Cluster-tilted algebras without clusters

We investigate the fibres of the surjective map from the class of tilted algebras to the class of cluster-tilted algebras given by forming the relation extension of tilted algebras. We introduce the notions of reflections and coreflections of tilted algebras and use them to construct all the tilted algebras in the fibre of a given cluster-tilted algebra in the case where the cluster-tilted algebra admits a local slice of tree type. Moreover, we give an explicit algorithm for the construction of the transjective component of the Auslander–Reiten quiver of the cluster-tilted algebra.