## Cluster-tilted algebras of finite representation type (2006)

Venue: | J. Algebra |

Citations: | 23 - 10 self |

### BibTeX

@ARTICLE{Buan06cluster-tiltedalgebras,

author = {Aslak Bakke Buan and Robert J. Marsh and Idun Reiten},

title = {Cluster-tilted algebras of finite representation type},

journal = {J. Algebra},

year = {2006},

pages = {412--431}

}

### OpenURL

### Abstract

Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

### Citations

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Citation Context ...t is well known that this holds true if and only if the underlying graph of Q is a Dynkin graph. Let D = D b (mod H) be the bounded derived category. It is equipped with a shiftfunctor [1]: D → D. By =-=[H]-=- D has AR-triangles with corresponding translationfunctor τ : D → D, with a quasi-inverse τ −1 . Let F = τ −1 [1] be the composition. It is an auto-equivalence on D. The cluster category is the orbit ... |

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Citation Context ...nfinite type, but (in terms of quivers) every full proper subquiver gives a cluster algebra of finite type. For general notions in the representation theory of finite dimensional algebras we refer to =-=[R]-=- and [ARS]. Some of the results in this paper have been presented at conferences in Uppsala (June 2004) and Mexico (August 2004). We would like to thank Claus Michael Ringel and Øyvind Solberg for som... |

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Citation Context ... algebras H over a field K (or more generally with Ext-finite hereditary abelian K-categories with tilting object) were introduced in [BMRRT]. An alternative description in Dynkin type A was given in =-=[CCS1]-=-. The motivation came from the theory of cluster algebras, introduced by Fomin and Zelevinsky in [FZ], and the connection between cluster algebras and quiver representations [MRZ]. A tilting theory in... |

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Citation Context ...type, but (in terms of quivers) every full proper subquiver gives a cluster algebra of finite type. For general notions in the representation theory of finite dimensional algebras we refer to [R] and =-=[ARS]-=-. Some of the results in this paper have been presented at conferences in Uppsala (June 2004) and Mexico (August 2004). We would like to thank Claus Michael Ringel and Øyvind Solberg for some helpful ... |

64 | algebras I: Foundations
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Citation Context ...ing object) were introduced in [BMRRT]. An alternative description in Dynkin type A was given in [CCS1]. The motivation came from the theory of cluster algebras, introduced by Fomin and Zelevinsky in =-=[FZ]-=-, and the connection between cluster algebras and quiver representations [MRZ]. A tilting theory in cluster categories was developed in [BMRRT], and the associated endomorphism algebras EndC(T) op for... |

60 | Cluster algebras as Hall algebras of quiver representations - Caldero, Chapoton |

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Citation Context ...following useful result of Assem-Brüstle-Schiffler and ReitenTodorov gives a major simplification of our original proof in this section. We include a proof for the convenience of the reader. See also =-=[KR]-=- for related considerations. Lemma 3.1. Let Γ be a cluster-tilted algebra over k and let S, S ′ be two simple Γ-modules. Then dimK Ext 1 (S ′ , S) ≥ dimK Ext 2 (S, S ′ ). Proof. We have Γ = EndC(T) op... |

45 | Cluster mutation via quiver representations
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(Show Context)
Citation Context ...t the clustertilted algebras of finite type are (up to Morita equivalence) uniquely determined by their quivers. It would be interesting to know to which extent this is true beyond finite type. Using =-=[BMR2]-=-, it is possible to obtain information on relations of cluster-tilted algebras in general, but we shall not deal with this here. Our main theorem on describing the relations of cluster-tilted algebras... |

21 |
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Citation Context ...llows from Proposition 4.1. That there are no other minimal commutativity relations follows from Lemma 4.12. □ Up to isomorphism of cluster-tilted algebras, we can by the multiplicative basis theorem =-=[BGRS]-=- assume that all λ’s in the above statement are −1. We have the following nice consequence, using Lemma 4.14. Recall that when there is an arrow i → j, we call a path from j to i shortest if it contai... |

