Quivers with relations arising from clusters (An 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 An. We associate to each cluster C of U an abelian category C-C such that the indecomposable objects of 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 C-C. Then, we generalize the "denominator theorem" of Fomin and Zelevinsky to any cluster.