On the Auslander-Reiten quiver of the representations of an infinite quiver

TL;DR

This paper investigates the structure of the Auslander-Reiten quiver for representations of infinite quivers, introducing a subcategory to better understand its components and proposing a conjecture for the remaining parts.

Contribution

It defines a new subcategory C(Q) within rep(Q) and provides a complete description of its Auslander-Reiten quiver components, linking them to those of rep(Q).

Findings

Connected components of C(Q) are also components of rep(Q)'s Auslander-Reiten quiver.

Complete description of the Auslander-Reiten quiver of C(Q).

Conjecture on components of rep(Q) not arising from C(Q).

Abstract

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of its Auslander-Reiten quiver. To achieve this goal, we define a category C(Q) which is a full, abelian and Hom-finite subcategory of rep(Q) containing all the almost split sequences of rep(Q). We give a complete description of the Auslander-Reiten quiver of C(Q) by describing its connected components. Finally, we prove that these connected components are also connected components of the Auslander-Reiten quiver of rep(Q). We end the paper by giving a conjecture describing the Auslander-Reiten components of rep(Q) that cannot be obtained as Auslander-Reiten components of C(Q).