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173
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 178 (3 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.
Clustertilted algebras are Gorenstein and stably CalabiYau
 CALABIYAU, ADV. MATH
, 2006
"... We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that cluster ..."
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Cited by 145 (18 self)
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We prove that in a 2CalabiYau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its CohenMacaulay modules is 3CalabiYau. We deduce in particular that clustertilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3CalabiYau duality over non stable endomorphism rings. This strengthens and generalizes the Extgroup symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative CalabiYau duality from 2CalabiYau to dCalabiYau categories. We show how to produce many examples of dcluster tilted algebras.
Cluster algebras as Hall algebras of quiver representations
, 2005
"... Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the ..."
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Cited by 135 (5 self)
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Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the
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 120 (10 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
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 108 (19 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
Cluster algebras, quiver representations and triangulated categories
, 2009
"... This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In additi ..."
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Cited by 106 (6 self)
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This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
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 88 (14 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 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 80 (19 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 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 71 (10 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
Semicanonical bases and preprojective algebras II: A multiplication formula
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
"... Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type a ..."
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Cited by 67 (13 self)
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Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky’s theory of cluster algebras. It was inspired by recent results of Caldero and Keller.