## On generalized cluster categories (2011)

Citations: | 12 - 4 self |

### Citations

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Citation Context ...ategory we mean an exact k-category with enough projectives and injectives ,where the projectives and the injectives coincide. If E is Frobenius, then the associated stable category E is triangulated =-=[Hap88]-=- and is by definition algebraic. For an object T in an additive k-category, we denote by add (T ) the additive closure of T . For a k-algebra A, we denote by ModA the category of right A-modules and b... |

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Citation Context ... cluster category CQ was defined to be the orbit category Db(kQ)/τ−[1], where τ is the Auslander-Reiten translation in the bounded derived category Db(kQ). The category CQ is Hom-finite, triangulated =-=[Kel05]-=-, and 2-Calabi-Yau (2-CY for short), that is, there is a functorial isomorphism DHomCQ(X, Y ) ≃ HomCQ(Y,X [2]), where D = Homk(−, k). A theory for a special kind of objects, called cluster-tilting obj... |

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176 | Quivers with potentials and their representations
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Citation Context ...er is organized as follows. In section 1 we recall some background material on cluster-tilting objects in 2-CY categories, on generalized cluster categories from [Ami08] and on Jacobian algebras from =-=[DWZ08]-=-, together with the generalization to frozen Jacobian algebras given in [BIRS09b]. In section 2 we construct a special triangle (Proposition 2.7), which is useful for our construction of a functor fro... |

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118 | Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59
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Citation Context ...LUSTER CATEGORIES CLAIRE AMIOT, IDUN REITEN, AND GORDANA TODOROV Abstract. Associated with a finite dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in =-=[Ami08]-=-. It was shown to be triangulated, and 2Calabi-Yau when it is Hom-finite. By definition, the cluster categories of [BMR+06] are a special case. In this paper we show that a large class of 2-Calabi-Yau... |

118 |
Mutation in triangulated categories and rigid Cohen-Macaulay modules
- Iyama, Yoshino
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Citation Context ...able and pairwise not isomorphic. Then for each i = 1, . . . , n there is a unique indecomposable object T ∗i not isomorphic to Ti, such that T ∗ = (T/Ti) ⊕ T ∗ i is a cluster-tilting object [BMR+06],=-=[IY08]-=-. The new object T ∗ is called the mutation of T at Ti. If T = T1 ⊕ . . . ⊕ Tn is a cluster-tilting object in a Frobenius stably 2-CY category, we can only mutate at the Ti which are not projective-in... |

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105 | Cluster structures for 2-Calabi-Yau categories and unipotent groups
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Citation Context ...quivalence. 4. 2-Calabi-Yau categories associated with elements in the Coxeter group In this section we apply Theorem 3.1 to the categories associated with elements in the Coxeter group introduced in =-=[BIRS09a]-=-. 4.1. Results of [BIRS09a] and [BIRS09b]. Let Q be a finite quiver without oriented cycles. We denote as usual by Q0 = {1, . . . , n} the set of vertices and by Q1 the set of arrows. The preprojectiv... |

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Citation Context ... 5.2. Example which is not associated with a word. Let E be the category modΛ where Λ is the preprojective algebra of type A3, which is one of the cases investigated by Geiss, Leclerc and Schröer in =-=[GLS06]-=-. This is a Frobenius category which is stably 2-Calabi-Yau and of the THE UBIQUITY OF GENERALIZED CLUSTER CATEGORIES 29 form Ew where w is the element in the Coxeter group of maximal length. Correspo... |

84 | Derived equivalences from mutations of quivers with potential - Keller, Yang |

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73 |
Sous les catégories dérivées
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Citation Context ...ALIZED CLUSTER CATEGORIES 7 are isomorphic. Here the category Db(P) is the thick subcategory of Db(E) generated by P. Thus the localization of Db(E) by Db(P) is equivalent to the stable category E by =-=[KV87]-=-, and this localization gives us the right vertical map of this diagram. This implies immediately that the functor L ⊗ΛM : D b(Λ) → E factors through the orbit category Db(Λ)/S[−2]. However the proof ... |

