## Tilting theory and cluster combinatorics

Venue: | 572–618. EQUIVALENCE AND GRADED DERIVED EQUIVALENCE 43 |

Citations: | 55 - 4 self |

### BibTeX

@INPROCEEDINGS{Buan_tiltingtheory,

author = {Aslak Bakke Buan and Robert Marsh and Markus Reineke and Idun Reiten},

title = {Tilting theory and cluster combinatorics},

booktitle = {572–618. EQUIVALENCE AND GRADED DERIVED EQUIVALENCE 43},

year = {}

}

### OpenURL

### Abstract

of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

### Citations

226 |
Triangulated categories in the representation theory of finite-dimensional algebras
- Happel
- 1988
(Show Context)
Citation Context ...τ is defined on ZQ, just taking (n, i) to (n −1, i). In this way ZQ is a stable translation quiver. We denote the corresponding mesh category by k(ZQ). We have the following: Proposition 1.1. (Happel =-=[Hap88]-=-,5.6) Let Q be any quiver of Dynkin type. Then the mesh category k(ZQ) is equivalent to ind D. It follows that (as a stable translation quiver), ZQ depends only on the underlying Dynkin diagram ∆, and... |

103 | Cluster Algebras II: Finite type classification
- Fomin, Zelevinsky
(Show Context)
Citation Context ... other words, we suppose that for all x, y ∈ x, bxy = 0 if and only if byx = 0, that bxy > 0 if and only if byx < 0, and that bxx = 0. Such a pair (x, B) is called a seed. Fomin and Zelevinsky [FZ1], =-=[FZ2]-=- have defined a certain subring A(x, B) of F associated to the seed (x, B), known as a cluster algebra. Given such a seed, and an element z ∈ x, define a new element z ′ ∈ F via the binary exchange re... |

83 | Quivers with relations arising from clusters (An case). Transaction od AMS
- Caldero, Chapoton, et al.
(Show Context)
Citation Context ...o use the new perspective on tilting theory afforded by cluster algebras and the cluster category to conjecture a generalisation of APR-tilting (see [APR79]). P. Caldero, F. Chapoton and R. Schiffler =-=[CCS04a]-=- have recently associated a category to the cluster algebra of type An, giving a definition via the combinatorics of the corresponding cluster algebra. They have shown that this category is equivalent... |

67 | Cluster Algebras III: Upper bounds and double Bruhat cells, preprint ArXiv:math.RT/0305434
- Berenstein, Fomin, et al.
(Show Context)
Citation Context ...oping algebra of a semisimple Lie algebra over C, and that it should model the (classical and quantised) coordinate rings of varieties associated to algebraic groups (now shown in several cases — see =-=[BFZ02]-=-, [Sco03].), with particular relevance to total positivity properties; there have already been many applications to other areas as well [CFZ02, BFZ02, FZ02c, FZ03, GSV02, MRZ03, P03]. The definition i... |

64 | algebras I: Foundations
- Fomin, Zelevinsky, et al.
(Show Context)
Citation Context ...ms are to show how this category can be used to study the tilting theory of H (and related algebras) and to show that it can be used as a model for the combinatorics of an associated Fomin–Zelevinsky =-=[FZ02a]-=- cluster algebra. Hom-configurations are certain collections of non-isomorphic indecomposable objects in D, and were considered in [Rie80a] in connection with the classification of self-injective alge... |

60 | Cluster algebras as Hall algebras of quiver representations
- Caldero, Chapoton
(Show Context)
Citation Context ...re 17. The quivers of the algebras Γ and Γ ′ Conjecture 9.4 was solved in [BMR04a]. Cluster categories have also proved useful in the Hall algebra approach to cluster algebras of Caldero and Chapoton =-=[CC]-=-. Acknowledgements We would like to thank A. Zelevinsky for many helpful discussions. R. Marsh would like to thank Northeastern University, Boston and NTNU, Trondheim, for their kind hospitality. Refe... |