21 |
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Citation Context ...type A was given in [CCS1]. The motivation came from the theory of cluster algebras, introduced by Fomin and Zelevinsky in [FZ], and the connection between cluster algebras and quiver representations =-=[MRZ]-=-. A tilting theory in cluster categories was developed in [BMRRT], and the associated endomorphism algebras EndC(T) op for a (cluster-)tilting object T in C, were investigated in [BMR1, BMR2]. The clu... |

19 |
Todorov G. Tilting theory and cluster combinatorics, preprint math.RT/0402054
- Buan, Marsh, et al.
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Citation Context ...ing. Lemma 1.7. Let Γ be a cluster-tilted algebra of finite representation type and let Pi, Pj be indecomposable projective Γ-modules. Then dimk Hom(Pi, Pj) ≤ 1. Proof. This follows from Lemma 8.2 in =-=[BMRRT]-=-, using that any tilting object in a cluster category C is the image of a tilting module for some algebra H ′ with ′ = C. □ CH 2. Double path avoiding quivers In this section we show some necessary co... |

6 |
Andrei Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices
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Citation Context ...rnating with 2n −2 vertices. If n = 2, then the reduced quiver is so by induction Q is not double path avoiding. • We note that this is shown in the finite type case in the cluster algebra context in =-=[BGZ]-=-. We will later see that if Q is an oriented cycle of length ≥ 3, then Q is double path avoiding. The motivation for the notion of double path avoiding quivers is the following. Proposition 2.2. Let K... |

2 |
with relations and cluster-tilted algebras, preprint math.RT/0411238
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(Show Context)
Citation Context ...ith this here. Our main theorem on describing the relations of cluster-tilted algebras of finite type answers Conjecture 9.2 in [BMRRT]. We remark that these relations appeared in [CCS1], and that in =-=[CCS2]-=- it is shown, independently, that they hold in a cluster-tilted algebra associated to a (simply-laced) Dynkin quiver. In type A it is shown also that they are defining relations. As an application of ... |

2 |
Recognizing cluster algebras of finite type’, preprint 2004, arXiv:math. CO/0406545. I. Assem Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada ibrahim.assem@usherbrooke.ca T. Brüstle Département de Mathématiques
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Citation Context ...ng results on the quivers of cluster-tilted algebras gives information on matrices associated with cluster algebras. Note that results on quivers of cluster algebras of finite type are given by Seven =-=[S]-=-. In fact Seven gives a list of all cluster algebras which are of minimal infinite type; i.e. they are of infinite type, but (in terms of quivers) every full proper subquiver gives a cluster algebra o... |

1 |
Cluster-tilted algebras, preprint math.RT/0402075
- Buan, Marsh, et al.
- 2004
(Show Context)
Citation Context ...1 C(T ∐ X, T ∐ X) = 0, then X is a direct summand in a direct sum of copies of T. The endomorphism-ring EndC(T) op of a tilting object T is called a cluster-tilted algebra. The following was shown in =-=[BMR1]-=-. Proposition 1.1. Let Γ = EndC(T) op be a cluster-tilted algebra with C = CH the cluster category for some hereditary algebra H, and T a tilting object in C. Then Γ is of finite representation type i... |

1 |
Algebras and quadratic forms
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(Show Context)
Citation Context ...hat ρ is a minimal zerorelation. Then there is an arrow i → j, and the induced oriented cycle is full and pure. Proof. Since there is a minimal zero-relation ρ given by a path from j to i, we know by =-=[Bo]-=- that Ext 2 (Sj, Si) ̸= 0, and hence Ext 1 (Si, Sj) ̸= 0 by Lemma 3.1. Hence there is an arrow from i to j. We now want to show that the cycle induced by the minimal zero-relation ρ from j to i and th... |