Algebras and Representation Theory

This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers.

The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel’s celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams.

Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.

Table of Contents

1 Algebras

2 Modules and Representations

3 Simple Modules and the Jordan-Holder Theorem

4 Semisimple Modules and Semisimple Algebras

5 The Structure of Semisimple Algebras: The Artin-Wedderburn Theorem

6 Semisimple Group Algebras and Maschke’s Theorem

7 Indecomposable Modules

8 Representation Type

9 Representations of Quivers

1 0 Diagrams and Roots

11 Gabriel’s Theorem

12 Proofs and Background

A Induced Modules for Group Algebras

B Solutions to Selected Exercises

About the Authors

Karin Erdmann‘s research focus lies on representation theory of finite groups, and finite-dimensional algebras. She has written many research articles, and is the author of a research monograph and a textbook.

Thorsten Holm is Professor of Mathematics at Leibniz Universität Hannover. His research interests include representation theory of finite groups and finite-dimensional algebras, and algebraic combinatorics.