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arXiv · 1201.4833

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

Charles Paquette
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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 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).

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Charles Paquette. 2012-09-06. On the Auslander-Reiten quiver of the representations of an infinite quiver. https://arxiv.org/abs/1201.4833

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