53 |
Large modules over Artin algebras
- Auslander
- 1976
(Show Context)
Citation Context ...ting object always has exactly two complements. We show further that, given one complement M to an almost complete basic tilting object T, the other can be constructed using approximation theory from =-=[AS80]-=-. Indeed, we show that there is a triangle M ∗ → B → M → M ∗ [1] in C, where B → M is a minimal right addT-approximation of M in C and M ∗ is the other complement to T. Dually, there is a triangle M →... |

49 |
The module theoretical approach to quasi-hereditary algebras. Representations of algebras and related topics (Kyoto
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- 1990
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Citation Context ...∆ be a simply-laced Dynkin diagram, and denote by Π(∆) the preprojective algebra of type ∆. Then it is known that Π(∆) has finite representation type if and only if ∆ is of type A1, A2, A3 or A4 (see =-=[DR92]-=-). In type A1, the stable module category of Π(∆) has only one indecomposable (simple) object. In types A2, A3 and A4, the stable module category of Π(∆) can be seen to coincide with the cluster categ... |

48 | The Laurent phenomenon - Fomin, Zelevinsky |

45 | Cluster mutation via quiver representations
- Buan, Marsh, et al.
(Show Context)
Citation Context ... Proof We give the following update on the status of the conjectures in Section 9. Conjecture 9.2 has been solved in [BMR05], and partially in [CCS04b]. A local version of Conjecture 9.3 is proved in =-=[BMR04b]-=-. Note that settling the still unsolved Conjecture 9.1 would give a proof of the conjecture as it is formulated in this paper. In the case of simply-laced Dynkin quivers it was solved independently in... |

45 | Polytopal realizations of generalized associahedra
- Chapoton, Fomin, et al.
(Show Context)
Citation Context ...inatorics of A could be obtained from the category of decorated representations of a quiver Q with underlying graph ∆. In particular, this allowed the generalised associahedra (Stasheff polytopes) of =-=[CFZ02]-=- to be constructed directly from the representation theory of Q, and gave, for the first time, a uniform formula for the number of basic tilting modules over kQ in terms of the degrees of the correspo... |

41 | Y-systems and generalized associahedra
- Fomin, Zelevinsky
(Show Context)
Citation Context ... Q be any quiver of type ∆. Then ∆(C) is isomorphic to the abstract simplicial complex ∆Q of [MRZ03],3.7,4.11. Corollary 4.4, together with [MRZ03],4.11,4.12, show that the simplicial complex ∆(Φ) of =-=[FZ03]-=-,p6, can be obtained in a natural way from the category C associated to Φ. Theorem 4.5. Let Q = Qalt be an alternating quiver of type ∆. Then the map α ↦→ MQalt (α) between Φ≥−1 and ind C induces a bi... |

37 | Cluster algebras and Poisson geometry - Gekhtman, Shapiro, et al. - 2003 |

29 | Covering spaces in representation-theory - Bongartz, Gabriel - 1982 |

27 |
Coxeter functors without diagrams
- Auslander, Platzeck, et al.
(Show Context)
Citation Context ...ter algebra directly from C. Finally, we are able to use the new perspective on tilting theory afforded by cluster algebras and the cluster category to conjecture a generalisation of APR-tilting (see =-=[APR79]-=-). P. Caldero, F. Chapoton and R. Schiffler [CCS04a] have recently associated a category to the cluster algebra of type An, giving a definition via the combinatorics of the corresponding cluster algeb... |

27 | with relations and cluster tilted algebras
- Caldero, Chapoton, et al.
- 2004
(Show Context)
Citation Context ... . . Figure 15. The AR-quivers of Γ and Γ ′ Note Added in Proof We give the following update on the status of the conjectures in Section 9. Conjecture 9.2 has been solved in [BMR05], and partially in =-=[CCS04b]-=-. A local version of Conjecture 9.3 is proved in [BMR04b]. Note that settling the still unsolved Conjecture 9.1 would give a proof of the conjecture as it is formulated in this paper. In the case of s... |

