{
  "id": "math/0703039",
  "version": "v4",
  "published": "2007-03-01T17:35:26.000Z",
  "updated": "2010-07-30T09:05:53.000Z",
  "title": "Cluster algebra structures and semicanonical bases for unipotent groups",
  "authors": [
    "Christof Geiss",
    "Bernard Leclerc",
    "Jan Schröer"
  ],
  "comment": "Some minor typos corrected. Problem 23.1 of v2 is now solved (see Sections 22.8, 22.9), 121 pages. v4: typo in arxiv title corrected",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Let Q be a finite quiver without oriented cycles, and let $\\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of $mod(\\Lambda)$. We show that $C_M$ is a Frobenius category,and that its stable category is a Calabi-Yau category of dimension 2. Then we develop a theory of mutations of maximal rigid objects of $C_M$, analogous to the mutations of clusters in Fomin and Zelevinsky's theory of cluster algebras. We show that $C_M$ yields a categorification of a cluster algebra $A(C_M)$, which is not acyclic in general. We give a realization of $A(C_M)$ as a subalgebra of the graded dual of the enveloping algebra $U(\\n)$, where $\\n$ is a maximal nilpotent subalgebra of the symmetric Kac-Moody Lie algebra $\\g$ associated to the quiver Q. Let $S^*$ be the dual of Lusztig's semicanonical basis $S$ of $U(\\n)$. We show that all cluster monomials of $A(C_M)$ belong to $S^*$, and that $S^* \\cap A(C_M)$ is a basis of $A(C_M)$. Next, we prove that $A(C_M)$ is naturally isomorphic to the coordinate ring of the finite-dimensional unipotent subgroup $N(w)$ of the Kac-Moody group $G$ attached to $\\g$. Here w = w(M) is the adaptable element of the Weyl group of $\\g$ which we associate to each terminal representation M of Q. Moreover, we show that the cluster algebra obtained from $A(C_M)$ by formally inverting the generators of the coefficient ring is isomorphic to the coordinate ring of the unipotent cell $N^w := N \\cap (B_-wB_-)$ of G. We obtain a corresponding dual semicanonical basis of this coorindate ring.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-07-30T09:05:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M99",
      "16G20",
      "17B35",
      "17B67",
      "20G05",
      "81R10"
    ],
    "keywords": [
      "cluster algebra structures",
      "unipotent groups",
      "semicanonical bases",
      "terminal representation",
      "symmetric kac-moody lie algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 121,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3039G"
    }
  }
}