---
tag: 125Z033VCQ
kind: concept
name: Auslander–Reiten quiver
msc: [16G70]
specializes: [RE9YFFS3TS]
sources:
  - url: https://en.wikipedia.org/wiki/Auslander%E2%80%93Reiten_theory
---

## Summary

The Auslander–Reiten quiver of a finite-dimensional algebra is a [[quiver|RE9YFFS3TS]] that maps out its module category.
Its vertices are the isomorphism classes of [[indecomposable|E2P29GCD9T]] finitely generated modules, and its arrows record the irreducible morphisms between them.
It also carries the Auslander–Reiten translation $\tau$, which makes it a translation quiver.
For algebras of finite representation type, it describes the whole module category.
Its meshes come from the [[almost split sequences|M5D95FN7G3]].

## Explanations

### L1 human

The Auslander–Reiten quiver is a map of all the indecomposable modules of an algebra.
Each module is a dot, and an arrow goes from one dot to another when there is a basic map between them that cannot be broken down further.
The almost split sequences show up as small diamond or triangle shapes, called meshes.
For the path algebra of $1 \to 2$ the map has three dots, $S_2 \to P_1 \to S_1$.
For an algebra with infinitely many indecomposables, the quiver is infinite, but it still has a regular structure in many cases.