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

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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. Overview of the Contents. The topics covered in ToR and the methods employed ∗ are resolutely algebraic. Lie groups, C -algebras, and other areas of representation theory requiring analysis are not covered. On the other hand, in keeping with the widely acknowledged truth that algebraic representation theory benefits from a geometric perspective, the discourse involves a modicum of algebraic geometry on occasion and I have also tried my hand at depicting various noncommutative geometric spaces throughout the book. No prior knowledge of algebraic geometry is assumed, however. Representations of algebras form the unifying thread running through ToR. Therefore, Part I is entirely written in the setting of associative algebras. Chap- ter 1 develops the basic themes of representation theory: irreducibility, complete reducibility, spaces of primitive ideals, characters, ...; thechapterestablishes notation to be used throughout the remainder of the book; and it furnishes the fun- damental general results of representation theory, such as Wedderburn’s Structure Theorem. Chapter 2 covers topics that are somewhat more peripheral to the main thrust of ToR: projective modules (Section 2.1) and Frobenius algebras (Section 2.2). Readers whose main interest is in groups or Lie algebras may skip this chapter at a first reading. However, Section 2.2 deploys some tools that are indispensable for the discussion of finite-dimensional Hopf algebras in Chapter 12. Parts II and III are respectively devoted to representations of groups and Lie algebras. To some degree, these two parts can be tackled in any order. However, I have made a deliberate effort at presenting the material on group representations in a palatable manner, offering it as an entryway to representation theory, while the part on Lie algebras is written in a slightly terser style demanding greater mathematical maturity from the reader. Most of Part II deals with finite-dimensional representations of finite groups, usually over a base field whose characteristic does Preface xiii not divide the order of the group in question. Chapter 3 covers standard territory, with the possible exception of some brief excursions into classical invariant theory (§§3.7.4, 3.8.4). Chapter 4, however, presents the representation theory of the symmetric groups in characteristic 0 via an elegant novel approach devised by Okounkov and Vershik rather than following the route taken by the originators of the theory, Frobenius, Schur, and Young. Much of this chapter elaborates on Chapter 2 of Kleshchev’s monograph [125], providing full details and some additional background. My presentation of the material on Lie algebras and their representations in Part III owes a large debt to the classics by Dixmier [63]and Humphreys [105] and also to Fulton and Harris [83] as well as the more recent monograph [69] by Erdmann and Wildon. The notation and terminology in this part are largely adopted from [105] and its Afterword (1994). Departing from tradition, the discussion of the Nullstellensatz and the Dixmier-Mœglin equivalence for enveloping algebras of Lie algebras in Section 5.6 relies on the symmetric ring of quotients rather than the classical ring of quotients; this minimizes the requisite background material from noncommutative ring theory, which is fully provided in Appendix E. Hopf algebra structures are another recurring theme throughout the book: they are first introduced for the special case of group algebras in Section 3.3; an analogous discussion for enveloping algebras of Lie algebras follows in §5.4.4; and Hopf algebras are finally tackled in full generality in Part IV. While this part of ToR is relatively dry in comparison with the rest of the book, the reader familiar with the earlier special cases will be amply prepared and hopefully willing to face up to what may initially seem like a profusion of technicalities. The effort is worthwhile: many facets of the representation theory of groups or Lie algebras, especially those dealing with tensor products of representations, take their most natural form when viewed through the lens of Hopf algebras and, of all parts of ToR,itisPartIVthat leads closest to the frontier of current research. On the other hand, I believe that students planning to embark on the investigation of Hopf algebras will profit from a grounding in the more classical representation theories of groups and Lie algebras, which is what ToR aims to provide. Prerequisites. The various parts of ToR differ rather significantly with regard to their scope and difficulty. However, much of the book was written for a readership having nothing but a first-year graduate algebra course under their belts: the basics of groups, rings, modules, fields, and Galois theory, but not necessarily anything beyond that level. Thus, I had no qualms assuming a solid working knowledge of linear algebra—after all, representation theory is essentially linear algebra with (quite a lot of) extra structure. Appendix B summarizes some formal points of linear algebra, notably the properties of tensor products. The prospective reader should also be well acquainted with elementary group theory: the isomorphism theorems, Sylow’s Theorem, and abelian, nilpotent, and xiv Preface solvable groups. The lead-in to group representations is rather swift; group al- gebras and group representations are introduced in quick succession and group representation theory is developed in detail from there. On the other hand, no prior knowledge of Lie algebras is expected; the rudiments of Lie algebras are presented in full, albeit at a pace that assumes some familiarity with parallel group-theoretic lines of reasoning. While no serious use of category theory is made in this book, I have frequently availed myself of the convenient language and unified way of looking at things that categories afford. When introducing new algebraic objects, such as group algebras or enveloping algebras of Lie algebras, I have emphasized their “functorial” proper- ties; this highlights some fundamental similarities of the roles these objects play in representation theory that may otherwise not be apparent. The main players in ToR k are the category Vectk of vector spaces over a field and the categories Repfin A of all finite-dimensional representations of various k-algebras A. Readers having had no prior exposure to categories and functors may wish to peruse Appendix A before delving into the main body of the text. Using this Book. ToR is intended as a textbook for a graduate course on represen- tation theory, which could immediately follow the standard abstract algebra course, and I hope that the book will also be useful for subsequent reading courses and for readers wishing to learn more about the subject by self-study. Indeed, the more advanced material included in ToR places higher demands on its readers than would probably be adequate for an introductory course on representation theory and it is unrealistic to aim for full coverage of the book in a single course, even if it spans two semesters. Thus, a careful selection of topics has to be made by the instructor. When teaching Abstract Algebra over the years, I found that finite groups have tended to be quite popular among my students—starting from minimal prerequisites, one quickly arrives at results of satisfying depth and usefulness. Therefore, I usually start the follow-up course Representation Theory by diving right into representations of groups (Part II), covering all of Chapter 3 and some of Chapter 4 in the first semester. Along the way, I add just enough material about algebras from Chapter 1 to explain the general underpinnings,often relegating proofs to reading assignments. In the second semester, I turn to representations of Lie algebras and try to cover as much of Part III as possible. Section 5.6 is generally only presented in a brief “outlook” format and Sections 8.6–8.8 had to be left uncovered so far for lack of time. Instead, in one or two lectures at the end of the second semester of Representation Theory or sometimes in a mini-course consisting of four or five lectures in our Algebra Seminar, I try to give the briefest of glimpses into the theory of Hopf algebras and their representations (Part IV). Alternatively, one could conceivably begin with a quick pass through the representation-theoretic fundamentals of algebras, groups, Lie algebras, and Hopf algebras before spiraling back to cover each or some of these topics in greater Preface xv depth. Teaching a one-semester course will most likely entail a focus on just one of Parts II, III, or IV depending on the instructor’s predilections and the students’ background. In order to enable the instructor or reader to pick and choose topics from various parts of the book, I have included numerous cross references and frequent reminders throughout the text. The exercises vary greatly in difficulty and purpose: some merely serve to unburden various proofs of unsightly routine verifications, while others present substantial results that are not proved but occasionally alluded to in the text. I have written out solutions for the majority of exercises and I’d be happy to make them available to instructors upon request. Acknowledgments. My original exposure to representation theory occurred in lectures by my advisor Gerhard Michler, who introduced me to the group-theoretic side of the subject, and by Rudolf Rentschler, who did the same with Lie algebras. My heartfelt thanks to both of them. I also owe an enormous debt to Don Passman, Lance Small, and Bob Guralnick who shared their mathematical insights with me and have been good friends over many years. While working on this book, I was supported by grants from the National Security Agency and from Temple University. For corrections, suggestions, en- couragement, and other contributions during the writing process, I express my gratitude to Zachary Cline, Vasily Dolgushev, Karin Erdmann, Kenneth Good- earl, Darij Grinberg, Istvan Heckenberger, Birge Huisgen-Zimmermann, James Humphreys, Yohan Kang, Alexander Kleshchev, Donald Passman, Brian Rider, Louis Rowen, Hans-Jürgen Schneider, Paul Smith, Philipp Steger, Xingting Wang, and Sarah Witherspoon. I should also like to thank the publishing staff of the American Mathematical Society, especially Barbara Beeton for expert LATEX sup- port, Sergei Gelfand for seeing this project through to completion, and Meaghan Healy and Mary Letourneau for careful copyediting. My greatest debt is to my family: my children Aidan, Dalia, Esther, and Gabriel and my wife Maria, to whom this book is dedicated.

