Introduction to the Representation Theory of Algebras

Introduction to the Representation Theory of Algebras

Michael Barot Introduction to the Representation Theory of Algebras 123

Michael Barot Instituto de Matemáticas Universidad Nacional Autónoma de México Mexico City Mexico ISBN 978-3-319-11474-3 ISBN 978-3-319-11475-0 (ebook) DOI 10.1007/978-3-319-11475-0 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014957158 Mathematics Subject Classification (2010): 16G20 Springer International Publishing Switzerland 2015 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Angélica

Preface The aim of these notes is to give a brief and elementary introduction to the representation theory of finite-dimensional algebras. The notes originated from an undergraduate course I gave in two occasions at Universidad Nacional Autonóma de México. The plan of the course was to try to cope with two competing demands: to expect as little as possible and to reach as much as possible: to expect only linear algebra as background and yet to make way to substantial and central ideas and results during its progress. Therefore some crucial decisions were necessary. We opted for the model case rather than the most general situation, for the most illustrating example rather than the most extravagant one. We sought a guideline through this vast field which conducts to as many important notions, techniques and questions as possible in the limited space of a one-semester course. So, it is a book written from a specific point of view and the title should really be Introduction to the theory of algebras, which are finite dimensional over some algebraically closed field. The book starts with the most difficult chapter: matrix problems. Conceptually there is little to understand in that chapter, but it requires a considerable effort from the reader to follow the argumentation within. However, this chapter is central: it prepares all the main examples which later will guide through the rest. In the following two chapters we consider the main languages of representation theory. Since there are several competing languages in representation theory, a considerable amount of our effort is directed towards mastering and combining all of them. As you will see each of these languages has its own advantages and it therefore not only enables the reader to consult the majority of all research articles in the field, but also enriches the way we may think about the notions themselves. The rest of the book is devoted to gain structural insight into the categories of modules of a given algebra. In the chapter about module categories some older results are proved, whereas in the next four chapters more recent developments are discussed. The last chapter is more of an open-minded collection concerning indecomposable modules, the building bricks of the module categories, with respect to certain invariants, called dimension vectors. vii

viii Preface My thanks go to Jan Schröer, for it was him who suggested to write jointly a book on representation theory but unfortunately gave up on it, to Juraj Hromkovič who invited me to ETH Zürich for a sabbatical in 2010 and made thus possible to finish the manuscript, to Manuela Tschabold who tirelessly read the whole manuscript carefully eliminating thus many errors and misprints, to Karin Baur who took it as a base for a course in 2011 suggesting many improvements, to Christof Geiss who was able to improve the manuscript still further, to Mario Aigner from Springer who tirelessly fought on my side for a good layout and to Angélica Herrera Loyo for her patience. I apologize to the reader for all the errors which still remain and for the poor English in which it is written. Mexico City, Mexico January 2012 Michael Barot

Contents 1 Matrix Problems... 1 1.1 Introduction... 1 1.2 The Two Subspace Problem... 3 1.3 Decomposition into Indecomposables... 4 1.4 The KroneckerProblem... 6 1.5 The ThreeKroneckerProblem... 12 1.6 Comments... 14 2 Representations of Quivers... 15 2.1 Quivers... 15 2.2 Representations... 17 2.3 Categories andfunctors... 20 2.4 The Path Category... 23 2.5 Equivalenceof Categories... 26 2.6 A New Example... 28 3 Algebras... 33 3.1 Definition and Generalities About Algebras... 33 3.2 The Path Algebra... 35 3.3 Quotients by Ideals... 37 3.4 Idempotents... 38 3.5 Morita Equivalence... 42 3.6 The Radical... 43 3.7 The Quiverof an Algebra... 46 3.8 Relations... 49 3.9 Summary... 51 4 Module Categories... 53 4.1 The CategoricalPoint of View... 53 4.2 Duality... 54 4.3 Kernels and Cokernels... 56 4.4 Unique Decomposition... 60 ix

x Contents 4.5 Projectives andinjectives... 62 4.6 ProjectiveCovers andinjectivehulls... 68 4.7 Simple Modules... 70 5 Elements of Homological Algebra... 73 5.1 Short Exact Sequences... 73 5.2 The Baer Sum... 78 5.3 Resolutions... 83 5.4 LongExact Sequences... 88 6 The Auslander-Reiten Theory... 93 6.1 The Auslander-ReitenQuiverandthe InfiniteRadical... 93 6.2 Example:The KroneckerAlgebra... 95 6.3 The Auslander-ReitenTranslate... 103 6.4 Auslander-ReitenSequences... 106 7 Knitting... 113 7.1 Sourceand Sink Maps... 113 7.2 The Knitting Technique... 116 7.3 Knitting with Dimension Vectors... 119 7.4 Limits of Knitting... 122 7.5 Preprojective Components... 126 8 Combinatorial Invariants... 129 8.1 GrothendieckGroup... 129 8.2 The Homological Bilinear Form... 131 8.3 The CoxeterTransformation... 134 8.4 QuadraticForms... 137 8.5 Roots... 141 9 Indecomposables and Dimensions... 147 9.1 The Brauer-ThrallConjectures... 147 9.2 Gabriel s Theorem... 150 9.3 Reflection Functors... 152 9.4 Kac s Theorem... 156 9.5 Tame and Wild... 158 9.6 Module Varieties... 163 References... 167 Index of Symbols... 169 Index of Notions... 171