Companion Bases for Cluster-Tilted Algebras

Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of ℤ-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type. We give a complete description of the relationship between different companion bases for the same quiver. We establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of type A may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel’s Theorem. In addition, we show how to mutate a companion basis for a quiver to produce a companion basis for a mutation of that quiver.