Philadelphia Martin Lorenz June 2018

Conventions

Functions and actions will generally be written on the left. In particular, unless otherwise specified, modules are left modules. Rings need not be commutative. Every ring R is understood to have an identity element, denoted by 1R or simply 1, → = and ring homomorphisms f : R S are assumed to satisfy f (1R) 1S.Wework over a commutative base field k. Any specific assumptions on k will be explicitly stated, usually at the beginning of a section or chapter. Here is some general notation frequently occurring in the text; a more compre- hensive list appears at the end of the book.

Z+ , R+ ,... nonnegative integers, reals, . . . N natural numbers, {1, 2,...} { } ∈ N [n] the set 1, 2,...,n for n disjoint union of sets #X number of elements if X is a finite set; otherwise ∞ X I set of functions f : I → X X (I) the subset of X I consisting of all finitely supported functions: f (i) = 0 for all but finitely many i ∈ I (for an abelian group X) G X short for G × X → X, a left action of the group G on X G\X the set of orbits for an action G X or, alternatively, a transversal for these orbits Vectk category of k-vector spaces ∗ V dual space of V ∈ Vectk,thatis,Homk(V, k) ∗  · , ·  evaluation pairing V × V → k ⊕ V I direct sum of copies of V labeled by I ⊗ V n nth tensor power of V GL(V ) group of invertible linear endomorphisms of V

xvii

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α-string, 388 anticommutative, 12, 22 abelianization, 583 artinian, 19, 40, 476 of a group, 126, 505, 583 associated graded, 269, 271, 276, 277, of a Lie algebra, 506 288, 311 of an algebra, 504 augmented, 564 action, 13, 441 central simple, 22, 76, 106 adjoint, 283, 286, 300, 393, 397, 400, 493, commutative, 4, 9, 40, 135 496, 546 connected, 507 on bimodules, 286, 468 defined over a finite field, 77 doubly transitive, 131 defined over a subfield, 65, 147 locally finite, 438, 443 filtered, 276, 277, 288, 311 multiplicative, 309, 366 finite-dimensional (or finite), 6 of a Lie algebra, 251 free, 6–9, 605 rational (of an algebraic group), 514 free commutative, 9 ad-nilpotent, 323, 334, 395 Frobenius, 96–109, 544, 564 ad-semisimple, 323 graded, 7, 8, 21, 178, 495 Adams operations, 491 graded-commutative, 12, 22 additive group, 456 local, 38, 96 functor, 506 noetherian, 19, 40, 271 adjunction, 583 opposite, 14, 96 admissible transposition, 208, 215, 224 PI (polynomial identity), 50, 64, 277, 440, Ado’s Theorem, 250 441 affine algebraic group, 503–540 prime, 63 connected, 520–525 primitive, 63 irreducible, 520–525 residually finite (or residually affine algebraic variety, 43 finite-dimensional), 440 affine group scheme, 504 self-injective, 564 affine scheme, 504 semiprime, 63, 64 algebra, 3 semiprimitive, 48, 63 affine, 9, 36, 40, 43, 95, 271, 497 semisimple, 51, 57, 63, 64 commutative, 601 split, 59, 102 almost commutative, 277, 288 separable, 65, 76, 105, 147, 553 alternating, 11, 22 simple, 22, 169

633 634 Subject Index

symmetric, 96–109, 122 associative, 316, 432 symmetric (of a vector space), 10, 138 G-invariant, 570 algebra of invariants, 174–178, 189, 282, invariant, 197, 568 394, 397–399, 401, 404, 408, 497, 499 nondegenerate, 316, 320, 432, 568 of an H-module algebra, 495 skew-symmetric, 568 algebra of polynomial functions, 175, 449, symmetric, 568 602 bilinear map, 592 algebraic n-torus, 511 balanced, 588, 592 algebraic number field, 222 bimodule, 15, 577, 589 ambiguity, 607 regular, 15, 95, 97, 577, 590 resolvable, 607 Borel subalgebra, 376 resolvable relative to ≤, 611 braid relation, 215 annihilator (of a subset of an algebra), 110 branching graph, 187, 192–197 anti-invariant, 368 branching rule, 200, 224, 229, 234 anticommutativity, 246 Brauer-Fowler Theorem, 164–166, 567 antipode, 451, 452 Bruhat order, 536, 538, 540 bijective, 477, 552 Burnside’s paqb-Theorem, 163, 164 of a group algebra, 135 Burnside’s Lemma, 134 of an enveloping algebra, 276, 279 Burnside’s Theorem, 60–63, 70, 77, 127, antisymmetrization map, 180 129, 161, 310, 564 antisymmetrizer, 180, 368, 484 Artin-Tate Lemma, 20, 41, 278, 499 Cartan homomorphism, 86, 88, 96 ascending central series, 254 of a finite-dimensional algebra, 92 associative law, 4, 15, 576 of a finite-dimensional Hopf algebra, 549, for tensor products, 590, 592 558 augmentation ideal, 447, 526 of the Sweedler algebra, 96 of a group algebra, 132 of the Taft algebra, 484 of an enveloping algebra, 275, 278 Cartan integers, 328, 333, 336, 339, 342 relative, 141, 277, 461, 479 Cartan matrix of a root system, 358, 362 augmentation map, 132, 275 type An, 336 automorphism type Bn, 339 − inner, 39 types An Dn, 358 of graphs, 199 Cartan subalgebra, 326 automorphism group, 576 Cartan’s Criterion, 263–265, 317 averaging operator, 134 Cartier-Gabriel-Kostant Theorem, 496 Axiom of Choice, 19, 582 Casimir element, 99, 101, 102, 123, 146, 160, 305, 306, 309, 558–561 base of a root system, 349 of a representation, 321 for the root system An, 350 universal, 321 for the root system Bn, 350 Casimir operator, 100, 101, 106, 110 for the root system Cn, 350 Casimir trace, 101, 102, 123, 558, 559 for the root system Dn, 351 Catalan number, 205 bi-ideal, 447 category, 575 bi-opposite concrete, 578 bialgebra, 448 locally small, 576 Hopf algebra, 451 monoidal, 137 bialgebra, 447–448 opposite, 579 bicommutant, 36 small, 576 bifunctor, 20, 136, 467, 469, 581, 588, 590, tensor, 137 593–595 Cauchon diagram, 536, 537 bilinear form Cauchy-Frobenius Lemma, 134 alternating, 246, 568 center, 248 Subject Index 635

