## Cluster structures for 2-Calabi-Yau categories and unipotent groups

Citations: | 33 - 6 self |

### BibTeX

@MISC{Buan_clusterstructures,

author = {A. B. Buan and O. Iyama and I. Reiten and J. Scott},

title = {Cluster structures for 2-Calabi-Yau categories and unipotent groups},

year = {}

}

### OpenURL

### Abstract

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

### Citations

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Citation Context ...deal T in Λ, the factor algebra Λ/T is Gorenstein of dimension at most one. Proof. (a) Consider the exact sequence 0 → ΩΛ(D(Λ/T)) → P → D(Λ/T) → 0 with a projective Λ-module P. Using Lemma II.2.1 and =-=[CE]-=-, we have Tor Λ 1 (Λ/T, D(Λ/T)) ≃ D Ext 1 Λop(Λ/T, Λ/T) = 0. Applying Λ/T ⊗Λ to the above exact sequence, we get the exact sequence 0 → Λ/T ⊗ΛΩΛ(D(Λ/T)) → Λ/T ⊗ΛP → Λ/T ⊗ΛD(Λ/T) → 0. The Λ/T-module Λ/... |

299 | Representation theory of artin algebras - Auslander, Reiten, et al. - 1997 |

228 |
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Citation Context ...ed 2-CY category. We say that an exact category C is stably 2-CY if it is Frobenius, that is, C has enough projectives and injectives, which coincide, and the stable category C, which is triangulated =-=[H1]-=-, is Hom-finite 2-CY. Recall that C is said to have enough projectives if for each X in C there is an exact sequence 0 → Y → P → X → 0 in C with P projective. Having enough injectives is defined in a ... |

185 | Tame algebras and integral quadratic forms - Ringel - 1984 |

136 |
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Citation Context ... 0, (iii) there exists an exact sequence 0 → Λ → T0 → · · · → Tn → 0 with Ti in addT.CLUSTER STRUCTURES FOR 2-CALABI-YAU CATEGORIES AND UNIPOTENT GROUPS 21 We say that T ∈ D(Mod Λ) a tilting complex =-=[Ri]-=- if (i ′ ) T is quasi-isomorphic to an object in the category K b (pr Λ) of bounded complexes of finitely generated projective Λ-modules prΛ, (ii ′ ) Hom D(mod Λ)(T, T[i]) = 0 for any i ̸= 0, (iii ′ )... |

120 | perverse sheaves, and quantized enveloping algebras - Lusztig - 1991 |

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90 | From triangulated categories to cluster algebras II, Annales Scientifiques de l’ École Normale Supérieure, 4ème série - Caldero, Keller |

83 | Double Bruhat cells and total positivity
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Citation Context ...skew-symmetric part v appearing in (6) of the unipotent element and ψ± = 1 ( ) [678] 2 [18] −[27]+[36] ±[45] . The functions and Pfaff [1234] Pfaff [1234] are examples of generalized minors of type D =-=[FZ1]-=-. In the notation of Example 3 in Section I.3, we have seen that T = M16 ⊕ M24 ⊕ M25 ⊕ M26 ⊕ M68 ⊕ M18 ⊕ M− ⊕ M+ ⊕ P2 is a cluster tilting object in B = SubP2, which can be extended to a cluster tilti... |

80 | Quivers with relations arising from clusters (An case
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Citation Context ... the Norwegian Research Council. 12 BUAN, IYAMA, REITEN, AND SCOTT finite dimensional hereditary algebras were introduced for this purpose in [BMRRT], and shown to be triangulated in [Ke1] (see also =-=[CCS]-=- for the An case), and the module categories mod Λ for Λ a preprojective algebra of a Dynkin quiver have been used for a similar purpose [GLS1]. This development has both inspired new directions of in... |

80 | On triangulated orbit categories
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Citation Context ...rant 167130 from the Norwegian Research Council. 12 BUAN, IYAMA, REITEN, AND SCOTT finite dimensional hereditary algebras were introduced for this purpose in [BMRRT], and shown to be triangulated in =-=[Ke1]-=- (see also [CCS] for the An case), and the module categories mod Λ for Λ a preprojective algebra of a Dynkin quiver have been used for a similar purpose [GLS1]. This development has both inspired new ... |

