Job Rock | Searcharxiv

arXiv2020

Continuous Quivers of Type A (IV) Continuous Mutation and Geometric Models of $\mathbf E$-clusters

This if the final paper in the seriesContinuous Quivers of Type $A$. In this part, we generalize existing geometric models of type $A$ cluster structures to the new $\mathbf E$-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use thes geometric models and isomorphisms of cluster theories to begin classifying continuous type $A$ cluster structures. We also introduce a continuous generalization of mutation. This encompasses mutation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure.

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arXiv2019

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 $A_n$. We provide a similar description for distinguished triangles in the bounded derived category whose first and third terms are indecomposable. 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. (Added Section 5 to version 2.)

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