of a bimodule, 468 grouplike, 429, 435, 439, 442, 449, 495 of a Lie algebra, 248 simple, 444 of an algebra, 3 coassociativity, 135, 276, 428, 431, 432 central closure, 525, 621 cobordism diagram, 431, 432 centralizer, 36, 130, 189, 224, 225, 238, 248, coefficient coalgebra, 446 400, 408 coform, 433 centralizing homomorphism, 50, 501 coideal, 428 chain condition left, 442 ascending (ACC), 19 right, 442, 447 for semiprime ideals, 49 coideal subalgebra descending (DCC), 19, 476, 606 left, 471, 564 for ideals, 620 right, 447, 467, 475 character, 66 coinduction functor, 28, 95, 117 central, 160, 291–292, 383–384, 400 coinvariants, 140 complex, 148 of a comodule, 466 formal, 173–174, 309, 391, 414 commutative diagram, 576 comodule, 13, 441 of an sl2-representation, 307–308 of Verma modules, 415, 416 irreducible, 444 irreducible, 70, 121 regular, 442 map, 74–75, 92, 486–487, 557 trivial, 466 of an induced representation, 123 comodule algebra, 492 of projectives, 92 complement, 18, 51 regular, 66, 76, 100, 102, 103, 107, 130, complete flag, 260 146, 552 composition (of a positive integer), 181, 230, character table, 121, 227 240 A weak, 181 of 4, 156 of A , 158 composition factor, 35, 65, 91 5 composition series, 32, 72, 91 of D , 128 4 comultiplication, 135, 275, 279, 428 of S , 129 3 conditional expectation, 40 of S , 154 4 conjugacy classes (of a group), 120 of S , 155 5 of A , 155 characteristic polynomial, 31, 174, 498 n of S , 153 Chevalley property, 483, 484, 497, 526 n p-regular, 121, 487 Chevalley Restriction Theorem, 399 Conjugacy Theorem, 395, 396, 399, 403 Chevalley’s Theorem, 483, 529 connected component (of a topological Chinese Remainder Theorem, 63, 70, 126, space), 523 127 content class equation, 561, 562 of a λ/μ-tableau, 223 class function, 120, 165 of a Young tableau, 206 class sum, 133, 160, 163 convolution, 434 classification of finite simple groups, 165, algebra, 434, 439 484, 497 core (of a prime ideal), 296 Clifford theory, 166–168 coroot, 349 Clifford’s Theorem, 157, 162, 167, 168 counit, 132, 275, 409, 428 co-opposite counit laws, 428, 432, 433 bialgebra, 448 Coxeter generators, 192, 538 coalgebra, 429, 442 crossed product, 166–169 coaction, 441 cyclotomic polynomial, 126 coalgebra, 428 co-opposite, 429, 442 Dedekind’s Lemma, 462 cocommutative, 428 degree 636 Subject Index

of a representation, 24 primitive, 448, 450, 452 of a word, 608 regular (of a Lie algebra), 396 total, 8, 9 unipotent, 258 deleted permutation representation, see elementary automorphism (of a Lie algebra), representation, deleted permutation 326, 395, 407 derivation, 20, 250, 281, 530 elementary divisor, 37 inner, 251 Engel’s Theorem, 257–260, 292, 297 left σ-, 20 enveloping algebra of a Lie algebra, outer, 251, 253, 257, 318 266–278, 436, 448, 612 derived series, 254 coalgebra structure, 430 descending central series, 254, 278 Hopf ideals of, 461 determinant, 238, 450 in positive characteristic, 277 diagonal action, 137 of sl2, 309–312, 605, 609 diagonal map, 275 restricted, 314, 449 diamond conditions, 607 equivalence Diamond Lemma, 534, 605–613 of categories, 14, 582, 583 diamond Lie algebra, 252, 253, 255, 257 of representations, 25, 39 dimension augmentation, 489 Erdős-Kaplansky Theorem, 469, 595 dimension of a representation, 24 exact sequence, 178, 593 direct product exponent of a group, 124 of algebras, 56, 109 extended center, 295, 617 of groups, 123 extension of scalars, 29, 39, 64, 65, 76 of Lie algebras, 251, 274 exterior algebra, 11, 138 of modules, 95 direct sum of representations, 24 Φ-convexity, 388 distinguished grouplike element, 543 Φ-extremal element, 388 division algebra, 30 f.c. (finite conjugate) center, 140, 501 Dixmier-Mœglin equivalence, 295, 297, 299, Feit-Thompson Theorem, 164 405, 615 Ferrers diagram, 198 domain, 20, 270, 276 field extension double centralizer, 36, 60 finite, 66 Double Centralizer Theorem, 235 separable, 65, 66, 76 dual bases, 81, 595 filtration for a Frobenius form, 98, 99 of a representation, 41 Dual Bases Lemma, 81, 83 of a vector space, 269 dual numbers, 527 of an algebra, 269, 270 dual root system, 349 finite dual, 68, 436, 440, 441 duality functor, 579 of a bialgebra, 453–457 Duflo isomorphism, 394 of a group algebra, 455 Dynkin diagram, 358 of a Hopf algebra, 453–457, 486, 509 − of a polynomial algebra, 455 types An Dn, 358 Finiteness Theorem for coalgebras and eigenspace, 54 comodules, 444, 476, 565 generalized, 262 fixed point, 130 Eilenberg swindle, 95 forgetful functor, 7, 578, 585 element formal linear combinations, 584 grouplike, 429, 448, 450, 451, 457, 459, forms of degree n, 183 475, 543, 550 Frobenius endomorphism, 69 modular (right or left), 543, 544, 549, 550, Frobenius extension, 110, 117 566 of Hopf algebras, 566 nilpotent, 258 Frobenius form, 98, 544 normal, 47, 276, 290, 298, 457, 620 Frobenius laws, 434, 447 Subject Index 637

Frobenius reciprocity, 28, 61, 119, 144, 200, abelian, 84, 115, 124–126, 131, 160, 162, 224, 231 168, 461 Frobenius’ Divisibility Theorem, 161–163, algebraic, see affine algebraic group 559, 560, 562 alternating, 127, 155–158, 164, 176 Frobenius’ formula, 169 cyclic, 69, 115, 125, 505, 506 Frobenius-Schur indicator, 165, 169, 567, dihedral, 127, 129, 131, 349, 408, 491 568, 571 Klein 4-group, 153 functor, 578, 588, 590, 593 quaternion, 131, 491 adjoint, 7, 10, 115, 267, 437, 440, 452, 583 simple, 148, 164–166, 169, 462, 484, 497 contravariant, 578, 595 solvable, 163 covariant, 578 symmetric, 12, 127–129, 131, 150–155, essentially surjective, 582 177, 179–185, 187–242, 283, 308, 346, exact, 81, 124, 589 388, 479, 536, 538 left, 81, 593 group algebra, 113, 114, 365, 451 right, 589 Hopf subalgebras and Hopf ideals of, 461 faithful, 578 group functor, 504 full, 578 group ring, 116 Fundamental Theorem integral, 485 S of n-invariants, 371, 404 group scheme, 503 on coalgebras, 443 grouplike element, see element, grouplike groupoid, 577 G-algebra, 138, 282, 492 g-algebra, 282, 492 G-core, 531 H-core, 500, 501, 538–540 G-ideal, 531 H-hull, 500 G-prime ideal, 531 H-ideal, 500 G-set, 129, 140, 141, 149 H-prime, 500 Galois descent, 64 H-simple, 501 Gelfand-Zetlin (GZ) algebra, 188, 189, 195 Harish-Chandra homomorphism, 400 Gelfand-Zetlin (GZ) basis, 194, 196, 197, Harish-Chandra isomorphism, 401, 404, 408 218 Hattori’s Lemma, 124 Hattori-Stallings rank, 87, 92, 94, 124 of Vn−1, 194, 197, 219, 221, 222 general linear group, 506 Hattori-Stallings trace, 83–84 general linear Lie algebra, 247 head (of a representation), 89, 381 generalized matrix algebra, 16 heart (of a prime ideal), 296 generating function, 170, 172 height (of a root), 356 generators Heisenberg Lie algebra, 250, 252, 254, 262, of a module, 18 265, 278, 293 of an algebra, 6 Hermite reciprocity, 313 Generic Flatness Lemma, 289, 601, 603 Hermitian inner product, 146 generic matrix, 457, 459 Hessian, 314 generic point, 45, 48, 291 highest root, 389 Goodearl-Letzter stratification, 533 highest weight, 379 grading, 21 representation, 379–385 Graph Isomorphism Theorem, 187, 188, vector, 380 198, 199, 205, 215–217 Higman ideal, 123 Grothendieck group Higman trace, 100, 101, 110 of finite-dimensional projectives, 85 Hilbert Basis Theorem, 19, 20, 271 of finite-dimensional representations, 72, Hilbert series, 175, 178 285, 308, 390, 485 Hilbert-Serre Theorem, 178 of finitely generated projectives, 84 Hochschild (co)homology, 287 group homogeneous 638 Subject Index