78 |
Cohen-Macaulay modules over Cohen-Macaulay rings
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Citation Context .... (2) The stable category of maximal Cohen-Macaulay modules CM(R) over a 3-dimensional complete local commutative noetherian Gorenstein isolated singularity R containing the residue field k [A2] (see =-=[Yo]-=-). (3) The preprojective algebra Λ associated to a finite connected quiver Q without loops is defined as follows: Let ˜ Q be the quiver constructed from Q by adding an arrow α∗ : i → j for each arrow ... |

70 |
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Citation Context ...we obtain a kind of classification of functorially finite extension closed subcategories of a triangulated 2-CY category in terms of functorially finite rigid subcategories, analogous to results from =-=[AR1]-=-. Theorem I.2.5. Let C be a 2-CY triangulated category. Then the functorially finite extension closed subcategories B of C are all obtained as preimages under the functor π: C → C / D14 BUAN, IYAMA, ... |

64 | Zelevinsky A. Cluster Algebras I: Foundations
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Citation Context ... The GLS ϕ-map with applications to the Dynkin case 39 III.3. Cluster Structure of the loop group SL2(L) 43 References 47 Introduction The theory of cluster algebras, initiated by Fomin-Zelevinsky in =-=[FZ2]-=-, and further developed in a series of papers, including [FZ3, BFZ, FZ4], has turned out to have interesting connections with many parts of algebra and other branches of mathematics. One of the links ... |

62 | Loop groups, Oxford Mathematical Monographs - Pressley, Segal - 1986 |

60 | Cluster algebras as Hall algebras of quiver representations
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Citation Context ... the simple module Sil . This map is known to be a strong cluster map [GLS1, GLS2]. 1.5. The Caldero-Chapoton ϕ-map. There is another interesting example of a strong cluster map. Caldero and Chapoton =-=[CC]-=- defined a map ϕ from a cluster category C = CkQ where Q is a quiver with n vertices to Q(u1, . . .,um) which also satisfies the properties of a strong cluster-map. The image of a cluster category und... |

56 | Cluster tilted algebras are Gorenstein and stably - Keller, Reiten |

54 |
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Citation Context ...T, and whenever X ∈ C satisfies Ext 1 (M, X) = 0 for all M ∈ T, then X ∈ T. Such T has been called cluster tilting subcategory in [BMRRT, KR1] if it is in addition functorially finite in the sense of =-=[AS]-=-, which is automatically true when T is finite. Such T has been called maximal 1-orthogonal subcategory in [I1, I2], and Ext-configuration in [BMRRT], without the assumption on functorially finiteness... |

54 | Generalized associahedra via quiver representations
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Citation Context ...o have interesting connections with many parts of algebra and other branches of mathematics. One of the links is with the representation theory of algebras, where a first connection was discovered in =-=[MRZ]-=-. A philosophy has been to model the main ingredients in the definition of a cluster algebra in a categorical/module theoretical setting. The cluster categories associated with All authors were suppor... |

53 | Chain Complexes and Stable Categories - Keller - 1990 |

50 |
Singular Loci of Schubert Varieties
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Citation Context ... related to Chapter II, where completely different methods are used. For general background on representation theory of algebras, we refer to [ARS, ASS, Rin1, H1, AHK], and for Lie theory we refer to =-=[BL]-=-. Our modules are usually left modules and composition of maps fg means first f, then g. The second author would like to thank William Crawley-Boevey and Christof Geiss for answering a question on ref... |

43 | Cluster mutation via quiver representations - Buan, Marsh, et al. |

40 |
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Citation Context ...bras are studied as analogs of Auslander algebras in [GLS1, I1, I2, KR1]. We denote by mod C the category of finitely presented C-modules. If C has pseudokernels, then mod C forms an abelian category =-=[A1]-=-. Proposition I.1.11. Let C be an exact stably 2-CY category. Assume that C has pseudokernels and the global dimension of mod C is finite. Let Γ = End(T) for a cluster tilting object T in C. (a) Γ has... |

