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87
From triangulated categories to cluster algebras
"... 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 ..."
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Cited by 181 (21 self)
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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.
Cluster structures for 2-Calabi-Yau categories and unipotent groups
"... 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 ..."
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Cited by 110 (18 self)
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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
Derived equivalences from mutations of quivers with potential
- ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
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Cluster algebras and quantum affine algebras
, 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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Cited by 74 (9 self)
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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
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 (12 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.
Mutation of cluster-tilting objects and potentials
- 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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Cited by 56 (10 self)
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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.
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 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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Cited by 51 (6 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 CQ-module M (these are certain preinjective CQ-modules), we attach a natural subcategory CM of mod(Λ). We show that CM is a
Quiver varieties and cluster algebras
, 2009
"... Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a Fomin-Zelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on ..."
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Cited by 51 (0 self)
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Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a Fomin-Zelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties [49]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [21] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize
Cluster tilting for one-dimensional hypersurface singularities
- 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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Cited by 37 (15 self)
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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.
Cluster tilting for higher Auslander algebras
"... Abstract. 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 ca ..."
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Cited by 33 (9 self)
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Abstract. 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. Contents 1. Our results 3 1.1. n-cluster tilting in module categories 4