23 |
Tilted algebras, in: Representations of Algebras
- Bongartz
(Show Context)
Citation Context ...g module if (a) Ext 1 H(T, T) = 0, that is T is exceptional, and there is an exact sequence 0 → H → T0 → T1 → 0 with T0 and T1 in addT (see [HR82]). There are some useful equivalent characterisations =-=[Bon80]-=-: (b) T is exceptional and has n non-isomorphic indecomposable direct summands10 BUAN, MARSH, REINEKE, REITEN, AND TODOROV (possibly with multiplicities), where n is the number of non-isomorphic simp... |

23 | Cluster-tilted algebras of finite representation type
- Buan, Marsh, et al.
(Show Context)
Citation Context ...(b) The AR-quiver of Γ ′ . . . Figure 15. The AR-quivers of Γ and Γ ′ Note Added in Proof We give the following update on the status of the conjectures in Section 9. Conjecture 9.2 has been solved in =-=[BMR05]-=-, and partially in [CCS04b]. A local version of Conjecture 9.3 is proved in [BMR04b]. Note that settling the still unsolved Conjecture 9.1 would give a proof of the conjecture as it is formulated in t... |

7 |
Critical simply connected algebras
- Bongartz
- 1984
(Show Context)
Citation Context ...g module if (a) Ext 1 H(T, T) = 0, that is T is exceptional, and there is an exact sequence 0 → H → T0 → T1 → 0 with T0 and T1 in addT (see [HR82]). There are some useful equivalent characterisations =-=[Bon80]-=-: (b) T is exceptional and has n non-isomorphic indecomposable direct summands10 BUAN, MARSH, REINEKE, REITEN, AND TODOROV (possibly with multiplicities), where n is the number of non-isomorphic simp... |

6 |
Selfinjective and simply connected algebras
- Bretscher, Läser, et al.
- 1981
(Show Context)
Citation Context ...Hom, and call the resulting collections Ext-configurations. We show that they exhibit a behaviour similar to that of Hom-configurations. In particular, they are invariant under the functor F (compare =-=[BLR81]-=-, where it is shown that Hom-configurations exhibit a similar kind of invariance in the Dynkin case). As a consequence we can show that they are in 1–1 correspondence with what we call basic cluster-t... |

5 |
H.: Tilting and cotilting for quivers of type Ãn
- Buan, Krause
(Show Context)
Citation Context ...3, can be regarded as the skeleton of a simplicial complex with simplices the faithful basic exceptional modules. This simplicial complex is in fact the Stasheff associahedron of dimension n − 1, see =-=[BK]-=-. 5. Complements of almost complete basic tilting objects Let H be a finite dimensional hereditary algebra with n non-isomorphic simple modules. An H-module T is said to be an almost complete basic ti... |

5 |
Complements to partial tilting modules
- Coelho, Happel, et al.
- 1994
(Show Context)
Citation Context ...s. Hence there is a more regular behaviour in C. Certain classes of hereditary categories exhibit a similar behaviour [HU03]. The analogous question has been investigated for arbitrary artin algebras =-=[CHU94]-=-. We say that a basic exceptional object T in C is an almost complete basic tilting object if there is an indecomposable object M in C such that T ∐ M is a basic tilting object. Then we have the follo... |

2 |
Generalised APR-tilting
- Buan, Marsh, et al.
- 2004
(Show Context)
Citation Context ...er of the AR-quivers of Γ and Γ ′ . . ◦ . . . ◦ . . • . • . • • . . . . • . . • (a) The quiver of Γ (b) The quiver of Γ ′ Figure 17. The quivers of the algebras Γ and Γ ′ Conjecture 9.4 was solved in =-=[BMR04a]-=-. Cluster categories have also proved useful in the Hall algebra approach to cluster algebras of Caldero and Chapoton [CC]. Acknowledgements We would like to thank A. Zelevinsky for many helpful discu... |

1 |
Tilting sets on cylinders
- Happel
- 1985
(Show Context)
Citation Context ...such that HomD(G i X, Y ) is not zero, there are also only finitely many values of i such that Ext 1 D(G i X, Y ) is not zero, for X, Y in D. We remark that the quotient D b (H)/[2] was considered in =-=[Hap85]-=-; however, this quotient has quite different properties and is not closely linked with cluster algebras. While several properties hold for arbitrary functors G satisfying (g1) and (g2), we shall mainl... |