40 |
Total positivity in Schubert varieties
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Citation Context ...be the torus. Recall that the Weyl group Norm(H)/H is isomorphic to the Coxeter group W associated with Q. For an element w ∈ W = Norm(H)/H and any lifting ˜w of w in G, define the unipotent cell U w =-=[BZ]-=- to be the intersection U w = U ∩ B− ˜wB−, where B− is the opposite Borel subgroup corresponding to U. Then Uw is independent of the choice of lifting of w. It is a quasi-affine algebraic variety of d... |

40 | Rigid modules over preprojective algebras
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Citation Context ... [BMRRT], and shown to be triangulated in [Ke1] (see also [CCS] for the An case), and the module categories mod Λ for Λ a preprojective algebra of a Dynkin quiver have been used for a similar purpose =-=[GLS1]-=-. This development has both inspired new directions of investigations on the categorical side, as well as interesting feedback on the theory of cluster algebras, see for example [ABS, BMR1, BMR2, BM, ... |

38 | Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories
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Citation Context ...e algebras are triangulated Calabi-Yau categories of dimension 2 (2-CY for short). They both have what is called cluster tilting objects/subcategories [BMRRT, KR1, IY] (called maximal 1-orthogonal in =-=[I1]-=-), which are important since they are the analogs of clusters. The investigation of cluster tilting objects/subcategories in 2-CY categories and related categories is interesting both from the point o... |

38 |
Defining relations of certain infinite-dimensional groups, Asterisque, hors serie
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Citation Context ...for each w in W the stably 2-CY category Sub Λ/Iw as investigated in Chapter II.40 BUAN, IYAMA, REITEN, AND SCOTT On the other hand, associated with the underlying graph of Q, is a Kac-Moody group G =-=[KP1]-=- with a maximal unipotent subgroup U. Let H be the torus. Recall that the Weyl group Norm(H)/H is isomorphic to the Coxeter group W associated with Q. For an element w ∈ W = Norm(H)/H and any lifting ... |

36 | Mutation in triangulated categories and rigid Cohen-Macaulay modules - Iyama, Yoshino |

35 | Semicanonical bases and preprojective algebras - Geiß, Leclerc, et al. |

29 | Dualizing complexes, Morita equivalence and the derived Picard group of a ring - Yekutieli - 1999 |

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21 | Clusters and seeds in acyclic cluster algebras - Buan, Marsh, et al. |

21 | On the exceptional fibres of Kleinian singularities
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Citation Context .... Note that Λ is uniquely determined up to isomorphism by the underlying graph of Q. When Λ is the preprojective algebra of a Dynkin quiver over k, the stable category modΛ is 2-CY (see [AR2, 3.1,1.2]=-=[CB]-=-[Ke2, 8.5]). When Λ is the completion of the preprojective algebra of a finite connected quiver without loops which is not Dynkin, the bounded derived category Db (f. l. Λ) of the category f. l. Λ of ... |

21 | Appendix: Some remarks concerning tilting modules and tilted algebras. Oringin. Relevance. Future.’, Handbook of tilting theory - Ringel |

20 | Cluster algebras IV - Fomin, Zelevinsky - 2007 |

20 | Acyclic cluster algebras via Ringel-Hall algebras, preprint
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Citation Context ... Actually by the work of Caldero and Keller [CK1, CK2] this gave a categorical interpretation of the cluster algebra structure (including multiplication) in case of acyclic cluster algebras (see also =-=[Hu]-=-). Moreover, we have the following. Proposition 1.3. Any subcluster algebra of an acyclic cluster algebra is acyclic. Proof. This follows from the fact that for any quiver Q which is mutation equivale... |

