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34
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 ..."
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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 ..."
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Cited by 100 (10 self)
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Root systems and generalized associahedra 1 Root systems and generalized associahedra 3
Rigid modules over PREPROJECTIVE ALGEBRAS
, 2005
"... Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λmodules, their endomorphism algebras and a mutation operation on these modules. This leads to a representationtheoretic construction of the cluster algebra structure on the ring C[N] of polynomial functions on ..."
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Cited by 91 (13 self)
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Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λmodules, their endomorphism algebras and a mutation operation on these modules. This leads to a representationtheoretic construction of the cluster algebra structure on the ring C[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type ∆. As an application we obtain that all cluster monomials
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 ..."
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Cited by 74 (9 self)
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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 ..."
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Cited by 63 (2 self)
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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
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 ..."
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Cited by 51 (6 self)
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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
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 ..."
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Cited by 48 (5 self)
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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.
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 ..."
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Cited by 37 (2 self)
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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