Maximal Rigid Objects as Noncrossing Bipartite Graphs

Abstract

We classify the maximal rigid objects of the Σ2τ-orbit category \({\mathcal{C}}(Q)\) of the bounded derived category for the path algebra associated to a Dynkin quiver Q of type A, where τ denotes the Auslander-Reiten translation and Σ2 denotes the square of the shift functor, in terms of bipartite noncrossing graphs (with loops) in a circle. We describe the endomorphism algebras of the maximal rigid objects, and we prove that a certain class of these algebras are iterated tilted algebras of type A.