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View This Volume's Front and Back Matter GRADUATE STUDIES IN MATHEMATICS 193 A Tour of Representation Theory Martin Lorenz 10.1090/gsm/193 A Tour of Representation Theory GRADUATE STUDIES IN MATHEMATICS 193 A Tour of Representation Theory Martin Lorenz EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 16Gxx, 16Txx, 17Bxx, 20Cxx, 20Gxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-193 Library of Congress Cataloging-in-Publication Data Names: Lorenz, Martin, 1951- author. Title: A tour of representation theory / Martin Lorenz. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu- ate studies in mathematics ; volume 193 | Includes bibliographical references and indexes. Identifiers: LCCN 2018016461 | ISBN 9781470436803 (alk. paper) Subjects: LCSH: Representations of groups. | Representations of algebras. | Representations of Lie algebras. | Vector spaces. | Categories (Mathematics) | AMS: Associative rings and algebras – Representation theory of rings and algebras – Representation theory of rings and algebras. msc | Associative rings and algebras – Hopf algebras, quantum groups and related topics – Hopf algebras, quantum groups and related topics. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Lie algebras and Lie superalgebras. msc | Group theory and generalizations – Representation theory of groups – Representation theory of groups. msc | Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups and related topics. msc Classification: LCC QA176 .L67 2018 | DDC 515/.7223–dc23 LC record available at https://lccn.loc.gov/2018016461 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 232221201918 For Maria Contents Preface xi Conventions xvii Part I. Algebras Chapter 1. Representations of Algebras 3 1.1. Algebras 3 1.2. Representations 24 1.3. Primitive Ideals 41 1.4. Semisimplicity 50 1.5. Characters 65 Chapter 2. Further Topics on Algebras 79 2.1. Projectives 79 2.2. Frobenius and Symmetric Algebras 96 Part II. Groups Chapter 3. Groups and Group Algebras 113 3.1. Generalities 113 3.2. First Examples 124 3.3. More Structure 131 3.4. Semisimple Group Algebras 143 3.5. Further Examples 150 3.6. Some Classical Theorems 159 vii viii Contents 3.7. Characters, Symmetric Polynomials, and Invariant Theory 170 3.8. Decomposing Tensor Powers 179 Chapter 4. Symmetric Groups 187 4.1. Gelfand-Zetlin Algebras 189 4.2. The Branching Graph 192 4.3. The Young Graph 197 4.4. Proof of the Graph Isomorphism Theorem 205 4.5. The Irreducible Representations 217 4.6. The Murnaghan-Nakayama Rule 222 4.7. Schur-Weyl Duality 235 Part III. Lie Algebras Chapter 5. Lie Algebras and Enveloping Algebras 245 5.1. Lie Algebra Basics 246 5.2. Types of Lie Algebras 253 5.3. Three Theorems about Linear Lie Algebras 257 5.4. Enveloping Algebras 266 5.5. Generalities on Representations of Lie Algebras 278 5.6. The Nullstellensatz for Enveloping Algebras 287 5.7. Representations of sl2 300 Chapter 6. Semisimple Lie Algebras 315 6.1. Characterizations of Semisimplicity 316 6.2. Complete Reducibility 320 6.3. Cartan Subalgebras and the Root Space Decomposition 325 6.4. The Classical Lie Algebras 334 Chapter 7. Root Systems 341 7.1. Abstract Root Systems 342 7.2. Bases of a Root System 349 7.3. Classification 356 7.4. Lattices Associated to a Root System 361 Chapter 8. Representations of Semisimple Lie Algebras 373 8.1. Reminders 374 8.2. Finite-Dimensional Representations 377 8.3. Highest Weight Representations 379 Contents ix 8.4. Finite-Dimensional Irreducible Representations 385 8.5. The Representation Ring 390 8.6. The Center of the Enveloping Algebra 393 8.7. Weyl’s Character Formula 408 8.8. Schur Functors and Representations of sl(V ) 418 Part IV. Hopf Algebras Chapter 9. Coalgebras, Bialgebras, and Hopf Algebras 427 9.1. Coalgebras 427 9.2. Comodules 441 9.3. Bialgebras and Hopf Algebras 447 Chapter 10. Representations and Actions 465 10.1. Representations of Hopf Algebras 466 10.2. First Applications 476 10.3. The Representation Ring of a Hopf Algebra 485 10.4. Actions and Coactions of Hopf Algebras on Algebras 492 Chapter 11. Affine Algebraic Groups 503 11.1. Affine Group Schemes 503 11.2. Affine Algebraic Groups 508 11.3. Representations and Actions 512 11.4. Linearity 515 11.5. Irreducibility and Connectedness 520 11.6. The Lie Algebra of an Affine Algebraic Group 526 11.7. Algebraic Group Actions