component (of a representation), 53–56, identity component (of an affine algebraic 60, 168, 197 group), 523 element (in a graded algebra), 8 identity element, 4, 577 homomorphism identity functor, 578 of algebras, 4 induction functor, 27, 39, 117, 123, 124, 166 of bialgebras, 447 inflation (of a representation), 27, 48, 58, 73 of coalgebras, 428 inner product, 332 of comodules, 441 Hermitian, 149, 570 of graded algebras, 8 integral of Hopf algebras, 451 in a Hopf algebra, 542 of Lie algebras, 246 of the representation ring, 490 of modules, 13 integral closure, 104 of representations, 25 integrality, 104, 160, 162, 163, 499, 560 hook, 201 intertwine, 25 length, 201 invariant theory, 170, 175 partition, 205 multiplicative, 309, 366 walk, 202 noncommutative, 492 Hook-Length Formula, 188, 201 invariants, 64, 398 Hopf algebra, 134, 273, 451 multiplicative, 309, 366 almost cocommutative, 474, 485 of the weights lattice for An, 371 bi-opposite, 451 of the weights lattice for Cn, 371 bisemisimple, 554 of a group representation, 132 cocommutative, 135, 276, 467, 484, 485, of a Hopf algebra representation, 466 490, 491, 494 of a Hopf representation, 466 involutory, 453, 460, 462, 475, 480, 549 of a Lie algebra representation, 258, 279 self-dual, 548 of a permutation representation, 132 trigonometric, 462 of an sl2-representation, 313 unimodular, 542, 543, 549, 562 of outer tensor products, 140 Hopf ideal, 451 polynomial, 175 Hopf kernel, 479 inversions (of a permutation), 209, 217, 355 Hopf module, 471 involution, 164, 393, 489 relative, 471, 475, 563 standard (of a group algebra), 135, 192 Hopf subalgebra, 451 standard (of an enveloping algebra), 276 Hopkins-Levitzki Theorem, 477 irreducible components icosahedron, 176 of a root system, 358 ideal of a topological space, 521 cofinite, 62, 436 irreducible constituent (of a representation), completely prime, 310, 311, 405, 534 55 locally closed prime, 290, 296, 311 isomorphism, 576 maximal, 44 functorial, 581 nil, 88, 289 natural, 581 nilpotent, 48, 63, 289 of categories, 582 of a Lie algebra, 248 of functors, 110, 581 prime, 44 of graphs, 199 primitive, 42, 287 of root systems, 342 right vs. left, 42, 294 Isomorphism Theorem (for Lie algebras), rational, 294, 296 248 semiprime, 44, 49, 287, 288, 600 isotropy group, 141, 159 idealizer, 620 Itô’s Theorem, 160, 166, 168 idempotent, 75, 81, 95, 105, 558 primitive central, 60, 103, 146, 196, 560 Jacobi identity, 246, 247, 250, 283, 613 Subject Index 639

Jacobson property, 287, 288, 290, 296, 300, reductive, 323, 325, 334, 529 522, 601 semisimple, 256, 315 Jacobson radical, 48, 63, 69, 289, 483, 496 simple, 256, 257, 315, 316, 319, 321, 331, Jacobson-Morozov Lemma, 314 336, 338, 339, 375 Jacobson-Zariski topology, 45–50, 290, 291, solvable, 253–257, 260, 262, 263, 265, 293, 300, 406, 531 266, 278, 297, 317, 319, 377 Jordan canonical form, 26, 262–263 Lie bracket, 246 Jordan decomposition, 263, 266 Lie commutator, 59, 67, 246, 250 abstract, 324, 327 Lie subalgebra, 247 preservation of, 324 Lie’s Theorem, 260–262, 264–266, 279, Jordan-Chevalley decomposition, 263 297–299, 323, 379 Jordan-Hölder Theorem, 33–35, 85, 91, 298 linear algebraic group, 517–519 Jucys-Murphy (JM) elements, 189 linear dual, 595 linear form, 595 k-field, 5, 22, 296 linear recursion, 456 kernel (of a representation), 24 locally finite part (of a representation), 443 Killing form, 316, 321, 409 longest element (of a Weyl group), 356 Kolchin’s Theorem, 258 Koszul sign rule, 22 MacMahon Master Theorem, 172 Kronecker product, 597 marked cycle shape, 190 Krull-Schmidt Theorem, 38, 40, 85, 91, 564, Markov chain, 202 565 Maschke’s Theorem, 116, 117, 123, 143, 147–150, 164, 179, 188, 238, 482, 483, λ-ring, 490 487, 552–554, 566 λ-tableau, 200 for Hopf algebras, 554, 556, 558, 559, 567 λ/μ-tableau, 222 relative, 148, 530, 566 Lagrange’s Theorem, 563 matrix lattice, 115 monomial, 526 Laurent polynomial algebra, 115, 366 orthogonal, 220 Le-diagram, 537 permutation, 128 Leibniz formula, 253, 395, 407 maximal vector, 379 Leibniz identity, 246, 250 maximum condition, see chain condition, Leibniz product rule, 250, 281, 282 ACC length Michler’s Theorem, 497 of a permutation, 209, 217, 355 minimal polynomial, 51 of a representation, 34, 53 minimum condition, see chain condition, of sl2, 303 DCC of a Weyl group element, 354 misordering index, 612 Levi decomposition, 256, 266 modular element, see element, modular Levitzki’s Theorem, 258 module, 13–16 Lie p-algebra, 448, 526 artinian, 19, 41 Lie algebra, 246 cyclic, 18, 53 2-dimensional nonabelian, 265, 266, 278, finite-length, 41 290, 293, 299 finitely generated, 17, 18 abelian, 246, 249, 251, 254–256, 267, flat, 589, 601 287, 325, 326 free, 17, 19, 80, 473, 584, 589, 591 classical, 334 graded, 21, 178 linear, 247, 249, 250, 257–265 graded free, 178 nilpotent, 253–255, 257–259, 262, 265, injective, 81, 94, 107, 110 278, 292–294, 296, 297, 319, 377 locally finite, 443 of an affine algebraic group, 526–530 noetherian, 19, 41 opposite, 274 over a PID, 37, 38 640 Subject Index

