{
  "id": "math/0208229",
  "version": "v2",
  "published": "2002-08-29T02:24:52.000Z",
  "updated": "2003-03-12T00:17:04.000Z",
  "title": "Cluster algebras II: Finite type classification",
  "authors": [
    "Sergey Fomin",
    "Andrei Zelevinsky"
  ],
  "comment": "50 pages, 18 figures. Version 2: new introduction; final version, to appear in Invent. Math",
  "categories": [
    "math.RA",
    "math.AG",
    "math.CO"
  ],
  "abstract": "This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-03-12T00:17:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M99"
    ],
    "keywords": [
      "cluster algebra",
      "finite type classification",
      "finite root systems",
      "essential combinatorial ingredient",
      "semisimple lie algebras"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00222-003-0302-y",
      "journal": "Inventiones Mathematicae",
      "year": 2003,
      "month": "Oct",
      "volume": 154,
      "number": 1,
      "pages": 63
    },
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003InMat.154...63F"
    }
  }
}