}_{\\Lambda_Q}$ induced by spherical twist functors. We also present a conjectural relation between certain Lie subalgebras of $\\mathfrak{g}(\\mathcal{R}_{\\Lambda_Q})$ and $\\mathfrak{g}(\\mathcal{R}_Q)$, where $\\mathfrak{g}(\\mathcal{R}_Q)$ is the Ringe-Hall Lie algebra associated to the root category $\\mathcal{R}_Q$ of $Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-19T17:02:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ringel-hall lie algebra",
      "weyl group",
      "preprojective algebra",
      "root category",
      "realization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}