Maximal rigid objects as noncrossing bipartite graphs

TL;DR

This paper classifies maximal rigid objects in a specific orbit category related to Dynkin quivers of type A, using bipartite noncrossing graphs, and describes their endomorphism algebras, linking them to iterated tilted algebras.

Contribution

It introduces a classification of maximal rigid objects via bipartite noncrossing graphs in the orbit category of type A Dynkin quivers and characterizes their endomorphism algebras.

Findings

Maximal rigid objects correspond to bipartite noncrossing graphs with loops.

Endomorphism algebras of these objects are described explicitly.

Some of these algebras are shown to be iterated tilted algebras of type A.

Abstract

Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid objects in the corresponding orbit category C(Q), in terms of bipartite noncrossing graphs (with loops) in a circle. We also 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.