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49
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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Cited by 91 (16 self)
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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.
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Tilting theory and cluster combinatorics
 572–618. EQUIVALENCE AND GRADED DERIVED EQUIVALENCE 43
"... of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsk ..."
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Cited by 55 (4 self)
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of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of selfinjective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APRtilting.
Cluster mutation via quiver representations
 Comment. Math. Helv
"... Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of ..."
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Cited by 45 (15 self)
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Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.
Cluster Categories for Algebras of Global Dimension 2 and . . .
, 2008
"... Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical cluster ..."
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Cited by 33 (1 self)
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Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical clustertilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by GeissLeclercSchröer and by BuanIyamaReitenScott. Our results rely on quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobifinite we prove that it is endowed with a clustertilting object whose endomorphism algebra is isomorphic
Quivers with potentials and their representations I: Mutations
, 2007
"... We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representationtheoretic interpretation of quiver mutations at arbitrary vertices. This gives a farreaching generalization of BernsteinGelfandPono ..."
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Cited by 33 (1 self)
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We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representationtheoretic interpretation of quiver mutations at arbitrary vertices. This gives a farreaching generalization of BernsteinGelfandPonomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, CalabiYau algebras, cluster algebras.
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 32 (6 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
Clustertilted algebras of finite representation type
 J. Algebra
, 2006
"... Abstract. We investigate the clustertilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) clustertilted algebra of finite type is uni ..."
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Cited by 23 (10 self)
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Abstract. We investigate the clustertilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) clustertilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of clustertilted algebras of finite representation type are obtained along the way.
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 22 (3 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
Clusters and seeds for acyclic cluster algebras with an appendix by Buan
 A., Caldero P., Keller B., Marsh R., Reiten I., Todorov G., Proc. Amer. Math. Soc
"... Abstract. We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky. We also obtain an interpretation of the monomial in the denominator of a nonpolynomial cluster vari ..."
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Cited by 21 (4 self)
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Abstract. We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky. We also obtain an interpretation of the monomial in the denominator of a nonpolynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.