, Volume 16, Issue 6, pp 1685-1715
Date: 25 Sep 2012

On the Auslander-Reiten Quiver of the Representations of an Infinite Quiver

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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 \(\overline{\rm {rep}}(Q)\) 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 \(\overline{\rm {rep}}(Q)\) 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 \(\overline{\rm {rep}}(Q)\) .

Presented by: Raymundo Bautista.