{
  "id": "1002.2762",
  "version": "v2",
  "published": "2010-02-14T11:11:21.000Z",
  "updated": "2010-02-19T15:17:52.000Z",
  "title": "A quantum cluster algebra of Kronecker type and the dual canonical basis",
  "authors": [
    "Philipp Lampe"
  ],
  "comment": "32 pages",
  "journal": "Int. Math. Res. Notices 2011, no. 13, 2970-3005",
  "doi": "10.1093/imrn/rnq162",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U(n). We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(C_M) is the specialization of the dual of an appropriate canonical basis element. Therefore, U_q^+(w) is a quantum cluster algebra in the sense of Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized cluster variables and for expansions of products of dual canonical basis elements.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-19T15:17:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B37",
      "13F60"
    ],
    "keywords": [
      "quantum cluster algebra",
      "kronecker type",
      "dual canonical basis elements",
      "subalgebra",
      "universal enveloping algebra"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.2762L"
    }
  }
}