AuslanderReiten

Auslander-Reiten theory is a framework in the representation theory of algebras that studies the morphisms between indecomposable modules, organized by almost split sequences and translated by the Auslander-Reiten translation. Developed in the 1970s by Maurice Auslander and Idun Reiten, the theory provides tools to understand how indecomposable modules fit together and how they can appear as extensions of one another.

A central concept in the theory is the almost split sequence. For an indecomposable non-projective module M over a finite-dimensional algebra A, there exists a short exact sequence 0 -> τM -> E -> M -> 0, where τM is the Auslander-Reiten translate of M. This sequence is characterized by universal properties: any morphism into M that is not split factors through the right-hand map, and any morphism from τM to another module that is not split factors through the left-hand map. The translate τ provides a way to pair non-projective modules with their “almost predecessors” in the category.

The Auslander-Reiten quiver is a directed graph whose vertices correspond to isomorphism classes of indecomposable modules (often excluding projective or injective ones in the stable version), and whose arrows represent irreducible morphisms. The AR-translate τ acts as a translation on the quiver, linking modules that appear in almost split sequences. The quiver encodes the global shape of the category and changes in a controlled manner under various functors.

Applications of Auslander-Reiten theory include classifying algebras by representation type (finite, tame, or wild), describing the module category for Dynkin and extended Dynkin diagrams, and informing structural results in triangulated categories through Auslander-Reiten triangles.