Auslander–Reiten quiver

Summary

The Auslander–Reiten quiver of a finite-dimensional algebra is a quiver that maps out its module category. Its vertices are the isomorphism classes of indecomposable 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.