On indecomposable modules over cluster-tilted algebras of type A

Gabriel's Theorem describes the dimension vectors of the finitely generated indecomposable modules over the path algebra of a simply-laced Dynkin quiver. It shows that they can be obtained from the expressions for the positive roots of the corresponding root system in terms of the simple roots. Here, we present a method for finding the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A.;It is known that the quiver of a cluster-tilted algebra of Dynkin type A is given by an exchange matrix of the corresponding cluster algebra. We define a companion basis for such a quiver to be a Z -basis of roots of the integral root lattice of the corresponding root system whose associated matrix of inner products is a positive quasi-Cartan companion of the corresponding exchange matrix.;Our main result establishes that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A arise from expressions for the positive roots of the corresponding root system in terms of a companion basis (for the quiver of that algebra). This can be regarded as a generalisation of part of Gabriel's Theorem in the Dynkin type A case. The proof uses the fact that the quivers of the cluster-tilted algebras of Dynkin type A have a particularly nice description in terms of triangulation of regular polygons.