Lie groups, lie algebras and their representations

Graduate Texts in Mathematics S Axler Springer Science+Business Media, LLC 102 Editorial Board F.W Gehring K.A Ribet Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 T AKEUTI/ZARINa Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topologica! 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102) Bibliography : p Includes index Lie groups Lie algebras Representations of groups Representations of algebras Title II Series QA387.V35 1984 512'.55 84-1381 Printed on acid-free paper This book was originally published in tbe Prentice-Hall Series in Modern Analysis, 1974 Selection from "Tree in Night Wind" copyright 1960 by Abbie Huston Evans Reprinted from her volume Fact of Crystal by permission of Harcourt Brace Jovanovich, Inc First published in The New Yorker © 1974, 1984 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 1984 Softcover reprint of the hardcover 1st edition 1984 Ali rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC 9876543 ISBN 978-1-4612-7016-4 ISBN 978-1-4612-1126-6 (eBook) DOI 10.1007/978-1-4612-1126-6 Yet here is no confusion: central-ru/ed Divergent plungings, run through with a thread Of pattern never snapping, cleave the tree Into a dozen stubborn tusslings, yieldings, That, balancing, bring the whole top alive Caught in the wind this night, the fu/l-leaved boughs, Tied to the trunk and governed by that tie, Find and hold a center that can rule With rhythm al/ the buffeting andjlailing, Tii/ in the end complex resolves to simple from Tree in Night Wind ABBIE HUSTON EYANS PREFACE This book has grown out of a set of lecture notes I had prepared for a course on Lie groups in 1966 When I lectured again on the subject in 1972, I revised the notes substantially It is the revised version that is now appearing in book form The theory of Lie groups plays a fundamental role in many areas of mathematics There are a number of books on the subject currently available -most notably those of Chevalley, Jacobson, and Bourbaki-which present various aspects of the theory in great depth However, feei there is a need for a single book in English which develops both the algebraic and analytic aspects of the theory and which goes into the representation theory of semisimple Lie groups and Lie algebras in detail This book is an attempt to fiii this need It is my hope that this book will introduce the aspiring graduate student as well as the nonspecialist mathematician to the fundamental themes of the subject I have made no attempt to discuss infinite-dimensional representations This is a very active field, and a proper treatment of it would require another volume (if not more) of this size However, the reader who wants to take up this theory will find that this book prepares him reasonably well for that task I have included a large number of exercises Many of these provide the reader opportunities to test his understanding In addition I have made a systematic attempt in these exercises to develop many aspects of the subject that could not be treated in the text: homogeneous spaces and their cohomologies, structure of matrix groups, representations in polynomial rings, and complexifications of real groups, to mention a few In each case the exercises are graded in the form of a succession of (Iocally simple, hope) steps, with hints for many Substantial parts of Chapters 2, and 4, together with a suitable selection from the exercises, could conceivably form the content of a one year graduate course on Lie groups From the student's point vii viii Preface of view the prerequisites for such a course would be a one-semester course on topologica! groups and one on differentiable manifolds The book begins with an introductory chapter on differentiable and analytic manifolds A Lie group is at the same time a group and a manifold, and the theory of differentiable manifolds is the foundation on which the subject should be built It was not my intention to be exhaustive, but I have made an effort to