Algebras and Representation Theory

✍ Scribed by Karin Erdmann, Thorsten Holm

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 304
• Series: Springer Undergraduate Mathematics Series
• Category: Library

✦ Synopsis

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

Front Matter ....Pages i-ix
Algebras (Karin Erdmann, Thorsten Holm)....Pages 1-27
Modules and Representations (Karin Erdmann, Thorsten Holm)....Pages 29-59
Simple Modules and the Jordan–Hölder Theorem (Karin Erdmann, Thorsten Holm)....Pages 61-84
Semisimple Modules and Semisimple Algebras (Karin Erdmann, Thorsten Holm)....Pages 85-102
The Structure of Semisimple Algebras: The Artin–Wedderburn Theorem (Karin Erdmann, Thorsten Holm)....Pages 103-116
Semisimple Group Algebras and Maschke’s Theorem (Karin Erdmann, Thorsten Holm)....Pages 117-127
Indecomposable Modules (Karin Erdmann, Thorsten Holm)....Pages 129-141
Representation Type (Karin Erdmann, Thorsten Holm)....Pages 143-162
Representations of Quivers (Karin Erdmann, Thorsten Holm)....Pages 163-184
Diagrams and Roots (Karin Erdmann, Thorsten Holm)....Pages 185-202
Gabriel’s Theorem (Karin Erdmann, Thorsten Holm)....Pages 203-238
Proofs and Background (Karin Erdmann, Thorsten Holm)....Pages 239-264
Back Matter ....Pages 265-298