Abstract

We describe how tilting modules are used to classify the representationfinite algebras and their indecomposable modules.

Introduction

Probably the first appearance of tilting modules in representation theory of finite-dimensional algebras was in 1973 the use of reflection functors when Bernstein, Gelfand and Ponomarev [5] reproved Gabriel's classification of the representation-finite hereditary quiver algebras. Dlab and Ringel [18] extended in 1976 the use of reflection functors to arbitrary representation-finite hereditary algebras. Next, the concept of reflection functors has been generalized in 1979 by Auslander, Platzeck and Reiten [2] (they called it ”Coxeter functors without diagrams”), and finally in 1980 by Brenner and Butler [13], who coined the term tilting and gave the first general definition of a tilting module, together with basic properties of tilting functors.

In a time where most people working with representation-finite algebras were knitting Auslander-Reiten sequences, this was a new approach: To study a class of modules which are given by abstract properties. Tilting modules have then been used very successfully by Bongartz [9] and by Happel and Vossieck [26] to find a far-reaching generalization of Gabriel's Theorem to representation-directed algebras, see Theorems 6 and 7 below.

The aim of these notes is to describe this powerful application of tilting theory in the classification of representation-finite algebras, and to add some more details on the representation-infinite case and the classification of tame algebras. These notes arose from two lectures which I gave at the meeting on ’Tilting Theory’ at the Fraueninsel near Munich.