projective, 79, 80, 107, 110, 548, 589 On-Line Encyclopedia of Integer Sequences, module algebra, 282, 492 197 Molien’s Theorem, 175–178, 408 open problem, 271, 553, 559 monoid, 116, 495, 577 opposite monoid algebra, 116, 448, 449, 495 algebra, 14 monoid ring, 116 bialgebra, 448 total, 116 category, 579 monomial, 6, 605 group, 123, 135 standard, 11, 12, 172, 272, 430, 436, 460 Lie algebra, 274 Morita context, 95 orbit sum, 133, 189, 366 associated to a Hopf action, 549 order morphism Bruhat, 538, 540 functorial, 580 degree-lexicographic, 608 natural, 580 isomorphism, 536 of functors, 580 lexicographic, 608 multilinear map, 592 of a basis, 272, 430 symmetric, 183 partial, 75, 198, 363, 375, 408 multiplication, 4 semigroup, 606 multiplicative group functor, 506 orthogonal idempotents, 88 multiplicity, 55, 145, 377 orthogonality relations, 92, 144, 145, 149, of a weight, 302, 307 158, 160, 165 of composition factor, 35, 91 column, 148 Multiplicity-Freeness Theorem, 187, 192, for Hopf algebras, 557, 558 193 generalized, 149 multiplier algebra, 620 orthonormal basis, 144 Murnaghan-Nakayama Rule, 188, 222, p-core, 148, 483 227–231, 233, 234 p-regular conjugacy class, 121, 169 Nakayama automorphism, 98–100, 546 element, 121 Nakayama relations, 119 group, 488 Nakayama twist, 108 pairing, 86, 449 natural equivalence, 581 partition (of a positive integer), 153, 198 natural transformation, 579 conjugate, 198 naturality condition, 580 PID, 222 Newton identities, 171, 228, 419 pipe dream, 537 Nichols-Zoeller Theorem, 541, 550, 559, place permutations, 179 562–565 plethysm, 135 nilpotency class (of a Lie algebra), 254 Poincaré series, 175 nilradical (of a finite-dimensional Lie Poincaré-Birkhoff-Witt Theorem, 269–275, algebra), 257 277, 278, 283, 286, 288, 293, 302, 311, Noether’s Finiteness Theorem, 497 380, 400, 401, 430, 612 Noether-Deuring Theorem, 29, 39 polarization, 183–185, 394 normal element, see element, normal polynomial function, 602 normalizer, 248 invariant, 394 of a Lie subalgebra, 259 on GLn, 457 Nullstellensatz, 43, 296, 300, 383, 510, 517, on SLn, 457 521, 522, 599, 600 poset, 19 for enveloping algebras, 287–290, 292, positive cone, 75 296, 310, 405 power series, 116, 436 weak, 36, 41, 45, 287–289, 291, 296, 521, power sum, 171 600 primitive element, see element, primitive Subject Index 641

principal indecomposable representations, simple, 354 91, 94, 564 regular elements of a Lie algebra, 396 principal open subset, 603 representation probability, 202 absolutely irreducible, 61, 145, 160–162, product of categories, 581 361, 375 projective, 79 adjoint, 251, 311, 320 cover, 91 of a group, 130, 133, 149, 497 representation, 79, 168 of a Hopf algebra, 493, 496, 548, 558 pseudo-reflection, 404 of a Lie algebra, 247, 249, 251, 256, group, 404 283, 389, 394 pulling back (a representation), 27, 48, 73 of an algebraic group, 530 pushing forward (a representation), 27 completely reducible, 51–53, 64, 141 complex, 113 q-binomial coefficient, 463 defining quantum of sl , 301 2 × 2-matrices, 609 2 of sln+1, 388 GLn, 459 deleted permutation, 128, 131 SLn, 459 faithful, 24, 64, 284, 302 n affine -space, 458, 533, 610 G-faithful, 479 affine n-torus, 534 g-faithful, 269, 284, 479 binomial formula, 463 inner faithful (of a Hopf algebra), 479 determinant, 459 of a group, 479 group, 299 of a Lie algebra, 479 plane, 23, 47, 50, 96, 300, 458, 460, 533 finite-dimensional, 24 torus, 23 finite-length, 33, 41, 52, 384 quasi-inverse, 582 finitely generated, 52 quasihomeomorphism, 300 fundamental (of a semisimple Lie quaternion algebra, 131, 571 algebra), 387 quotient representation, 24 of sln+1, 387 R-points, 504 indecomposable, 37, 91, 306 Rabinowitsch trick, 289, 600 induced, 123 Radford’s formula, 541, 549, 562 injective, 94, 110, 564 radical injecture, 107 Jacobson, see Jacobson radical irreducible, 30, 247 of a finite-dimensional Lie algebra, 256, degree of, 41, 105, 119, 159–162, 201, 266 205, 496, 497, 559–561 of a representation, 63 finite-dimensional, 60–62 of the Killing form, 316 Frobenius-Schur indicator of, 569, 571 semiprime (of an ideal), 49, 67, 599 of sl2, 302, 313, 314 random walk, 202 of so5, 417 F rank of SL2( p) in characteristic p, 169 of a finitely generated projective, 83 of a p-group in characteristic p, 126 of a free module, 591 of a commutative algebra, 43 of a linear map, 595 of a Hopf algebra, 480, 482, 484, 496 of a root system, 342 of a Lie algebra in characteristic p, 278 of a semisimple Lie algebra, 326, 332 of a semisimple Lie algebra, 373–423 rational canonical form, 26 of a solvable Lie algebra, 260 reduced trace, 106 of the 2-dimensional nonabelian Lie reduced word, 538 algebra, 265, 299 A reduction system, 606 of the alternating group 4, 156 A reflection, 151, 344, 404 of the alternating group 5, 157–158 642 Subject Index

C Q of the cyclic group n over , 125 ring of algebraic integers, 222 D of the dihedral group 4, 127 ring of quotients of the general linear group, 235–242 classical, 620 of the Heisenberg Lie algebra, 252 left, 619 of the polynomial algebra, 31 maximal, 620 of the quantum plane, 50 right, 619 of the Sweedler algebra, 96 symmetric, 295, 615, 619 S root, 328 of the symmetric group 3, 129 S negative, 350 of the symmetric group 4, 153 S positive, 350 of the symmetric group 5, 154, 159 S simple, 350 of the symmetric group n, 131, 150, 187–242 root lattice, 361 of the Weyl algebra, 30, 41, 252, 262 for Bn, 370 locally finite, 303, 320, 396, 438, 443 for Dn, 370 modular, 113, 121 for the root system Cn, 370 of a group, 113 root space, 328 of a Lie algebra, 247, 257 decomposition, 325, 328, 376 of an algebra, 24 root string, 328 permutation, 130, 132, 137, 140, 141 root system, 333, 342 projective, 79, 107, 110, 168 irreducible, 331, 356 rational (of an affine algebraic group), of rank 2, 343 513, 529 of type A1, 343 regular, 25, 30, 66, 130, 146 of type A2, 337 bimodule, 25, 97 roots of unity, 506 right, 25 Schur division algebra, 36, 41, 56, 58, 124, self-dual, 137, 150, 197, 469 235, 288 semisimple, 51 Schur functor, 239, 419 simple, 30 Schur’s Double Centralizer Theorem, 238, standard 239, 241, 242, 419 S of 3, 147, 177 Schur’s Lemma, 35, 36, 55, 56, 58, 61, 92, S of n, 128, 131, 150–153, 194, 197, 117, 143, 144, 194, 227, 288, 322, 324, 205, 216, 217, 219, 221, 222, 228, 334, 529, 557, 569, 570, 618 392 semi-invariant, 132, 279 of the Weyl algebra, 26, 30, 41, 265 semidirect product, 278 S standard permutation (of n), 128, 130, of Lie algebras, 251, 286 150, 217, 392 semisimplification, 63, 69 trivial, 466 Shephard-Todd-Chevalley Theorem, 404, of a group, 117, 130 408 of a Hopf algebra, 466 short exact sequence, 18, 19, 33, 66, 81, 124, of a Lie algebra, 279 446, 490, 589 of the representation algebra, 561 split, 18 representation algebra, 485, 556 sign representation, 128 representation ring of Weyl groups, 368 of sl2, 308 skeleton (of a category), 582 of a Hopf algebra, 485 skew hook, 227 of a Lie algebra, 286, 308, 390 skew Laurent polynomial algebra, 20, 23 representative functions, 455, 462 skew polynomial algebra, 20, 278 restitution, 183–185 skew shape, 222 restriction functor, 27, 39, 44, 117, 166 Skolem-Noether Theorem, 555 Reynolds operator, 40, 118, 566 sl2-triples, 188, 213, 300, 328, 336 Rieffel’s Theorem, 480 smash product, 286, 495 Subject Index 643

