Quivers with relations arising from clusters $(A_n$ case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let $U$ be a cluster algebra of type $A_n$. We associate to each cluster $C$ of $U$ an abelian category $\mathcal{C}_C$ such that the indecomposable objects of $\mathcal{C}_C$ are in natural correspondence with the cluster variables of $U$ which are not in $C$. We give an algebraic realization and a geometric realization of $\mathcal{C}_C$. Then, we generalize the ``denominator theorem'' of Fomin and Zelevinsky to any cluster.