Quivers with relations and cluster tilted algebras Philippe Caldero Frederic Chapoton Ralf Schi er

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Niveau: Supérieur, Doctorat, Bac+8
ar X iv :m at h. RT /0 41 12 38 v 1 1 0 N ov 2 00 4 Quivers with relations and cluster tilted algebras Philippe Caldero, Frederic Chapoton, Ralf Schi?er Abstract Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations. 0 Introduction Cluster algebras were introduced in the work of S. Fomin and A. Zelevinsky, [FZ02, FZ03a, FZ03b]. This theory appeared in the context of dual canonical basis and more particularly in the study of the Berenstein-Zelevinsky conjec- ture. Cluster algebras are now connected with many topics: double Bruhat cells, Poisson varieties, total positivity, Teichmuller spaces. The main results on cluster algebras are on the one hand the classification of finite cluster alge- bras by root systems and on the other hand the realization of algebras of regular functions on double Bruhat cells in terms of cluster algebras.

  • let

  • p1 ? p2

  • roots ?

  • cluster algebras

  • all corresponding

  • then fg

  • auslander-reiten translate

  • cluster category


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Quivers with relations and cluster tilted algebras

0

Philippe Caldero, Fr´d´ric Chapoton, Ralf Schiffler

Abstract
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in
connection with dual canonical bases.To a cluster algebra of simply
laced Dynkin type one can associate the cluster category.Any cluster of
the cluster algebra corresponds to a tilting object in the cluster category.
The cluster tilted algebra is the algebra of endomorphisms of that tilting
object. Viewingthe cluster tilted algebra as a path algebra of a quiver
with relations, we prove in this paper that the quiver of the cluster tilted
algebra is equal to the cluster diagram.We study also the relations.
As an application of these results, we answer several conjectures on the
connection between cluster algebras and quiver representations.

Introduction

Cluster algebras were introduced in the work of S. Fomin and A. Zelevinsky,
[FZ02, FZ03a, FZ03b].This theory appeared in the context of dual canonical
basis and more particularly in the study of the Berenstein-Zelevinsky
conjecture. Clusteralgebras are now connected with many topics:double Bruhat
cells, Poisson varieties, total positivity, Teichm¨ller spaces.The main results
on cluster algebras are on the one hand the classification of finite cluster
algebras by root systems and on the other hand the realization of algebras of regular
functions on double Bruhat cells in terms of cluster algebras.
Recently, many new results have been established relating cluster algebras
of simply laced finite type to quiver representations.It has been shown in
+
arXiv:math.RT/0411238 v110 Nov 2004
[CCS] (typeAa] (types) and [BMRA, D, E) that the set of cluster variables
is in bijection with the set of indecomposable objects in the so called cluster
−1
categoryC, which is the quotient categoryD/τ[1] of the bounded derived
categoryDof quiver representations by the inverse Auslander-Reiten translate
−1
τcomposed with the shift [1].
For typeAthe authors associated in [CCS] a quiver with relations to each
cluster in such a way that the indecomposable representations of that quiver
with relations are in bijection with all cluster variables outside the cluster.A
result of this approach was the description of the denominator of the Laurent
polynomial expansion of any cluster variable in the variables of any cluster.In
this paper, we generalize this result to the typesDandE(Theorem 4.4).
+
Buan, Marsh, Reineke, Reiten and Todorov [BMRa] used tilting theory to
relate the cluster algebra to the cluster category; each cluster corresponds to a

1