20 |
den Bergh: Noetherian hereditary abelian categories satisfying Serre duality
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Citation Context ...C, where D = Homk( , k). A Hom-finite triangulated category is 2-CY if and only if it has almost split triangles with translation τ and τ : C → C is a functor isomorphic to the shift functor [1] (see =-=[RV]-=-). We have the following examples of 2-CY categories. (1) The cluster category CH associated with a finite dimensional hereditary k-algebra H is by definition the orbit category D b (H)/τ −1 [1], wher... |

19 |
Derived categories and their uses, Handbook of algebra
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Citation Context ...rbit category D b (H)/τ −1 [1], where D b (H) is the bounded derived category of finitely generated H-modules, and τ is the AR-translation of D b (H) [BMRRT]. It is a Hom-finite triangulated category =-=[Ke2]-=-, and it is 2-CY since τ = [1]. (2) The stable category of maximal Cohen-Macaulay modules CM(R) over a 3-dimensional complete local commutative noetherian Gorenstein isolated singularity R containing ... |

18 | Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras
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(Show Context)
Citation Context ...of finite length modules over the completion of the preprojective algebra of a non-Dynkin quiver with no loops. Our main tool is to extend the tilting theory developed for Λ in the noetherian case in =-=[IR]-=-. This turns out to give a large class of 2-CY categories associated with elements in the corresponding Coxeter groups. For these categories we construct cluster tilting objects associated with each r... |

17 |
Todorov G. Tilting theory and cluster combinatorics
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Citation Context ...l authors were supported by a STORFORSK-grant 167130 from the Norwegian Research Council. 12 BUAN, IYAMA, REITEN, AND SCOTT finite dimensional hereditary algebras were introduced for this purpose in =-=[BMRRT]-=-, and shown to be triangulated in [Ke1] (see also [CCS] for the An case), and the module categories mod Λ for Λ a preprojective algebra of a Dynkin quiver have been used for a similar purpose [GLS1]. ... |

16 | Auslander algebras and initial seeds for cluster algebras
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(Show Context)
Citation Context ... → τX when X is nonprojective in P, we get the quiver of the cluster tilting object given by the above reduced expression. That this quiver is the quiver of a cluster tilting object was also shown in =-=[GLS3]-=- for P being the AR quiver of a Dynkin quiver, and in [GLS6] in the general case. Lemma II.4.4. The word associated with P obtained in this way is reduced. Proof. The word satisfies the following cond... |

15 |
Fomin S., Zelevinsky A. Cluster Algebras III: Upper bounds and double Bruhat cells
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(Show Context)
Citation Context ...on both i and j occur an infinite number of times if the expression is infinite, each connected set of i’s or j’s is finite.) Note that in the Dynkin case essentially the same quiver has been used in =-=[BFZ]-=-. For a finite reduced expression si1 · · · sik we denote by Q(i1, · · ·ik) the quiver obtained from Q(i1, · · · ik) by removing the last i for each each i in Q0. We denote by Λ = T0 � T1 � · · · the ... |

15 | Cluster algebras IV: Coefficients - Fomin, Zelevinsky |

13 |
The preprojective algebra of a tame hereditary Artin algebra
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Citation Context ... the completion of the associated preprojective algebra. In [IR] the tilting Λ-modules of projective dimension at most one were investigated in the noetherian case, that is, when Q is extended Dynkin =-=[BGL]-=- (and also the generalized ones having loops). In this section we generalize some of these results to the non-noetherian case, concentrating on the aspects that will be needed for our construction of ... |

12 |
Notes on the no loop conjecture
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Citation Context ...position I.1.7. Hence we have pdΓ(T, Fi) ≤ 1 and consequently pdΓX ≤ m + 1. It follows that Γ has finite global dimension. (b) By (a), Γ is a finite dimensional algebra of finite global dimension. By =-=[Le, Ig]-=-, the quiver of Γ has no loops. We shall show the second assertion. Our proof is based on [GLS1, 6.4]. We start with showing that Ext 2 Γ (S, S) = 0 for any simple Γ-module S, assumed to be the top of... |

11 | On a partial order of tilting modules - Happel, Unger - 2005 |