{
  "id": "1008.3444",
  "version": "v2",
  "published": "2010-08-20T06:53:46.000Z",
  "updated": "2013-11-29T11:29:15.000Z",
  "title": "Cluster algebras arising from cluster tubes",
  "authors": [
    "Yu Zhou",
    "Bin Zhu"
  ],
  "comment": "21 pages, title changed, rewrite the proof of the main theorem in Section 3, add Section 5, final version to appear in Jour. London Math. Soc",
  "categories": [
    "math.RT",
    "math.CO",
    "math.RA"
  ],
  "abstract": "We study the cluster algebras arising from cluster tubes with rank bigger than $1$. Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object $T$ in the cluster tube $\\mathcal{C}_n$ of rank $n$. For any indecomposable rigid object $M$ in $\\mathcal{C}_n$, we define an analogous $X_M$ of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of $M$. We show that $X_M, X_{M'}$ satisfy the mutation formula when $M,M'$ form an exchange pair, and that $X_{?}: M\\mapsto X_M$ gives a bijection from the set of indecomposable rigid objects in $\\mathcal{C}_n$ to the set of cluster variables of cluster algebra of type $C_{n-1}$, which induces a bijection between the set of basic maximal rigid objects in $\\mathcal{C}_n$ and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube $\\mathcal{C}_n$ encode the combinatorics of the cluster algebra of type $B_{n-1}$ since the combinatorics of cluster algebras of type $B_{n-1}$ or of type $C_{n-1}$ are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type $C$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-29T11:29:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "16G70"
    ],
    "keywords": [
      "cluster tube",
      "cluster algebras arising",
      "indecomposable rigid object",
      "palus cluster character formula",
      "basic maximal rigid objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.3444Z"
    }
  }
}