snake relations, 433 complete, 170 socle, 53, 108, 377 elementary, 170, 392 series, 110 symmetrization map, 180, 284 special linear group, 161, 507 symmetrizer, 180, 484 special linear Lie algebra, 249 symmetry group, 127, 129, 176 spectrum, 44–49, 520–522 action of an algebraic group on, 530–538 Taft Hopf algebra, 460, 484, 542, 543, 548 tensor cofinite, 62 algebra, 6–9, 138, 281, 494 connectedness of, 508 antisymmetric, 179, 484 of U(g) for semisimple g, 406 category, 280 of U(sl ), 310–312 2 power, 8, 138, 281, 494, 590, 592 of Ug for nilpotent g, 292 product, 577, 587 of Ug for the 2-dimensional nonabelian of algebras, 5, 15 Lie algebra g, 290 outer, 67, 77, 137, 140, 230 of Ug for the Heisenberg Lie algebra, 293 product formula (for group of k[x, y], 46 representations), 142 of k[x], 45 symmetric, 179, 484 of a convolution algebra, 439 tensor category, 280 of quantum affine space, 533–536 tensor power, 6 of quantum matrices, 536–538 topological space of the Gelfand-Zetlin algebra, 196–197, connected, 520 211–214 irreducible, 49, 520 of the quantum plane, 47 totally nonnegative, 554 splitting field, 36, 37, 59, 61, 65, 70, 74, 77, trace, 26, 59, 66, 100–101 92, 93, 117, 124, 131, 144, 148, 155, form, 67–69, 106, 109, 120, 122, 486, 219, 497, 557 543, 548, 549 splitting map, 18 Hattori-Stallings, 83–84 splitting principle, 488 map, 83, 249, 475, 480, 596–597 stable isomorphism, 85, 96 relative, 148, 566 standard filtration (of an enveloping algebra), universal, 68, 76, 100 270–272 transporter, 248 standard Young tableau, 200, 222 transpose of a linear map, 139, 579, 595 Stirling numbers, 536 triangular decomposition, 375, 376 Structure Theorem for Hopf Modules, twist (of a representation), 29, 39, 108, 142, 471–474, 476, 478 150, 167, 423, 470, 475, 548 subalgebra, 4 cofinite, 40 unipotent, 258 subbialgebra, 447 unit, 4 subcategory, 577 axioms (laws), 4, 576 full, 577 of an adjunction, 267, 276 subcoalgebra, 428 universal property, 18, 114 subcomodule, 442 of enveloping algebras, 266 subgroup scheme, 506 of the free algebra, 9 subrepresentation, 24, 247 of the polynomial algebra, 9 Sweedler (-Heyneman) notation, 431 of the tensor algebra, 7 Sweedler (Hopf) algebra, 96, 109, 461, 474, of the tensor product, 588 500, 566 Vandermonde determinant, 412 Sweedler dual, see finite dual vector partition function, 380 Sweedler power, 491, 567 Verma module, 381, 415 symbol, 270, 273, 311 symmetric polynomial, 170 walls, 353 644 Subject Index

Wedderburn’s Structure Theorem, 3, 30, 51, 57, 59, 60, 62, 63, 91, 102, 103, 146, 153, 154, 236, 316, 554, 561 weight dominant, 364 for Dn, 370 for the root system Bn, 417 for the root system Cn, 370 fundamental, 362, 374 for An, 363 for Bn, 370 strongly dominant, 364, 368 weight diagram, 388 weight lattice, 362, 365, 374, 378 weight space, 54, 132, 142, 147, 260, 298, 302, 377, 515 generalized, 303 weight vector, 132, 196, 260, 279, 377 weights (of a representation), 54, 196, 197, 211–214, 217, 302, 307, 377 Weyl algebra, 12–13, 23, 26, 31, 41, 205, 262, 293, 314, 605, 608 standard representation of, 26, 30, 265 Weyl chamber, 353 fundamental, 354, 364 Weyl group, 345, 375, 398 of An, 346 of Bn, 348 of Cn, 348 of Dn, 349 Weyl’s Character Formula, 242, 368, 384, 391, 408–410, 413, 414, 416–418 Weyl’s Theorem, 320, 323, 324, 373, 380, 391, 405, 418, 529 winding automorphism, 142, 401, 475 Witt algebra, 253, 271 words, 6, 605

Young diagram, 187, 198 Young graph, 187, 197–204 Young module, 181 Young subgroup, 152, 181, 230, 231, 239, 422 Young’s orthogonal form, 188, 221, 225

Zariski dense, 377, 603 Zariski topology, 45, 182, 377, 399, 402, 516, 521, 602, 603 Zariski’s Lemma, 600 Zorn’s Lemma, 19, 52, 565 Index of Names

Adams, J. Frank, 491 Coxeter, Donald, 192 Ado, Igor, 250 Cuadra, Juan, 560 Adámek, Jiří, 575 Curtis, Charles, 79, 159 Amitsur, Shimshon, 619 Anan’in, Alexander, 441 Dedekind, Richard, 462 Artin, Emil, 19, 20 Demazure, Michel, 505 Aschbacher, Michael, 165 Derksen, Harm, 175 Auslander, Maurice, 79 Deuring, Max, 39 Dixmier, Jacques, 293, 296, 297, 299, 538 Bavula, Vladimir, 310 Drinfel’d, Vladimir, 474 Bergman, George, 50, 525, 605, 607 Duflo, Michel, 288, 394 Birkhoff, Garrett, 272 Dynkin, Eugene, 358 Block, Richard, 30 Eilenberg, Samuel, 95, 287 Borel, Armand, 376, 503 Engel, Friedrich, 258 Borho, Walter, 296, 297, 406 Erdős, Paul, 595 Bourbaki, Nicolas, 263, 341, 366, 449 Erdmann, Karin, 141, 341 Brauer, Richard, 121, 124, 164, 169 Etingof, Pavel, 554 Brown, Kenneth, 299, 459 Burnside, William, 60, 134, 163, 164, 480 Ferrer Santos, Walter, 498 Fowler, Kenneth A., 164 Cartan, Élie, 263 Frame, J. Sutherland, 201 Cartan, Henri, 263, 287 Frobenius, Ferdinand Georg, 97, 98, 134, Cartier, Pierre, 496 160, 187, 568 Casimir, Hendrik, 99, 100, 305, 320 Fuchs, Jürgen, 568 Catoiu, Stefan, 310 Fulton, William, 538 Cauchon, Gérard, 536 Cauchy, Augustin-Louis, 134 Gabriel, Pierre, 296, 297, 310, 496, 505 Chevalley, Claude, 263, 399, 404, 483, 526, Ganchev, Alexander, 568 529 Gelaki, Shlomo, 554 Chin, William, 531 Gelfand, Israel, 188, 189, 194 Clebsch, Alfred, 313 Goodearl, Kenneth, 299, 459, 533, 538 Clifford, Alfred, 166 Goodman, Roe, 235, 418 Connell, Ian, 476 Gordan, Paul, 313

