Results 1  10
of
16
From triangulated categories to cluster algebras
"... Abstract. In the acyclic case, we establish a onetoone correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator ..."
Abstract

Cited by 91 (16 self)
 Add to MetaCart
Abstract. In the acyclic case, we establish a onetoone correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the CalabiYau property of the cluster category. 1.
Ysystems and generalized associahedra
 Ann. of Math
"... Root systems and generalized associahedra 1 Root systems and generalized associahedra 3 ..."
Abstract

Cited by 61 (8 self)
 Add to MetaCart
Root systems and generalized associahedra 1 Root systems and generalized associahedra 3
RIGID MODULES OVER PREPROJECTIVE ALGEBRAS II: THE Kacmoody Case
, 2007
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials ..."
Abstract

Cited by 40 (7 self)
 Add to MetaCart
Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig’s semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric KacMoody Lie algebra g associated to the quiver Q.
Cluster algebra structures and semicanonical bases for unipotent groups
, 2008
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQmodule M (these are certain preinjective CQmodules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
Cluster algebras and quantum affine algebras
, 2009
"... Let C be the category of finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
Let C be the category of finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C
Cluster algebras and triangulated surfaces. Part I: Cluster complexes
"... Abstract. We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
Abstract. We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate. Contents
Laurent expansions in cluster algebras via quiver representations
, 2006
"... We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster alg ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra.
Cambrian Fans
"... Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc betwee ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc between csortable elements and cclusters induces a combinatorial isomorphism of fans. In particular, the cCambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the cCambrian fan are generated by certain vectors in the Worbit of the fundamental weights, while the rays of the ccluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the cCambrian fan is linearly isomorphic to the ccluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on cclusters by the cCambrian lattice. We give a simple bijection from cclusters to cnoncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric
Cluster algebras IV: coefficients
"... Abstract. We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of “principal ” coe ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Abstract. We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of “principal ” coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multigradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V. Fock and A. Goncharov. The coefficient dynamics leads to a natural generalization of Al. Zamolodchikov’s Ysystems. We establish a Laurent phenomenon for such Ysystems, previously known in finite type only, and sharpen the periodicity result from an earlier paper. For cluster algebras of finite type, we identify a canonical “universal ” choice
ON THE PROPERTIES OF THE EXCHANGE GRAPH OF A CLUSTER ALGEBRA
, 2008
"... Abstract. We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coeffi ..."
Abstract
 Add to MetaCart
Abstract. We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and results To state our results, recall the definition of a cluster algebra; for details see [FZ1, FZ4]. Let P be a semifield, that is, a free multiplicative abelian group of a finite rank m with generators g1,..., gm endowed with an additional operation ⊕, which is commutative, associative and distributive with respect to the multiplication. As an ambient field we take the field F of rational functions in n independent variables with coefficients in the field of fractions of the integer group ring ZP. A square matrix B is called skewsymmetrizable if DB is skewsymmetric for a nondegenerate nonnegative diagonal matrix D. A seed is a triple Σ = (x,y, B), where x = (x1,...,xn) is a transcendence basis of F over the field of fractions of ZP, y = (y1,..., yn) is an ntuple of elements of P and B is a skewsymmetrizable integer n × n matrix. The components of the seed are called the cluster, the coefficient tuple and the exchange matrix, respectively; the entries of x are called cluster variables. A seed mutation in direction k ∈ [1, n] takes Σ to an adjacent seed Σ ′ = (x′,y ′, B ′) whose components are defined as follows. The adjacent cluster x ′ is given by x ′ = (x \ {xk}) ∪ {x ′ k}, where the new cluster variable x ′ k is defined by the exchange relation (1) xkx ′ k = yj yj ⊕ 1 bki>0 x bki i