## Cluster algebras as Hall algebras of quiver representations

Citations: | 60 - 3 self |

### BibTeX

@MISC{Caldero_clusteralgebras,

author = {Philippe Caldero and Frédéric Chapoton},

title = {Cluster algebras as Hall algebras of quiver representations},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the

### Citations

298 |
Representation theory of Artin algebras
- Auslander, Smalo
- 1995
(Show Context)
Citation Context ...luster variables. We will see in this section properties of the Auslander-Reiten translation τ in this correspondence. First of all, let us recall some basic facts on the Auslander-Reiten theory, see =-=[ARS95]-=-. Let ΓQ be the Auslander-Reiten quiver of modk(Q). Recall that its set of vertices is Indk(Q) and the arrows are given by irreducible morphisms of the category. The AR-quiver ΓC of C is defined in th... |

91 | From triangulated categories to cluster algebras - Caldero, Keller |

83 | Quivers with relations arising from clusters (An case). Transaction od AMS
- Caldero, Chapoton, et al.
(Show Context)
Citation Context ...1 Date: February 1, 2008. 12 PHILIPPE CALDERO AND FRÉDÉRIC CHAPOTON algebras and cluster categories, but the properties of the correspondence are mostly conjectural, see [BMR + , Conjecture 9.3]. In =-=[CCS04]-=-, the authors prove that the denominators of cluster variables can be calculated from C in type A. They give a combinatorial/geometric approach of C in the spirit of Teichmüller spaces, [FG03]. The im... |

81 |
PBW-bases of quantum groups
- Ringel
- 1996
(Show Context)
Citation Context ... vector is a finite sum of the cardinalities of sets of triples (N, M, M/N) where the isomorphism classes are fixed. These cardinalities are known to be polynomials in q, called Hall polynomials, see =-=[Rin90]-=-. 3.3. We prove here that EQ is a subalgebra of F. Actually, we will prove the following: Proposition 3.6. Fix g in Q. For all M, N in modk(Q), we have (23) χ(Grg(M ⊕ N)) = ∑ e+f=g χ(Gre(M))χ(Grf(N)).... |

80 | On triangulated orbit categories
- Keller
(Show Context)
Citation Context ...gory modk(Q) of finite dimensional kQ-modules, such that the set of indecomposable objects of C is in bijection with Φ≥−1. The category C is not abelian in general, but it is a triangulated category, =-=[Kel]-=-. In [BMR+], this category is studied in depth. The authors give a correspondence between cluster variables and indecomposable objects of C. They prove that the compatibility of two cluster variables ... |

70 | Goncharov, “Moduli Spaces of Local Systems and Higher Teichmüller Theory
- Fock, B
(Show Context)
Citation Context ...3]. In [CCS04], the authors prove that the denominators of cluster variables can be calculated from C in type A. They give a combinatorial/geometric approach of C in the spirit of Teichmüller spaces, =-=[FG03]-=-. The implicit question behind all articles [MRZ03, BMR + , CCS04] dealing with cluster algebras and quiver theory is: can one realize the cluster algebra as a “Hall algebra” of the category C in some... |

67 | Cluster Algebras III: Upper bounds and double Bruhat cells, preprint ArXiv:math.RT/0305434
- Berenstein, Fomin, et al.
(Show Context)
Citation Context ...rizable matrix B is the subalgebra of F generated by all x such that (u, B) ∼ (x, B ′ ). Such x are called clusters and the elements of x are called cluster variables. Remark 2.2. More generally, see =-=[BFZ05]-=-, cluster algebras are associated to rectangular matrices in Mn,m(Z). We will not be concerned with such algebras in this article. Note the so-called Laurent phenomenon, see [FZ02]: Theorem 2.3. Let B... |

59 | Y -systems and generalized associahedra
- Fomin, Zelevinsky
(Show Context)
Citation Context ...iezes [CC73]. We would like to note that one of the starting points for the experimental work leading to this article was the combinatorial expressions for some Y -system Laurent polynomials given in =-=[FZ03b]-=- for multiplicity-free roots. Although these are definitely not the same as cluster Laurent polynomials, the combinatorics is quite similar. Acknowledgments: The first author would like to thank Marku... |

55 | Generalized associahedra via quiver representations - Marsh, Reineke, et al. |

54 |
Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory I
- Gabriel
- 1980
(Show Context)
Citation Context ... sequence is almost split in the following sense: each morphism N → M which is not a split epimorphism factors through σ. The AR-quivers of modk(Q) are well known and can be explicitly described, see =-=[Gab80]-=-. The AR-quiver ΓC is a slight extension of ΓQ. Indeed, see [BMR + ], each exact sequence as in (7) gives rise in the triangulated category C to a triangle (8) τM → B → M → SτM, where the first two mo... |

25 | Introduction to Algebraic Geometry and Algebraic Groups, North-Holland - Demazure, Gabriel - 1980 |

18 |
Triangulated polygons and frieze patterns
- Conway, Coxeter
- 1973
(Show Context)
Citation Context ...late in a combinatorical way the cluster variables in terms of any cluster. To conclude, we give a connection between our theorem, the geometric realization of [CCS04], and the Coxeter-Conway friezes =-=[CC73]-=-. We would like to note that one of the starting points for the experimental work leading to this article was the combinatorial expressions for some Y -system Laurent polynomials given in [FZ03b] for ... |

12 | Counting rational points of quiver moduli
- Reineke
(Show Context)
Citation Context ...nly on ∆. The subsections below are devoted to the proof of this Theorem. 3.2. In order to calculate the Euler-Poincaré characteristic of Grassmannians, we will use the following classical Lemma, see =-=[Rei]-=-, which is an application of Grothendieck-Lefschetz’s fixed point formula for the Frobenius in étale cohomology. Lemma 3.5. Let X be a variety defined over some ring of algebraic integers. We denote b... |

10 |
Finite type classification
- algebras
- 2003
(Show Context)
Citation Context ...the same cardinality, which is the rank of the cluster algebra. A cluster algebra is of finite type if the number of cluster variables is finite. The classification of cluster algebras of finite type =-=[FZ03a]-=- is a fundamental step in the theory. The main result is that these cluster algebras come from an antisymmetrized Cartan matrix of finite type, see Section 2.2. Moreover, in this case the cluster vari... |

5 | Tilting theory and cluster combinatorics. arXiv:math.RT/0402054 - Buan, Marsh, et al. |

1 | Grassmannians and Cluster Algebras, 2003. Institut Camille Jordan, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France E-mail address: caldero@igd.univ-lyon1.fr Institut Camille Jordan, Université Claude Bernard Lyon I, 69622 Villeurbanne C - Scott |