Representation-finite algebras

Abstract

For a long time, researchers in the representation theory of algebras concentrated on finding criteria allowing us to verify whether a given algebra is representation-finite or not, and, if this was the case, of computing all its (isoclasses of) indecomposable modules. Indeed, it was believed that this class of algebras would be relatively easy to classify and that their indecomposable modules have a relatively simple structure. This approach was largely successful. Actually, one of the first important results of modern-day representation theory was Gabriel’s theorem, which says that a hereditary algebra over an algebraically closed field is representation-finite if and only if it is the path algebra of a quiver whose underlying graph is one of the well-known Dynkin diagrams \(\mathbb {A}\), \(\mathbb {D}\) or \(\mathbb {E}\). Nowadays, there exists a reasonable global theory of representation-finite algebras. At present, we do not have a similar theory for studying representation-infinite algebras, but the ideas and techniques developed for representation-finite algebras still show their usefulness when applied to the understanding of new classes. The aim of this chapter is to prove some of the most important known results on representation-finite algebras highlighting of the methods that led to their proofs.