645 646 Index of Names

Gorenstein, Daniel, 165 Liu, Chia-Hsin, 476 Green, James, 418 Greene, Curtis, 201 Mac Lane, Saunders, 276, 575, 583 Grothendieck, Alexandre, 490, 498, 601 Manin, Yuri, 458 Markov, Andrey, 202 Harish-Chandra, 401 Martindale, Wallace, 619 Hattori, Akio, 83 Maschke, Heinrich, 143, 552 Heckenberger, Istvan, 566 Masuoka, Akira, 562 Hermite, Charles, 313 Meir, Ehud, 560 Herrlich, Horst, 575 Michler, Gerhard, 497 Hesse, Ludwig, 314 Molien, Theodor, 175, 176 Higman, Donald, 100, 106 Montgomery, M. Susan, 449, 568 Hilbert, David, 19, 43, 175, 178, 497, 599 Morita, Kiiti, 95 Hochschild, Gerhard, 287, 503, 529, 620 Morozov, Vladimir, 314 Hopf, Heinz, 451 Murnaghan, Francis, 227 Hopkins, Charles, 477 Murphy, Gwendolen, 189 Humphreys, James, 341, 503 Mœglin, Colette, 296, 299 Hölder, Otto, 33 Nakayama, Tadasi, 98, 108, 119, 227, 546 Irving, Ronald, 299 Newton, Isaac, 171 Itô, Noboru, 162 Nichols, Warren, 461, 563 Nijenhuis, Albert, 201 Jacobi, Carl, 246 Noether, Emmy, 19, 39, 497, 555 Jacobson, Nathan, 45, 48, 222, 277, 287, Nouazé, Yvon, 310 314, 601 Jantzen, Jens Carsten, 503 Okounkov, Andrei, 187, 188 Jordan, Camille, 26, 33, 262, 263, 323 Olshanski˘ı, Grigori˘ı, 189 Jucys, Algimantas, 189 Passman, Donald, 168, 620 Kac, George, 561 Poincaré, Henri, 175, 272 Kaplansky, Irving, 258, 553, 559, 562, 595 Procesi, Claudio, 183, 235, 418 Kashina, Yevgenia, 567 Quillen, Daniel, 288 Kemper, Gregor, 175 Kharchenko, Vladislav, 615 Radford, David, 541, 549, 554, 555 Killing, Wilhelm, 316 Rainich, George, 600 Kirillov, Alexandre, 310 Reiner, Irving, 79 Klein, Felix, 153 Reiten, Idun, 79 Kock, Joachim, 434 Remak, Robert, 37 Kolchin, Ellis, 258 Rentschler, Rudolf, 296, 297, 299 Kostant, Bertram, 496 Reynolds, Osbourne, 40 Koszul, Jean-Louis, 22 Rieffel, Marc, 480 Kronecker, Leopold, 597 Robinson, Gilbert de Beauregard, 201 Krull, Wolfgang, 37 Rowen, Louis, 440

Larson, Richard, 541, 554, 555 Schneider, Hans-Jürgen, 541, 547 Launois, Stéphane, 536 Schur, Issai, 161, 168, 187, 235, 418, 568 Leibniz, Gottfried Wilhelm, 246, 250, 253 Serre, Jean-Pierre, 178, 341, 481 Lenagan, Thomas, 538 Shephard, Geoffrey, 404 Letzter, Edward, 533 Sierra, Susan, 271 Levi, Eugenio, 256, 266 Skolem, Thoralf, 555 Levitzki, Jacob, 258, 289, 477 Smalø, Sverre, 79 Lie, Sophus, 246, 260 Small, Lance, 299 Linchenko, Vitaly, 568 Solomon, Ronald, 165 Index of Names 647

Sommerhäuser, Yorck, 567 Springer, Tonny, 503 Stallings, John, 83 Strecker, George, 575 Sweedler, Moss, 96, 428, 431, 436, 461, 491, 567 Szlachányi, Kornél, 568

Taft, Earl, 459 Tate, John, 20, 161 Thrall, Robert, 201 Todd, John, 404

Vecsernyés, Péter, 568 Verma, Daya-Nand, 381 Vershik, Anatoly, 187, 188

Wallach, Nolan, 235, 418 Walton, Chelsea, 271 Waterhouse, William, 503 Wedderburn, Joseph, 57 Weyl, Hermann, 12, 183, 235, 320, 344, 353, 408, 418 Wildon, Mark, 341 Wilf, Herbert, 201 Witt, Ernst, 253, 271, 272

Yakimov, Milen, 536 Young, Alfred, 152, 187

Zariski, Oscar, 45, 516, 521, 600, 602 Zetlin, Michael, 188, 189, 194 Zhang, James, 476 Zhu, Shenglin, 560 Zhu, Yongchang, 561, 562, 567 Zoeller, M. Bettina, 563

Notation

The list below contains the principal notation used in this book, organized by the context in which they arise. Parenthetical references at the end of an entry indicate where the notation is introduced; if an entire section is referenced, then the opening paragraph is understood.

General

Z+ , R+ ,... nonnegative integers, reals, . . . N natural numbers, {1, 2,...} F q field with q elements k base field ⊗ tensor product over k [n] the set {1, 2,...,n} for n ∈ N th μn group of n roots of unity disjoint union of sets

Sets. For given sets X and I, we use the following notation:

#X number of elements if X is finite; otherwise ∞ X I set of functions f : I → X X (I) the subset of X I , for an abelian group X, consisting of all finitely supported functions: f (i) = 0 for almost all i ∈ I kX or k[X] k-vector space of formal k-linear combinations of X (Example A.5) kX free k-algebra generated by the set X (§1.1.2)

649 650 Notation

Categories. Names of categories are generally chosen to be self-explanatory—e.g., AffineAlgebraicGroupsk for the category of affine algebraic groups over k.More economical notation is adopted for some frequently occurring categories.

AbGroups abelian groups Δ Algk, Algk k-algebras, Δ-graded k-algebras (Exercise 1.1.12) CommAlgk commutative k-algebras HopfAlgk Hopf k-algebras Liek Lie k-algebras AMod, ModA left, right A-modules AModB (A, B)-bimodules AProj projective left A-modules (§2.1.1) Aproj finitely generated projective left A-modules (§2.1.1) Aprojfin finite-dimensional projective left A-modules (§2.1.1) Rep A, Repfin A representations, finite-dimensional representations of A Δ Vectk, Vectk k-vector spaces, Δ-graded k-vector spaces (Exercise 1.1.12)

Vector Spaces. Let V be a k-vector space.

∗ V dual space, Homk(V, k) (§B.3.2) ∗  · , ·  evaluation pairing V × V → k ⊕ ⊕ V I , V n direct sum of copies of V labeled by the set I and n-fold direct sum of V ⊗ V n nth tensor power of V GL(V ) group of invertible endomorphisms of V SL(V ) group of endomorphisms of V having determinant 1 ⊗ TV, TnV tensor algebra of V and its nth component, V n (§1.1.2) STnV, AT nV spaces of symmetric and antisymmetric n-tensors (§3.8.1) Sym V, SymnV symmetric algebra of V and its nth component (§1.1.2)   V, nV exterior algebra of V and its nth component (§1.1.2) ∗ O(V ) = Sym(V ) algebra of polynomial functions on V (Section C.3)

Algebras. Let A be a k-algebra.

× A group of units (invertible elements) of A Aop opposite algebra Z A center of A { ∈ | = ∈ } ⊆ CA(X) a A ax xa for all x X , the centralizer of X A ∞ Areg , χreg regular representation of A and its character (if dimk A < ) Irr A,Irrfin A set of equivalence classes (or a full representative set) of Notation 651

irreducible and finite-dimensional irreducible representations R(A), Rk(A) Grothendieck group of finite-dimensional representations and R(A) ⊗ k (§1.5.5) MaxSpec A set of maximal ideals of A (§1.3.2, §1.3.3) Spec A primeidealsofA (§1.3.3) Prim A primitive ideals of A (Section 1.3) rad A Jacobson radical (§1.3.5) s.p. A = A/ rad A semiprimitive quotient (“semisimplification” if dimk A < ∞) ◦ A finite dual (§1.5.2) ∗ ◦ Atrace , Atrace spaces of trace forms and finite trace forms of A (§1.5.2) C(A) trace forms vanishing on a cofinite semiprime ideal (§1.5.2) Tr : A  A/[A, A] universal trace (§1.5.2) × Matn(A) monoid of n n-matrices over A × GLn(A) group of invertible n n-matrices over A ALie underlying Lie algebra of A (Example 5.2) gr A graded algebra associated to a filtration of A (§5.4.3)

Representations. Let V be a representation of some algebra A and let S be an irreducible representation of A.