71 | Generalized associahedra via quiver representations
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Citation Context ...torial isomorphism DHomCQ(X, Y ) ≃ HomCQ(Y,X [2]), where D = Homk(−, k). A theory for a special kind of objects, called cluster-tilting objects, was developed in [BMR+06]. This work was motivated via =-=[MRZ03]-=- by the Fomin-Zelevinsky theory of cluster algebras [FZ02], where the cluster-tilting objects are the analogs of clusters. Another category where a similar theory was developed is the category modΛ of... |

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Citation Context ...se Lemma 2.1 to obtain exact sequences of B̄-B-bimodules which permit to get the functorial minimal projective and injective resolutions of any B̄-module when viewed as a Bmodule. This is inspired by =-=[Boc08]-=-. Proposition 2.2. There exist exact sequences in mod (B̄op ⊗B) (a) 0 // ⊕ i/∈F0 Pi,i // ⊕ a/∈F1 Ps(a),t(a) // ⊕ a/∈F1 Pt(a),s(a) // ⊕ i/∈F0 Pi,i // B̄ // 0 (b) 0 // B̄ // ⊕ i/∈F0 Ii,i // ⊕ a/∈F1 Is(a... |

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54 | Mutation of cluster-tilting objects and potentials
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Citation Context ...imension of Ā 15 3.3. Construction of a triangle functor 16 3.4. Proof of the equivalence 18 4. 2-Calabi-Yau categories associated with elements in the Coxeter group 22 4.1. Results of [BIRS09a] and =-=[BIRS09b]-=- 22 4.2. Definition of the grading 24 4.3. Meaning of the grading 25 5. Examples 25 5.1. Standard cluster-tilting object associated to a reduced word 26 5.2. Example which is not associated with a wor... |

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Citation Context ...a generalized cluster category CΛ to some stable category E . It can be deduced from the universal property of piΛ given in subsection 4.1 of [Ami08] (see also section 9 of [Kel05] or the appendix of =-=[IO09]-=- for more details). This criterion, which is given in the next proposition, is a key step for proving the equivalence of the main theorem of this paper. For a Frobenius category E and an algebra Λ, we... |

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6 |
algebra structures and semicanonical bases for unipotent groups
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Citation Context ...f 2-CY triangulated categories with cluster-tilting objects. An important class is the stable categories Ew of the Frobenius categories Ew associated with elements w in Coxeter groups [BIRS09a], (see =-=[GLS07b]-=- for independent work when w is adaptable). This class contains both the cluster categories CQ and modΛ discussed above as special cases (see [BIRS09a], [GLS07b]). In Ew and Ew, there are standard clu... |

6 |
125–180, With an appendix by Michel Van den
- Math
- 2011
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Citation Context ... has no oriented cycles, where T is a cluster-tilting object in an algebraic 2-CY THE UBIQUITY OF GENERALIZED CLUSTER CATEGORIES 3 category C, then C is triangle equivalent to the cluster category CQ =-=[KR08]-=-. A crucial step in this paper for proving the equivalence CĀ ≃ E is the construction of a triangle functor from CĀ to E , sending Ā to T . This is done by first constructing a triangle functor fro... |

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3 |
Preprojective algebras and c-sortable words, arXiv:1002.4131
- Amiot, Iyama, et al.
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Citation Context ...oxeter groups from [BIRS09a], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in [Ami08] and then more generally for c-sortable elements in =-=[AIRT10]-=-. Contents Introduction 2 Notations 3 1. Background 3 1.1. Cluster-tilting objects 3 1.2. Generalized cluster categories 5 1.3. Jacobian algebras and generalizations 7 2. A useful triangle 8 2.1. Basi... |

2 | Algebras of tame acyclic cluster type - Amiot, Oppermann |

2 | The image of the derived category in the cluster category - Amiot, Oppermann |

1 | in press (arXiv:math. RT/09083499). ha l-0 2, v er sio n - 1 No v On generalized cluster categories 53 - Math |

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1 | Cluster categories. Preprint 2010, arXiv - Reiten |

1 | Calabi-Yau algebras and superpotentials. Talk at Séminaire d’algèbre at Institut Henri Poicare - Bergh - 2010 |