on Prime Spectra 530 Chapter 12. Finite-Dimensional Hopf Algebras 541 12.1. Frobenius Structure 541 12.2. The Antipode 549 12.3. Semisimplicity 552 12.4. Divisibility Theorems 559 12.5. Frobenius-Schur Indicators 567 Appendices Appendix A. The Language of Categories and Functors 575 A.1. Categories 575 x Contents A.2. Functors 578 A.3. Naturality 579 A.4. Adjointness 583 Appendix B. Background from Linear Algebra 587 B.1. Tensor Products 587 B.2. Hom-⊗ Relations 593 B.3. Vector Spaces 594 Appendix C. Some Commutative Algebra 599 C.1. The Nullstellensatz 599 C.2. The Generic Flatness Lemma 601 C.3. The Zariski Topology on a Vector Space 602 Appendix D. The Diamond Lemma 605 D.1. The Goal 605 D.2. The Method 606 D.3. First Applications 608 D.4. A Simplification 611 D.5. The Poincaré-Birkhoff-Witt Theorem 612 Appendix E. The Symmetric Ring of Quotients 615 E.1. Definition and Basic Properties 615 E.2. The Extended Center 617 E.3. Comparison with Other Rings of Quotients 619 Bibliography 623 Subject Index 633 Index of Names 645 Notation 649 Preface In brief, the objective of representation theory is to investigate the different ways in which a given algebraic object—such as an algebra, a group, or a Lie algebra—can act on a vector space. The benefits of such an action are at least twofold: the structure of the acting object gives rise to symmetries of the vector space on which it acts; and, in the other direction, the highly developed machinery of linear algebra can be brought to bear on the acting object itself to help uncover some of its hidden properties. Besides being a subject of great intrinsic beauty, representation theory enjoys the additional benefit of having applications in myriad contexts other than algebra, ranging from number theory, geometry, and combinatorics to probability and statistics [58], general physics [200], quantum field theory [212], the study of molecules in chemistry [49], and, more recently, machine learning [127]. This book has evolved from my lecture notes for a two-semester graduate course titled Representation Theory that I gave at Temple University during the academic years 2012/13 and 2015/16. Some traces of the informality of my original notes and the style of my lectures have remained intact: the text makes rather copious use of pictures and expansively displayed formulae; definitions are not numbered and neither are certain key results, such as Schur’s Lemma or Wedderburn’s Structure Theorem, which are referred to by name rather than number throughout the book. However, due to the restrictions imposed by having to set forth the material on the page in a linear fashion, the general format of this book does not in fact duplicate my actual lectures and it only locally reflects their content. I will comment more on this below. The title A Tour of Representation Theory (ToR) is meant to convey the panoramic view of the subject that I have aimed for.1 Rather than offering an 1The choice of title is also a nod to the Tour de France, and “Tor” in German is “gate” as well as “goal” (scored) and “fool”. xi xii Preface in-depth treatment of one particular area, ToR gives an introduction to three distinct flavors of representation theory—representations of groups, Lie algebras, and Hopf algebras—and all three are presented as incarnations of algebra representations. The book loops repeatedly through these topics, emphasizing similarities and con- nections. Group representations, in particular, are revisited frequently after their initial treatment in Part II. For example, Schur-Weyl duality is first discussed in Sec- tion 4.7 and later again in Section 8.8; Frobenius-Schur indicators are introduced in §3.6.3 in connection with the Brauer-Fowler Theorem and they are treated in their proper generality in Section 12.5; and Chapter 11, on affine algebraic groups, brings together groups, Lie algebras, and Hopf algebras. This mode of exposition owes much to the “holistic” viewpoint of the monograph [72] by Etingof et al., although ToR forgoes the delightful historical intermezzos that punctuate [72] and it omits quivers in favor of Hopf algebras. Our tour does not venture very far into any of the areas it passes through, but I hope that ToR will engender in some readers the desire to pursue the subject and that it will provide a platform for further explorations.
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