{
  "id": "math/0305434",
  "version": "v3",
  "published": "2003-05-30T03:58:41.000Z",
  "updated": "2004-01-21T22:13:58.000Z",
  "title": "Cluster algebras III: Upper bounds and double Bruhat cells",
  "authors": [
    "Arkady Berenstein",
    "Sergey Fomin",
    "Andrei Zelevinsky"
  ],
  "comment": "39 pages. Minor editorial changes, a reference added. This is the final version, to appear in Duke Mathematical Journal",
  "categories": [
    "math.RT",
    "math.AC",
    "math.AG"
  ],
  "abstract": "We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of \"acyclicity\", a cluster algebra coincides with its \"upper\" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2004-01-21T22:13:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "double bruhat cell",
      "upper bounds",
      "upper cluster algebra",
      "semisimple complex lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5434B"
    }
  }
}