Quiver Representations

This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The text has two parts. In Part I, the theory is studied in an elementary way using quivers and their representations. This is a very hands-on approach and requires only basic knowledge of linear algebra. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible. Part II then uses the language of algebras and modules to build on the material developed before. The equivalence of the two approaches is proved in the text. The last chapter gives a proof of Gabriel’s Theorem. The language of category theory is developed along the way as needed.

Part I: Quivers and their representations.- Representations of quivers.- Projective and injective representations.- Examples of Auslander-Reiten quivers.- Part II: Path algebras.- Algebras and modules.- Bound quiver algebras.- New algebras from old.- Auslander-Reiten theory.- Quadratic forms and Gabriel’s theorem.

Ralf Schiffler is a Professor in the Department of Mathematics at the University of Connecticut.