Auslander–Reiten quiver
Generated by GPT-5-miniExpansion Funnel Raw 1 → Dedup 0 → NER 0 → Enqueued 0
| Auslander–Reiten quiver | |
|---|---|
| Name | Auslander–Reiten quiver |
| Field | Representation theory |
| Introduced | 1970s |
| Introduced by | Maurice Auslander; Idun Reiten |
Auslander–Reiten quiver is a combinatorial and homological invariant arising in representation theory that organizes indecomposable modules and irreducible morphisms into a directed graph enriched by translation operators. It connects methods from algebraic geometry, homological algebra, and category theory to classify modules over algebras and to study morphism patterns through distinguished exact sequences and graph-theoretic components. The concept has influenced work by Bernstein, Gelfand, Ponomarev, Gabriel, Ringel, Happel, and Keller in the study of quiver representations, derived categories, and tilting theory.
Definition and basic properties
An Auslander–Reiten quiver encodes the isomorphism classes of indecomposable objects in a Krull–Schmidt category together with arrows corresponding to irreducible morphisms. The construction depends on homological invariants studied by Maurice Auslander and Idun Reiten and is used in the contexts of Artin algebras, path algebras of quivers studied by Pierre Gabriel and Gabriel's theorem, and triangulated categories influenced by Jean-Louis Verdier and Alex Heller. Fundamental properties interplay with the Rad-socle filtration studied by Claus Ringel, the Nakayama functor related to Takeshi Nakayama, and Serre duality techniques pioneered in work by Jean-Pierre Serre and Alexander Grothendieck.
Auslander–Reiten sequences and translations
Auslander–Reiten sequences, also called almost split sequences, are short exact sequences that are non-split and characterize irreducible morphisms between indecomposables; key contributors include Maurice Auslander, Idun Reiten, and Sonia Bromberg in expository developments. The Auslander–Reiten translation τ links indecomposable non-projective objects to their almost split predecessors, drawing on ideas from Sergei Bernstein and Pavel Deligne in derived contexts and from Dieter Happel in triangulated settings. The interplay with almost split triangles appears in work by Bernhard Keller and Dmitri Orlov, relating τ to Serre functors studied by Amnon Neeman and Alexander Beilinson in derived categories associated to coherent sheaves on varieties like those considered by David Mumford.
Construction for Artin algebras and quiver representations
For Artin algebras and finite quivers without oriented cycles, the Auslander–Reiten quiver is built from indecomposable modules classified by Gabriel, reflecting the representation type studied by Claus Ringel and Idun Reiten. Representations of Dynkin quivers from the classification of Vladimir Drinfeld and Eugene Dynkin are finite and yield finite Auslander–Reiten quivers, while work of William Crawley-Boevey, Harm Derksen, and Jerzy Weyman connects to moduli problems and invariant theory. Techniques from Maurice Auslander, Idun Reiten, and Claus Ringel guide the construction via almost split sequences, while homological dimensions considered by Maurice Auslander and Raphaël Rouquier provide constraints on projective and injective shapes influenced by Jean-Pierre Serre and Pierre Deligne.
Components and classification (tame, wild, tree, tubes)
Components of the Auslander–Reiten quiver fall into discrete families such as preprojective, regular, and preinjective components, with classification efforts by Gabriel, Nazarova, Roiter, and Drozd distinguishing tame and wild representation types. Tame cases studied by Vlastimil Dlab and Chin-Lung Wang include tube components and Euclidean extended Dynkin diagrams linked to Oleg Viro and Anatoly Maltsev, whereas wild behavior relates to Yuri Drozd's wild dichotomy and work by Dmitry Piontkovski. Tree classes and tube components appear in analyses by Claus Ringel and Idun Reiten, and connections to the McKay correspondence and work by John McKay and Miles Reid relate component shapes to singularity theory addressed by Michael Artin.
Applications in representation theory and tilting
The Auslander–Reiten quiver informs tilting theory developed by Bernhard Keller, Dieter Happel, and Toshiyuki Miyashita, where tilting modules and derived equivalences transform components and mutate quivers in the spirit of Sergey Fomin and Andrei Zelevinsky's cluster algebra program. Applications extend to the study of cluster categories introduced by Aslak Buan, Robert Marsh, and Idun Reiten, relating to mirror symmetry perspectives by Maxim Kontsevich and Paul Seidel, and to stability conditions influenced by Tom Bridgeland. Connections to homological conjectures of Maurice Auslander and Joseph Bernstein illuminate categorical actions studied by Mikhail Khovanov and Igor Frenkel.
Examples and computations (Dynkin, Euclidean, wild)
Explicit Auslander–Reiten quivers are computed for Dynkin types A_n, D_n, E6, E7, E8 classified by Eugene Dynkin and Pierre Gabriel, producing finite oriented graphs used by Klaus Bongartz and Ian Reiten in classification. Euclidean extended Dynkin types Ã_n, Ď_n, Ě_6, Ě_7, Ě_8 yield tubular and Euclidean components studied by Claus Ringel and William Crawley-Boevey, while wild quivers analyzed by Yuri Drozd and Alexander Kirillov produce intricate infinite quivers reflecting Gerstenhaber and Schack cohomology patterns. Computational approaches utilize algorithms and software developed under projects like GAP influenced by Joachim Neumann and Richard Parker, while continued research by Raphaël Rouquier and Bernhard Keller expands explicit computations in derived and triangulated settings.