BIRS 10w5069: Test problems for the theory of finite dimensional algebras

The roots of representation theory go far back into the history of mathematics: the study of symmetry, starting with the Platonic solids and the development of group theory; the study of matrices and the representation theory of groups by Klein, Schur and others which led to the development of the concepts of rings, ideals and modules; the study of normal forms in analysis, in the work of Weierstrass, Jordan and Kronecker, among others; the development of Lie theory. Some of the famous Hilbert’s problems relate representation theory with fundamental geometric concepts. Starting in the middle 60’s of last century, the ‘modern’ Representation Theory of finite dimensional algebras had a very fast start with three main driving forces: The categorical point of view, represented by Maurice Auslander and his school, leading to the concepts of almost-split sequences, Auslander-Reiten duality, and Auslander-Reiten quivers. The introduction of the concept of quiver representations by Pierre Gabriel, which is now a main tool in the analysis of the representation theory of finite dimensional algebras. The reformulation of problems from representation theory as matrix problems, associated to the Ukrainian school of A. Roiter lead to classification results in certain representation-infinite situations and the conceptual dichotomy of algebras according to their representation type as tame (including representation-finite) or wild. This ‘modern’ Representation Theory of finite dimensional algebras, typically over an algebraically closed field, is thus characterized in its early stages by the dominance of linear methods, functor categories and homological theory. A specific flavor, in that form not present in any other subject, is constituted by Auslander-Reiten theory (almost-split sequences, Auslander-Reiten quivers, Auslander-Reiten duality), an aspect reflecting a deep combinatorial structure of homological nature. The area has by now reached a highly mature stage, leaving behind the original motivation of determining (if possible) all the indecomposable representations relating to the concepts of representation-finite, tame and wild type, respectively. The present situation is best described by strong interactions of representation theory with other mathematical subjects, like Graph Theory, Combinatorics, Lie Theory, Algebraic and Differential Geometry, Singularity Theory, Quantum Groups, and Mathematical Physics. Moreover, as reflected by conferences like the biannual series of ICRAs (International Conferences on the representations of algebras and related topics), the vitality of the subject is characterized by continuously conquering new topics of neighboring areas.