  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 Cat_C such that the indecomposable
objects of Cat_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 Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin
and Zelevinsky to any cluster.