treat the main results of manifold theory that are used subsequently, especially the construction of global solutions to involutive systems of differential equations on a manifold In taking this approach have followed Chevalley, whose Princeton book was the first to develop the theory of Lie groups globally My debt to Chevalley is great not only here but throughout the book, and it will be visible to anyone who, Iike me, learned the subject from his books The second chapter deals with the general theory AII the basic results and concepts are discussed: Lie groups and their Lie algebras, the correspondence between subgroups and subalgebras, the exponential map, the Campbell-Hausdorff formula, the theorems known as the fundamental theorems of Lie, and so on The third chapter is almost entirely on Lie algebras The aim is to examine the structure of a Lie algebra in detail With the exception of the last part of this chapter, where applications are made to the structure of Lie groups, the action takes place over a field of characteristic zero The main results are the theorems of Lie and Engel on nilpotent and solvable algebras; Cartan's criterion for semisimplicity, namely that a Lie algebra is semisimple if and only if its Cartan-Killing form is nonsingular; Weyl's theorem asserting that ali finite-dimensional representations of a semisimple Lie algebra are semisimple; and the theorems of Levi and Mal'cev on the semidirect decompositions of an arbitrary Lie algebra into its radical and a (semisimple) Levi factor Although the results of Weyl and Levi-Mal'cev are cohomological in their nature (at least from the algebraic point of view), l have resisted the temptation to discuss the general cohomology theory of Lie algebras and have confined myself strictly to what is needed (ad hoc Iowdimensional cohomology) The fourth and final chapter is the heart of the book and is a fairly complete treatment of the fine structure and representation theory of semisimple Lie algebras and Lie groups The root structure and the classification of simple Lie algebras over the field of complex numbers are obtained As for representation theory, it is examined from both the infinitesimal (Cartan, Weyl, Harish-Chandra, Chevalley) and the global (Weyl) points of view First present the algebraic view, in which universal enveloping algebras left ideals, highest weights, and infinitesimal characters are put in the foreground have followed here the treatment of Harish-Chandra given in his early papers and used it to prove the bijective nature of the correspondence Preface IX between connected Dynkin diagrams and simple Lie algebras over the complexes This algebraic part is then followed up with the transcendental theory Here compact Lie groups come to the fore The existence and conjugacy of their maxima! tori are established, and Weyl's classic derivation of his great character formula is given It is my belief that this dual treatment of representation theory is not only illuminating but even essential and that the infinitesimal and global parts of the theory are complementary facets of a very beautiful and complete picture In order not to interrupt the main flow of exposition, have added an appendix at the end of this chapter where have discussed the basic results of finite reflection groups and root systems This appendix is essentially the same as a set of unpublished notes of Professor Robert Steinberg on the subject, and am very grateful to him for allowing me to use his manuscript It only remains to thank ali those without whose help this book would have been impossible am especially grateful to Professor M Singer for his help at various critica! stages Mrs Alice Hume typed the entire manuscript, and cannot describe my indebtedness to the great skill, tempered with great patience, with which she carried out this task would like to thank Joel Zeitlin, who helped me prepare the original 1966 notes; and Mohsen Pazirandeh and Peter Trombi, who looked through the entire manuscript and corrected