Cluster structures for 2-Calabi-Yau categories and unipotent groups (2009)

by A B Buan, O Iyama, I Reiten, J Scott
Venue:Compos. Math
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Cluster Categories for Algebras of Global Dimension 2 and . . .

by Claire Amiot , 2008
"... Let k be a field and A a finite-dimensional k-algebra 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 Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster ..."
Abstract - Cited by 120 (10 self) - Add to MetaCart
Let k be a field and A a finite-dimensional k-algebra 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 Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott. 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 Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic
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Cluster algebras, quiver representations and triangulated categories

by Bernhard Keller , 2009
"... This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau 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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This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau 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

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 ..."
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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 ..."
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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

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 ..."
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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
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Mutation of cluster-tilting objects and potentials

by A. B. Buan, O. Iyama, I. Reiten, D. Smith - Amer. Journal Math. (2008
"... Abstract. We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cl ..."
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Abstract. We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.
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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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Cluster tilting for one-dimensional hypersurface singularities

by Igor Burban, Osamu Iyama, Bernhard Keller, Idun Reiten - Adv. Math
"... Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

The periodicity conjecture for pairs of Dynkin diagrams

by Bernhard Keller , 2010
"... We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorification via triangulated categories. ..."
Abstract - Cited by 36 (0 self) - Add to MetaCart
We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.
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Cluster tilting for higher Auslander algebras

by Osamu Iyama , 2008
"... The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander-Reiten translation functor τn plays an important role in the study of n-cluster tilting subcategories. We study the category M ..."
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The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander-Reiten translation functor τn plays an important role in the study of n-cluster tilting subcategories. We study the category Mn of preinjective-like modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ n-complete if Mn = add M for an n-cluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)-complete. This gives an inductive construction of n-complete algebras. For example, any representation-finite hereditary algebra Λ (1) is 1-complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2-complete. Moreover, for any n ≥ 1, we have an n-complete algebra Λ (n) which has an n-cluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct n-cluster tilting subcategories of derived categories of n-complete algebras.