{
  "id": "math/0511382",
  "version": "v2",
  "published": "2005-11-15T13:50:33.000Z",
  "updated": "2006-06-19T13:01:40.000Z",
  "title": "Equivalences between cluster categories",
  "authors": [
    "Bin Zhu"
  ],
  "comment": "second version",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a hereditary abelian category $\\mathcal{H}$, we verify that the tilting functor Hom$_\\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\\mathcal{C(H)}$ to the cluster category $\\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\\mathcal{H},$ $\\mathcal{H}'$ is triangle equivalent to the derived category of a hereditary algebra, we prove that the cluster categories $\\mathcal{C(H)}$ and $\\mathcal{C(H')}$ are triangle equivalent to each other if and only if $\\mathcal{H}$ and $\\mathcal{H}'$ are derived equivalent, by using the precise relation between cluster-tilted algebras (by definition, the endomorphism algebras of tilting objects in cluster categories) and the corresponding quasi-tilted algebras proved previously. As an application, we give a realization of \"truncated simple reflections\" defined by Fomin-Zelevinsky on the set of almost positive roots of the corresponding type [FZ2, FZ5], by taking $\\mathcal{H}$ to be the representation category of a valued Dynkin quiver and $T$ a BGP-tilting (or APR-tilting, in other words).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-06-19T13:01:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70"
    ],
    "keywords": [
      "cluster category",
      "hereditary abelian category",
      "hereditary algebra",
      "triangle equivalent",
      "derived category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....11382Z"
    }
  }
}