many errors would also like to thank Ms Judy Burke, whose guidance was indispensable in preparing the manuscript for publication would like to end this on a personal note My first introduction to serious mathematics was from the papers of Harish-Chandra on semisimple Lie groups, and almost everything know of representation theory goes back either to his papers or the discussions have had with him over the past years My debt to him is too immense to be detailed V S VARADARAJAN Pacific Palisades PREFACE TO THE SPRINGER EDITION (1984) Lie Groups, Lie Algebras, and Their Representations went out of print recently However, many of my friends tqld me that it is stiU very useful as a textbook and that it would be good to have it available in print So when Springer offered to republish it, agreed immediately and with enthusiasm wish to express my deep gratitude to Springer-Verlag for their promptness and generosity am also extremely grateful to Joop Kolk for providing me with a comprehensive list of errata CONTENTS Preface vii Chapter Differentiable and Analytic Manifolds 1.1 Ditferentiable Manifolds 1.2 Analytic Manifolds 20 1.3 The Frobenius Theorem 1.4 Appendix 31 Exercises 35 Chapter 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 25 Lie Groups and Lie Algebras Definition and Examples of Lie Groups 41 Lie Algebras 46 The Lie Algebra of a Lie Group 51 The Enveloping Algebra of a Lie Group 55 Subgroups and Subalgebras 57 Locally isomorphic Groups 61 Homomorphisms 67 The Fundamental Theorem of Lie 72 Closed Lie Subgroups and Homogeneous Spaces 74 Orbits and Spaces of Orbits The Exponential Map 84 The Uniqueness of the Real Analytic Structure of a Real Lie Group 92 Taylor Series Expansions on a Lie Group 94 The Adjoint Representations of !J and G JOI The Differential of the Exponential Map 107 xi 41 Bibliography 418 E Sur la structure des groupes de transformations ftnis et continus, These, Paris, Nony, 1894 (Oeuvres Completes, Partie 1, Voi 1, pp 137-287) CARTAN, [1] [2] Les groupes projectifs qui ne laissent invariante aucune multiplicite plane, Bulletin de la Sociere mathematique de France, 41 (1913), 53-96 (Oeuvres Completes, Partie 1, Voi 1, pp 355-398) [3] Sur les invariants integraux de certains espaces homogenes closet les proprietes topologiques deces espaces, Ann Soc Pol Math., (1929), 181-225 (Oeuvres Completes, Partie 1, Voi 2, pp 1081-1125) [4] Lecons sur la theorie des spineurs, Hermann, Paris, 1938 [5] Les representations lineaires des groups de Lie, J Math pures et appliquees, 17 (1938), 1-12 [1] C Theory of Lie Groups, 1, Princeton University Press, Princeton, 1946 [2] Theorie des Groupes de Lie, II: Groupes Algebriques, Hermann, Paris, 1951 [3] Theorie des Groupes de Lie, III: Theoremes Generaux sur les algebres de Lie, Hermann, Paris, 1955 [4] The Algebraic Theory ofSpinors, Columbia University Press, New York, 1954 [5] A new kind of relationship between matrices, Amer J Math 65 (1943), 521531 [6] Sur la classification des algebres de Lie simples et de leurs representations, C R Acad Sci., Paris 227 (1948), 1136-1138 [7] lnvariants of finite groups generated by reflexions, Amer J Math 77 (1955), 778-782 CHEVALLEY, E B The structure of semisimple Lie algebras, Uspekhi Mat Nauk (N.S.) (1947), 59-127 (Amer Math Soc Translations No 17 (1950)) DYNKIN, [1] [2] Normed Lie algebras and analytic groups, Amer Math Soc Transl (1) 9, 470-534 HARISH-CHANDRA [1] On representations of Lie algebras, Ann Math 50 (1949), 900-915 [2] Faithful representations of Lie algebras, Ann Math 50 (1949), 68-76 [3] Faithfu1 representations of Lie groups, Proc Amer Math Soc (1950), 205-210 [4] On some applications of the universal enveloping algebra of a semi-simple Lie algebra, Trans Amer Math Soc 70 (1951), 28-96 [5] Representations of a semi-simple Lie group on a Banach space, 1, Trans Amer Math Soc 75 (1953), 185-243 [6] A formula for semi-simple Lie groups, Amer J Math 19 (1957), 733-760 Bibliography 419 [7] Spherical functions on a semi-simple Lie group, 1, Amer