Cluster algebras, quiver representations and triangulated categories

Keller, Bernhard

doi:10.1017/cbo9781139107075.004

Cited by 176 publications

(273 citation statements)

References 92 publications

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“…Note that the category D Mod Q T is not abelian in general. By a result of Bernhard Keller [9], it is a triangulated category. We can also construct the total category C generated by all the diagonals of the (n + 3) polygon.…”

Section: The Orbit Categorymentioning

confidence: 99%

Quivers with relations arising from clusters (𝐴_{𝑛} case)

Caldero

Chapoton

Schiffler

2005

Trans. Amer. Math. Soc.

329

448

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Abstract. Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A n . We associate to each cluster C of U an abelian category C C such that the indecomposable objects of C C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of C C . Then, we generalize the "denominator theorem" of Fomin and Zelevinsky to any cluster. IntroductionCluster algebras were introduced in the work of A. Berenstein, S. Fomin, and A. Zelevinsky [2,5,6,7]. This theory appeared in the context of dual canonical basis and more particularly in the study of the Berenstein-Zelevinsky conjecture. Cluster algebras are now connected with many topics: double Bruhat cells, Poisson varieties, total positivity, Teichmüller spaces. The main results on cluster algebras are on one hand the classification of finite cluster algebras by root systems and on the other hand the realization of algebras of regular functions on double Bruhat cells in terms of cluster algebras.Recall some facts about cluster algebras. Cluster algebras U of rank n form a class of algebras defined axiomatically in terms of a distinguished set of generators {u 1 , . . . , u n }. A cluster is a set of "cluster variables" {w 1 , . . . , w n } obtained combinatorially from {u 1 , . . . , u n }. The so-called Laurent phenomenon asserts that each cluster variable is a Laurent polynomial in the set of variables given by a cluster. For each cluster C, one can define combinatorially an oriented quiver Q C .Suppose from now on that U is a finite cluster algebra, i.e. there only exists a finite number of cluster variables. It is known that U can be described by the data of a root system X n . Moreover, there exists a cluster Σ such that Q Σ is the alternating quiver on X n . S. Fomin and A. Zelevinsky give in [6] a more precise description of the Laurent phenomenon: there exists a one-to-one correspondence α → w α between the set of almost positive roots of X n , i.e. positive roots and simple negative roots, and the set of cluster variables, such that the denominator of w α as a Laurent polynomial in Σ is given by the decomposition of α in the basis of simple roots. Via Gabriel's Theorem, this property suggests a link between cluster algebras and representation theory of artinian rings. This relation was already investigated in [10] and is the core subject of the recent papers [3,4].

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Section: The Orbit Categorymentioning

confidence: 99%

Quivers with relations arising from clusters (𝐴_{𝑛} case)

Caldero

Chapoton

Schiffler

2005

Trans. Amer. Math. Soc.

329

448

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“…Our way of understanding the cluster algebra C[N ] via the category mod(Λ) is very similar to the approach of [6], [7], [8], [10], [11], [28] which study cluster algebras attached to quivers via some new cluster categories. There are however two main differences.…”

Section: A λ-Module M Is Called Rigid Provided Extmentioning

confidence: 99%

Rigid modules over preprojective algebras

Leclerc²,

2006

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Abstract. 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 of C[N ] belong to the dual semicanonical basis.

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“…Fomin and Zelevinsky proved it for a special case (level 2 case in our terminology) [FZ3] by the cluster algebra approach [FZ1,FZ2,FZ4]. Since then, a remarkable link has been established between cluster algebras and cluster categories of the quiver representations (See [BMRRT,BMR,CC,CK1,CK2,Kel2] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Periodicities of T-systems and Y-systems

Inoue¹,

Iyama²,

Kuniba³

et al. 2010

Nagoya Math. J.

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Abstract. The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

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Cluster algebras, quiver representations and triangulated categories

Cited by 176 publications

References 92 publications

Quivers with relations arising from clusters (𝐴_{𝑛} case)

Quivers with relations arising from clusters (𝐴_{𝑛} case)

Rigid modules over preprojective algebras

Periodicities of T-systems and Y-systems

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