Continuous Quivers of Type A (II) The Auslander-Reiten Space
This work is the sequel to Continuous Quivers of Type A (I). In this paper we define the Auslander-Reiten space of a continuous type $A$ quiver, which generalizes the Auslander-Reiten quiver of type $A_n$ quivers. We prove that extensions, kernels, and cokernels of representations of type $A_{\mathbb R}$ can be described by lines and rectangles in a way analogous to representations of type $A_n$. Furthermore, we provide a complete classification of Auslander-Reiten sequences in the category of finitely generated representations of $A_{\mathbb R}$. This is part of a longer work; the other papers in this series are with Kiyoshi Igusa and Gordana Todorov. The goal of this series is to generalize cluster categories, clusters, and mutation for type $A_n$ quivers to continuous versions for type $A_{\mathbb R}$ quivers.