∗ A ↓A → φ V,ResB V, V B pullback (restriction) along φ: B A (§1.2.2) B ↑B → φ∗V,IndA V, V A pushforward (induction) along φ: A B (§1.2.2) Vλ weight space (eigenspace, semi-invariants; Example 1.30) αV α-twist of V (§1.2.2) ∞ χV character of V (if dimk V < ) Ker V {a ∈ A | a.V = 0}, the kernel of V rad V radical of V (Exercise 1.4.1) head V = V/ rad V head of V (§2.1.4) soc V socle of V (§1.4.2) length V (composition) length of V (§1.2.4) μ(S, V ) multiplicity of S in a composition series of V (§1.2.4) V (S) S-homogeneous component of V (§1.4.2) m(S, V ) length V (S) (§1.4.2) BiEndA(V ) bicommutant (double centralizer) of V (§1.2.5) D(S) Schur division algebra, EndA(S) (§1.2.5) PV projective cover (§2.1.4) th νn(V ) n Frobenius-Schur indicator of V (Lemma 3.31, §12.5.1) V A A-invariants in V (for an augmented algebra A; §10.1.1) 652 Notation

Groups. The cyclic group of order n, the dihedral group of order 2n, the symmetric C D group of degree n, and its alternating subgroup are respectively denoted by n, n, S A n,and n. In general, for a group G, the following notation is used:

Gab abelianization of G (§3.2.2) Op (G) p-core of G (Exercise 3.4.6) G X short for a left action G × X → X on the set X ∈ Gx isotropy group (stabilizer) of x X for an action G X G\X the set of orbits for an action G X or, alternatively, a transversal for these orbits G/H the collection of all left cosets gH (g ∈ G) of a subgroup H ≤ G; alternatively, a transversal for the left cosets Gg conjugacy class of g ∈ G cfk(G) vector space of k-valued class functions on G (§3.1.5) kG or k[G] group algebra of G over k V G G-invariants of a representation V (§3.3.1)

Symmetric Groups

S S m Young subgroup of n associated to a composition m of n (§3.8.2) P n set of partitions of n λ % n λ is a partition of n + ∈S si the transposition (i, i 1) n Z Z kS kS n ( n), the center of the group algebra n GZ kS n Gelfand-Zetlin subalgebra of n (Section 4.1) d dimk GZ = ∈ S dimk V (Theorem 4.4) n n V Irr n GZ X1,...,Xn Jucys-Murphy generators of n (§4.1.1) { n | ∈ GZ k }  GZ Spec(n) (φ(Xi))1 φ HomAlgk ( n, ) Spec n (§4.2.5) Cont(n) set of contents of standard Young tableaux with n boxes (§4.4.1) sgn sign representation (§3.2.4) S Mn standard permutation representation of n (§3.2.4) S Vn−1 standard (deleted permutation) representation of n (§3.2.4) λ λ/μ ∈ S % V , V the representation of Irr n corresponding to λ n and the representation given by the skew shape λ/μ (§4.3.2, §4.6.2) f λ, f λ/μ the numbers of λ-tableaux and λ/μ-tableaux; equal to λ λ/μ dimk V and dimk V , respectively (§4.3.3, §4.6.1) χλ, χλ/μ characters of V λ and V λ/μ (§4.6.3) Sλ Schur functor (§4.7.3) Notation 653

Lie Algebras and Root Systems. Let g be a Lie k-algebra.

gad adjoint representation (Example 5.3) Der g Lie algebra of derivations of g (§5.1.5) C i C g , ig terms of the descending and ascending central series (§5.2.1) D ig ith term of the derived series (§5.2.1) rad g radical of g (§5.2.2) Ug or U(g) enveloping algebra of g R(g) R(Ug), the representation ring of g (§5.5.8) V g g-invariants of a representation V (§5.3.1) B( · , · ) Killing form (§6.1.1)

The following notation applies to a semisimple Lie algebra g: h Cartan subalgebra of g (§6.3.1) gα root subspace (§6.3.2) Φ roots of g (§6.3.2) or an abstract root system (Section 7.1) Δ base of Φ (Section 7.2) Φ+ set of positive roots (Section 7.2) ρ half-sum of positive roots (§7.2.3) W = W Weyl group (§7.1.3) Φ ∗ w · λ shifted action: w(λ + ρ) − ρ for w ∈W,λ ∈ h (§8.3.3) = L LΦ root lattice of Φ (§7.4.1) = Λ ΛΦ weight lattice (§7.4.1) Λ+ dominant weights (§7.4.2) ∗ * partial order on RΦ(§7.4.2)oronh (§8.1.3) M(λ) Verma module with highest weight λ (§8.3.2) V (λ) head M(λ), the unique irreducible image of M(λ) (§8.3.2) χλ central character of M(λ) (§8.3.3) chλ formal character of V (λ) (§8.5.2 and §8.7.3) wλ aλ the anti-invariant w∈W sgn(w) x (§7.4.5 and Section 8.7)

Coalgebras, Bialgebras, and Hopf Algebras. The comultiplication of a k- → ⊗ → ⊗ coalgebra C is denoted by Δ: C C C, c c(1) c(2), and the counit by ε : C → k, c →ε, c.IfH is a Hopf algebra, then S: H → H denotes the antipode. Below, C and H retain their meaning and B denotes a k-bialgebra.

Ccop, Bbi op co-opposite coalgebra (§9.1.1) and bi-opposite bialgebra (§9.3.1) GC grouplike elements of C (§9.1.2) LB primitive elements of B (§9.3.2) 654 Notation

+ B Ker ε, the augmentation ideal of B (§9.3.1) ◦ B finite dual (§9.1.6, §9.3.6) l r ∫H , ∫ H , ∫H spaces of left, right, and two-sided integrals (§12.1.1) V H invariants of V ∈ Rep H (§10.1.1) V coH coinvariants of an H-comodule V (§10.1.1) ∈ Vad adjoint representation associated to V H ModH (§10.1.1)

Affine Algebraic Groups. Let G be an affine algebraic k-group and let A be a k-algebra equipped with a rational action G A by algebra automorphisms.

O(G) Hopf algebra (affine, commutative, reduced) of G (§11.2.2) G1 identity component (§11.5.3) Lie G Lie algebra of G (§11.6.1) G-Spec A collection of G-prime ideals of A (§11.7.3) I:GG-core of an ideal I of A (§11.7.1) Representation theory investigates the different ways in which a given algebraic object—such as a group or a Lie algebra—can act on a vector space. Besides being a subject of great intrinsic FIEYX]XLIXLISV]IRNS]WXLIEHHMXMSREPFIRI½XSJLEZMRKETTPM- cations in myriad contexts outside pure mathematics, including UYERXYQ½IPHXLISV]ERHXLIWXYH]SJQSPIGYPIWMRGLIQMWXV] Adopting a panoramic viewpoint, this book offers an introduction

XSJSYVHMJJIVIRX¾EZSVWSJVITVIWIRXEXMSRXLISV]VITVIWIRXEXMSRW University Temple Ryan Brandenberg, Photo by of algebras, groups, Lie algebras, and Hopf algebras. A separate part of the book is HIZSXIHXSIEGLSJXLIWIEVIEWERHXLI]EVIEPPXVIEXIHMRWYJ½GMIRXHITXLXSIREFPI and hopefully entice the reader to pursue research in representation theory. The book is intended as a textbook for a course on representation theory, which could immediately follow the standard graduate abstract algebra course, and for subsequent more advanced reading courses. Therefore, more than 350 exercises at ZEVMSYWPIZIPWSJHMJ½GYPX]EVIMRGPYHIH8LIFVSEHVERKISJXSTMGWGSZIVIH[MPPEPWS make the text a valuable reference for researchers in algebra and related areas and a source for graduate and postgraduate students wishing to learn more about repre- sentation theory by self-study.

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