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108
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 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 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
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 58 (7 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
Mutation of clustertilting objects and potentials
 Amer. Journal Math. (2008
"... Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that cl ..."
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Cited by 56 (11 self)
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Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that clustertilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3CY algebras. The nearly Morita equivalence for 2CYtilted algebras is shown to hold for the finite length modules over Jacobian algebras.
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 49 (7 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
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional 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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Cited by 38 (16 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional 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 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
The periodicity conjecture for pairs of Dynkin diagrams
, 2010
"... We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories. ..."
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Cited by 36 (0 self)
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We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.
Cluster tilting for higher Auslander algebras
, 2008
"... The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category M ..."
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Cited by 30 (9 self)
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The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster 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 ncluster tilting subcategories of derived categories of ncomplete algebras.