[Paper Review] Continuous Quivers of Type A (II) The Auslander-Reiten Space

Job Daisie Rock|arXiv (Cornell University)|Oct 9, 2019

Algebraic structures and combinatorial models26 references4 citations

TL;DR

This paper introduces the Auslander-Reiten space for continuous type $A_{\mathbb{R}}$ quivers, generalizing the classical Auslander-Reiten quiver. It establishes that nontrivial extensions and distinguished triangles in the derived category correspond to rectangles and almost complete rectangles in this space, with slopes $\pm(1,1)$, providing a geometric classification of extensions and triangles via geometric configurations in the AR-space.

ABSTRACT

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.)

Motivation & Objective

• To generalize the Auslander-Reiten quiver to a continuous setting for $A_{\mathbb{R}}$ quivers by defining the Auslander-Reiten space.
• To classify all Auslander-Reiten sequences in the category of finitely generated representations of $A_{\mathbb{R}}$.
• To establish a geometric correspondence between nontrivial extensions of indecomposable representations and rectangles in the AR-space.
• To extend this correspondence to the bounded derived category $\mathcal{D}^b(A_{\mathbb{R}})$, linking distinguished triangles to geometric configurations.
• To provide a complete classification of 16 types of Auslander-Reiten sequences in $\operatorname{rep}_k(A_{\mathbb{R}})$, with conditions on indecomposable representations.

Proposed method

• Define the Auslander-Reiten topology and a generalized metric on isomorphism classes of indecomposable representations using a mapping $\mathop{\boldsymbol{\Gamma}}$ to $\mathbb{R} \times [-\pi/2, \pi/2]$.
• Introduce the concept of 'good' slopes in the AR-space, defined as $\pm(1,1)$, to characterize valid geometric configurations.
• Use the $\mathop{\boldsymbol{\Gamma}}$ mapping to translate representation-theoretic properties into geometric properties in $\mathbb{R}^2$.
• Characterize extensions via rectangles and almost complete rectangles in the AR-space, where corners correspond to indecomposable representations.
• Extend results from the derived category by shifting triangles and using homological properties such as $\operatorname{Hom}(W[-1], V) \cong k$.
• Apply shift functors and use properties of kernels and cokernels in the hereditary category $\operatorname{rep}_k(A_{\mathbb{R}})$ to reconstruct triangles in $\mathcal{D}^b(A_{\mathbb{R}})$.

Experimental results

Research questions

• RQ1 How can the Auslander-Reiten quiver be generalized to continuous quivers of type $A_{\mathbb{R}}$?
• RQ2 What is the complete classification of Auslander-Reiten sequences in $\operatorname{rep}_k(A_{\mathbb{R}})$?
• RQ3 How do nontrivial extensions of indecomposable representations correspond to geometric configurations in the AR-space?
• RQ4 What is the geometric characterization of distinguished triangles in $\mathcal{D}^b(A_{\mathbb{R}})$ with indecomposable first and third terms?
• RQ5 What role do slopes $\pm(1,1)$ play in the correspondence between geometric rectangles and representation-theoretic triangles?

Key findings

• There exists a complete classification of 16 types of Auslander-Reiten sequences in $\operatorname{rep}_k(A_{\mathbb{R}})$, with a unique sequence containing any indecomposable $M_{|a,b|}$ that is not projective, injective, simple, or supported on adjacent sink-source pairs.
• Nontrivial extensions $V \hookrightarrow E \twoheadrightarrow W$ of indecomposable representations exist if and only if there is a rectangle or almost complete rectangle in the AR-space with $V$ as the left-most and $W$ as the right-most corner.
• If the rectangle is complete, $E$ is a direct sum of two indecomposables; if almost complete, $E$ is indecomposable.
• There is a bijection between nontrivial extensions (up to scaling and isomorphism) and rectangles or almost complete rectangles with 'good' slopes in the AR-space of $\operatorname{rep}_k(A_{\mathbb{R}})$.
• In the bounded derived category $\mathcal{D}^b(A_{\mathbb{R}})$, nontrivial distinguished triangles $V \to U \to W \to$ exist if and only if there is a rectangle or almost complete rectangle in the AR-space with $V$ as the left-most and $W$ as the right-most corner.
• For such triangles, if the rectangle is complete, $U$ is a direct sum of two indecomposables; if almost complete, $U$ is indecomposable, and there is a bijection with such geometric configurations in the AR-space of $\mathcal{D}^b(A_{\mathbb{R}})$.