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-
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Karin Erdmann Thorsten Holm
Algebras and Representation Theory
Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain, University of St. Andrews A. MacIntyre, Queen Mary University of London S. Scott, King’s College London N. Snashall, University of Leicester E. Süli, University of Oxford M.R. Tehranchi, University of Cambridge J.F. Toland, University of Bath
More information about this series at http://www.springer.com/series/3423
Karin Erdmann • Thorsten Holm
Algebras and Representation Theory
123
Karin Erdmann Mathematical Institute University of Oxford Oxford, United Kingdom
Thorsten Holm Fakult¨at f¨ur Mathematik und Physik Institut f¨ur Algebra, Zahlentheorie und Diskrete Mathematik Leibniz Universit¨at Hannover Hannover, Germany
ISSN 1615-2085 ISSN 2197-4144 (electronic) Springer Undergraduate Mathematics Series ISBN 978-3-319-91997-3 ISBN 978-3-319-91998-0 (eBook) https://doi.org/10.1007/978-3-319-91998-0 Library of Congress Control Number: 2018950191 Mathematics Subject Classification (2010): 16-XX, 16G10, 16G20, 16D10, 16D60, 16G60, 20CXX © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Introduction
Representation theory is a beautiful subject which has numerous applications in mathematics and beyond. Roughly speaking, representation theory investigates how algebraic systems can act on vector spaces. When the vector spaces are finitedimensional this allows one to explicitly express the elements of the algebraic system by matrices, and hence one can exploit basic linear algebra to study abstract algebraic systems. For example, one can study symmetry via group actions, but more general
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