Auslander–Reiten theory

Auslander–Reiten theory is a framework in representation theory developed to analyze the morphism and extension structure of modules over associative algebras. It organizes indecomposable modules and irreducible morphisms into combinatorial and homological invariants that connect to tilting theory, cluster categories, and the classification of finite-dimensional algebras. The theory arose from interactions among researchers working on the Auslander correspondence, homological conjectures, and the representation type of algebras in the 1970s and 1980s.

History and motivation

The development of Auslander–Reiten ideas is associated with figures such as Maurice Auslander, Idun Reiten, Claus M. Ringel, Sheila Brenner, Michael C. R. Butler, and contemporaries responding to problems posed in the work of Emmy Noether, Philip Hall, Israel Gelfand, and Hyman Bass. Early motivation came from classification problems studied by researchers at institutions like University of Toronto, Norwegian University of Science and Technology, University of Chicago, and conferences including the International Congress of Mathematicians where homological methods were prominent. Influences trace through collaborations with mathematicians such as Jean-Pierre Serre, Alexander Grothendieck, David J. Benson, Benson Farb, Hugh Thomas, and Dieter Happel, reflecting cross-currents among categories, homological algebra, and algebraic geometry.

Auslander–Reiten sequences and almost split exact sequences

Central to the theory are almost split exact sequences introduced by Auslander and Reiten in papers and lectures circulated through seminars at University of Oslo and University of Pennsylvania. These sequences characterize non-split extensions between indecomposable modules studied by Maurice Auslander and Idun Reiten and further investigated by Claus M. Ringel and Dieter Happel. The existence and uniqueness statements relate to the work of Albrecht Dold, Christian T. C. Wall, and homological methods used by Jean-Louis Verdier and Henri Cartan. Proofs and constructions employ techniques developed by David Eisenbud, Daniel Quillen, Jonathan Rosenberg, and scholars in algebraic topology and category theory such as Saunders Mac Lane and Max Karoubi.

Auslander–Reiten quivers and components

The Auslander–Reiten quiver encodes indecomposable objects and irreducible morphisms; its study connects with classification programs advanced by William Crawley-Boevey, Idun Reiten, and Claus M. Ringel. The combinatorial and geometric properties of quivers were influenced by work of Gabriel Gabriel, Pierre Gabriel, Bernard Keller, and Hugh Thomas on quiver representations and by contributions from Kenji Ueno, Iain Gordon, Victor Ginzburg, and Ernest Vinberg. Analysis of components—preprojective, regular, and preinjective—uses ideas related to the Dynkin diagram classifications explored by Eugène Dynkin, Nathan Jacobson, and Richard Brauer. Techniques draw from studies by George Lusztig, Robert Moody, Peter Slodowy, and G. Harold Hudson in Lie theory and singularity theory, while computational work leverages algorithms developed in groups at Max Planck Institute for Mathematics and Institute for Advanced Study.

Auslander–Reiten translate and functors τ and τ⁻¹

The Auslander–Reiten translate τ and its inverse τ⁻¹ are endofunctors that express almost split phenomena, with formalism informed by categorical frameworks from Grothendieck, Jean-Pierre Serre, and Alexander Grothendieck’s students including Pierre Deligne and Bernard Malgrange. The functors relate to derived category constructions popularized by Grothendieck’s school and developed by Amnon Neeman, Dieter Happel, and Raphaël Rouquier. Connections to Serre duality and Nakayama functors reflect influences from Tadao Oda, Matsumoto, and work by Hyman Bass and Masahiro Yoshida. Applications of τ appear in studies by Idun Reiten, Marta Asaeda, Christof Geiss, and Bernard Leclerc within the contexts of homological dualities and triangulated categories.

Applications and connections (representation theory, tilting, cluster categories)

Auslander–Reiten techniques underpin advances in tilting theory pioneered by Joachim Rickard, Tomasz Brzezinski, Alistair Savage, and Steffen Koenig, and in cluster categories initiated by Bernhard Keller, Andrei Zelevinsky, Sergey Fomin, and Hugh Thomas. The theory feeds into the categorification programs involving George Lusztig, Bernard Leclerc, and Kashiwara and intersects with work on Hall algebras by Philip Hall and Ringel. Further applications reach mathematical physics via links to Namikawa-type symplectic singularities studied by Alastair King, Tom Bridgeland, Yukinobu Toda, and Paul Seidel. Modern research connects Auslander–Reiten methods with derived equivalences explored by Rickard, cluster-tilting objects analyzed by Idun Reiten and Bernard Keller, and with geometric representation theory pursued by David Kazhdan, George Lusztig, and Alexander Beilinson.

Category:Representation theory