{
  "id": "1101.0580",
  "version": "v3",
  "published": "2011-01-03T19:15:35.000Z",
  "updated": "2013-05-23T12:13:31.000Z",
  "title": "Quantum cluster algebras of type A and the dual canonical basis",
  "authors": [
    "Philipp Lampe"
  ],
  "comment": "48 pages",
  "doi": "10.1112/plms/pds098",
  "categories": [
    "math.RT"
  ],
  "abstract": "The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis under Kashiwara's bilinear form.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-23T12:13:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "17B37",
      "16G20"
    ],
    "keywords": [
      "quantum cluster algebra",
      "dual canonical basis",
      "quantum cluster variables",
      "kashiwaras bilinear form",
      "ordinary cluster algebra geiss-leclerc-schroeer"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.0580L"
    }
  }
}