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© 2018

Algebras and Representation Theory

Textbook

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-ix
  2. Karin Erdmann, Thorsten Holm
    Pages 1-27
  3. Karin Erdmann, Thorsten Holm
    Pages 29-59
  4. Karin Erdmann, Thorsten Holm
    Pages 61-84
  5. Karin Erdmann, Thorsten Holm
    Pages 85-102
  6. Karin Erdmann, Thorsten Holm
    Pages 117-127
  7. Karin Erdmann, Thorsten Holm
    Pages 129-141
  8. Karin Erdmann, Thorsten Holm
    Pages 143-162
  9. Karin Erdmann, Thorsten Holm
    Pages 163-184
  10. Karin Erdmann, Thorsten Holm
    Pages 185-202
  11. Karin Erdmann, Thorsten Holm
    Pages 203-238
  12. Karin Erdmann, Thorsten Holm
    Pages 239-264
  13. Back Matter
    Pages 265-298

About this book

Introduction

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.

Keywords

MSC (2010) 16-XX, 16G10, 16G20, 16D10, 16D60, 16G60, 20CXX algebras representations modules simple modules Jordan-Hoelder theorem Artin-Wedderburn theorem Maschke's theorem indecomposable modules Krull-Schmidt theorem representation type representations of quivers Gabriel's theorem

Authors and affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Fakultät für Mathematik und Physik, Institut für Algebra, Zahlentheorie und Diskrete MathematikLeibniz Universität HannoverHannoverGermany

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.

Bibliographic information

Reviews

“The book under review is a text-book for higher undergraduate mathematics students or graduate students who have previous knowledge of results from linear algebra, and basic properties of rings and groups. … It is also useful for non-experts (in representation theory of quivers), they may benefit from this book in several ways: by examining the numerous worked examples, or by working out the many exercises.” (Bin Zhu, zbMATH 1429.16001, 2020)