Cluster categories and duplicated algebras

Assem, Ibrahim; Brüstle, Thomas; Schiffler, Ralf; Todorov, Gordana

doi:10.1016/j.jalgebra.2005.12.002

Cited by 31 publications

(144 citation statements)

References 18 publications

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“…Suppose that [1]σ 0 = 0 and, finally, σ 0 = 0, a contradiction. Now we prove the final statement: we have that…”

Section: Corollary 45 With the Notation Of Theorem 44 We Have Thatmentioning

confidence: 97%

“…Write X 0 as Z [1], where Z is an indecomposable representation in H. (C d (H)) the subset of E (C d (H)) consisting of all indecomposable exceptional objects other than…”

Section: Proof (Of Theorem 46)mentioning

confidence: 99%

“…This connection between tilting theory and cluster combinatorics leads Buan, Marsh, Reineke, Reiten and Todorov [6] to introduce cluster categories for a categorical model for cluster algebras, see also [9] for type A n . Cluster categories are the orbit categories D/τ −1 [1] of derived categories of hereditary categories arising from the action of subgroup τ −1 [1] of the automorphism group. They are triangulated categories [19] and now they have become a successful model for acyclic cluster algebras [5,7,8], see also the surveys [4,24] and references therein for recent developments and background of cluster tilting theory.…”

mentioning

confidence: 99%

“…d-cluster categories D/τ −1 [d], as a generalization of cluster categories, were introduced by Keller [19] and Thomas [25] for d ∈ N. They are studied by Keller and Reiten [20], Palu [1,23]; see also [3] for a geometric description of d-cluster categories of type A n . d-cluster categories are triangulated categories with Calabi-Yau dimension d + 1.…”

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confidence: 99%

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Generalized cluster complexes via quiver representations

Zhu

2007

J Algebr Comb

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We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (− −) on any pair of "colored" almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their dcompatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.

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“…Suppose that [1]σ 0 = 0 and, finally, σ 0 = 0, a contradiction. Now we prove the final statement: we have that…”

Section: Corollary 45 With the Notation Of Theorem 44 We Have Thatmentioning

confidence: 97%

“…Write X 0 as Z [1], where Z is an indecomposable representation in H. (C d (H)) the subset of E (C d (H)) consisting of all indecomposable exceptional objects other than…”

Section: Proof (Of Theorem 46)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Generalized cluster complexes via quiver representations

Zhu

2007

J Algebr Comb

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“…Since then cluster categories have been the subject of many investigations, see, for instance, [ABST1,ABST2,BMR1,BMR2,BMRT,CC,CCS2,CK1,CK2,K,KZ,Z1].…”

Section: Introductionmentioning

confidence: 99%

A geometric model for cluster categories of type D n

Schiffler

2007

J Algebr Comb

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We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. , which is only valid in type A n , the cluster category is realized by an ad-hoc method as a category of diagonals of a regular polygon with n + 3 vertices. The morphisms between diagonals are constructed geometrically using so called elementary moves and mesh relations. In that realization, clusters are in one-to-one correspondence with triangulations of the polygon and mutations are given by flips of diagonals in the triangulation. Recently, Baur and Marsh [BM] have generalized this model to m-cluster categories of type A n . IntroductionIn this paper, we give a geometric realization of the cluster categories of type D n in the spirit of [CCS1]. The polygon with (n + 3) vertices has to be replaced by a polygon with n vertices and one puncture in the center, and instead of looking at diagonals, which are straight lines between two vertices, one has to consider homotopy classes of paths between two vertices, which we will call edges. This punctured polygon model has appeared recently in the work of Fomin, Shapiro and Thurston [FST] on the relation between cluster algebras and triangulated surfaces. Let us point out that they work in a vastly more general context and the punctured polygon is only one example of their theory. We define the cluster category by an ad-hoc method as the category 1

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