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30
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.
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 ..."
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Cited by 40 (7 self)
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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 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 33 (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
ON CLUSTER ALGEBRAS WITH COEFFICIENTS AND 2CALABIYAU CATEGORIES
"... Abstract. Building on work by GeissLeclercSchröer and by BuanIyamaReitenScott we investigate the link between certain cluster algebras with coefficients and suitable 2CalabiYau categories. These include the clustercategories associated with acyclic quivers and certain Frobenius subcategories ..."
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Cited by 23 (6 self)
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Abstract. Building on work by GeissLeclercSchröer and by BuanIyamaReitenScott we investigate the link between certain cluster algebras with coefficients and suitable 2CalabiYau categories. These include the clustercategories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in ‘Cluster algebras IV: Coefficients’. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always
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 22 (1 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 19 (2 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 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 18 (1 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
Preprojective algebras and cluster algebras
, 2008
"... We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups. ..."
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Cited by 10 (0 self)
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We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
Verma modules and preprojective algebras
"... We give a geometric construction of the Verma modules of a symmetric KacMoody Lie algebra g in terms of constructible functions on the varieties of nilpotent finitedimensional modules of the corresponding preprojective algebra Λ. 1 ..."
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Cited by 8 (4 self)
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We give a geometric construction of the Verma modules of a symmetric KacMoody Lie algebra g in terms of constructible functions on the varieties of nilpotent finitedimensional modules of the corresponding preprojective algebra Λ. 1
Cluster categories and selfinjective algebras: type A
, 2006
"... We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories. ..."
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Cited by 7 (2 self)
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We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories.