Semicanonical bases and preprojective algebras II: a multiplication formula, (2007)

by C GEISS, B LECLERC, J SCHROER
Venue:Compositio Math.
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On differential graded categories

by Bernhard Keller - 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, Dugger-Shipley,..., Toën and Toën-Vaquié. ..."
Abstract - Cited by 190 (4 self) - Add to MetaCart
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, Dugger-Shipley,..., Toën and Toën-Vaquié.
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From triangulated categories to cluster algebras

by Philippe Caldero, Bernhard Keller
"... Abstract. In the acyclic case, we establish a one-to-one 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 173 (20 self) - Add to MetaCart
Abstract. In the acyclic case, we establish a one-to-one 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 Calabi-Yau property of the cluster category. 1.
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Cluster structures for 2-Calabi-Yau categories and unipotent groups

by A. B. Buan, O. Iyama, I. Reiten, J. Scott
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This c ..."
Abstract - Cited by 108 (19 self) - Add to MetaCart
Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau 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

Rigid modules over PREPROJECTIVE ALGEBRAS

by Christof Geiss, Bernard Leclerc, Jan Schröer , 2005
"... Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring C[N] of polynomial functions on ..."
Abstract - Cited by 88 (14 self) - Add to MetaCart
Let Λ be a preprojective algebra of simply laced Dynkin type ∆. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic 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

by David Hernandez, Bernard Leclerc , 2009
"... Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(̂g) of simply-laced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C ..."
Abstract - Cited by 71 (10 self) - Add to MetaCart
Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(̂g) of simply-laced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C

Cluster algebras and triangulated surfaces. Part I: Cluster complexes

by Sergey Fomin, Michael Shapiro, Dylan Thurston
"... 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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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
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ON CLUSTER ALGEBRAS WITH COEFFICIENTS AND 2-CALABI-YAU CATEGORIES

by Changjian Fu, Bernhard Keller
"... Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories ..."
Abstract - Cited by 58 (7 self) - Add to MetaCart
Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories 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

by Christof Geiss, Bernard Leclerc, Jan Schröer , 2008
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a ..."
Abstract - Cited by 49 (7 self) - Add to MetaCart
Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. To each terminal CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
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Laurent expansions in cluster algebras via quiver representations

by Philippe Caldero, Andrei Zelevinsky , 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 46 (5 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.
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On the combinatorics of rigid objects in 2-Calabi-Yau categories

by Raika Dehy, Bernhard Keller - INT. MATH. RES. NOT. IMRN, (11):ART. ID RNN029 , 2008
"... Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposab ..."
Abstract - Cited by 26 (2 self) - Add to MetaCart
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T ′ form a basis of the Grothendieck group of T and that, if T and T ′ are related by a mutation, then the indices with respect to T and T ′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin-Zelevinsky’s g-vectors.
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