This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

• Self contained • Treats both finite and infinite dimensional cases • Based on courses given by the authors

### Contents

Introduction; 1. Algebras and modules; 2. Quivers and algebras; 3. Representations and modules; 4. Auslander-Reiten theory; 5. Nakayama algebras and representation-finite group algebras; 6. Tilting theory; 7. Representation-finite hereditary algebras; 8. Tilted algebras; 9. Directing modules and postprojective components; Appendix: categories, functors and homology.

### Review

' ... a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. ... presented from the perspective of linear representations of quivers and homo logical algebra. The treatment is self contained and provides an elementary and up-to-date introduction to the subject using quiver-theoretical techniques and the theory of almost split sequences ...' L'enseignement mathematique