J Math 80 (1958), 241-310 [8] Spherical functions on a semi-simple Lie group, Il, Amer J Math 80 (1958), 553-613 HAUSDORFF, F [1] Die symbolische Exponential Forme/ in der Gruppen theorie, Berichte Ober die Verhandlungen, Leipzig 1906, 19 48 HELGASON, S [1] Differential Geometry and Symmetrii: Spaces, Academic Press, New York and London, 1962 HOCHSCHILD, G [1] The Structure of Lie Groups, Holden-Day lnc., 1965 JACOBSON, N [1] Lie Algebras, lnterscience Publishers, New York and London, 1962 KOBAYASHI, S., and K NOMIZU [1] Foundations of Di.fferentia/ Geometry, lnterscience Publishers, New York and London, 1963 KOSTANT, B [1] The principal three-dimensional subgroups and the Betti numbers of a complex simple Lie group, Amer J Math 81 (1959), 973-1032 [2] A formula for the multiplicity of a weight, Trans Amer Math Soc 93 (1959), 53-73 [3] Lie group representations on polynomial rings, Amer J Math 85 (1963), 327-404 KoszuL, J L [1] Homologie et cohomologie des algebres de Lie, Bul/ de la Societe mathematique de France 78 (1950), 65-127 LAZARD, M [1] Theorie des repliques Critere de Cartan Seminarie Sophus Lie: Theorie des Algebres de Lie, Topologie des groupes de Lie (1955), 6°, 1-9 MAL'CEV, A [1] On the simple connectedness of invariant subgroups of Lie groups, Doklady Akademii Nauk SSSR 34 (1942), 10-13 MoNTGOMERY, D., and L ZIPPIN [1] Topologica/ Transforma/ion Groups, lnterscience Publishers, New York and London, 1955 NARASIMHAN, R [1] Analysis on Real and Complex Manifolds, North Holland Publishing Company, Amsterdam, 1968 PALAIS, R S [1) On the existence of slices for actions of noncompact Lie groups, Ann Math 72 (1961), 295-323 420 [2] Bibliography A global formulation of the Lie theory of transformation groups Memoirs of the American Mathematical Society, Number 22, 1957 PARTHASARATHY, K R., R RANGA RAO, and V S VARADAP.AJAN [1] Representations of complex semi-simple Lie groups and Lie algebras, Ann of Math 85 (1967), 383-429 PONTRYAGIN, L S [1] Topologica! Groups, 2nd ed., Gordon and Breach, New York, 1966 SAMELSON, H [1] Topology of Lie groups, Bul! Amer Math Soc 58 (1952), 2-37 SEMINAIRE SOPHUS LIE [1] Theorie des Algebres de Lie, Topologie des groupes de Lie, Paris, 1955 SERRE, J P Algebres de Lie Semi-Simple Complexes, W A Benjamin, Inc., New York, 1966 [1] SHEPHARD, G C., and I A Tooo [1] Finite unitary reflexion groups, Canadian J Math (1954), 274-304 SPANIER, E H [1] Algebraic Topology, McGraw-Hill Book Company, New York, Toronto, London, 1966 STEINBERG R [1] Finite rejfexion groups (Unpublished notes) VARADARAJAN, V S [1] On the ring of invariant polynomials on a semi-simple Lie algebra, Amer J Math 90 (1968), 308-317 VERMA, DAY-NAND [l] Structure of certain induced representations of complex semisimple Lie algebras, Bul! Amer Math Soc 74 (1968), 160-166 WEIL, A L'integration dans les groups topologiques et se applications, Hermann, Paris, 1940 [1] WEYL, H [1] The Classical Groups, Princeton University Press, Princeton, 1946 [2] Theorie der Darstellung kontinuerlicher halb-einfacher Gruppen durch lineare transformationen, Teil 1, Mathematische Zeitschrift 23 (1925), 271-309 [3] Theorie der Darstellung kontinuerlicher halb-einfacher Gruppen durch lineare transformationen, Teil Il, Mathematische Zeitschrift 24 (1926), 328-376 [4] Theorie der Darstellung kontinuerlicher halb-einfacher Gruppen durch lineare transformationen, Teil III, Mathematische Zeitschrift 24 (1926), 377-395 [5] The theory ofgroups and quantum mechanics, Dover Press WHITNEY, H [1] Elementary structure of real algebraic varieties, Ann of Math 66 (1957), 545556 INDEX A Adjoint representation, of a Lie algebra, 50 of a Lie algebra in its universal enveloping algebra, 172 of a Lie group, 104 Admissible topological group, 62 Ado's theorem, 237 Algebraic group: complex, 45 real, 45 Algebraic set, 23 Analytic group, 41 Analytic manifold: complex, 21 real, 20 Analytic structure, real, 20 Analytic subgroup of a Lie group, 42 Analytic system of tangent spaces, 25 Baker-Campbell-Hausdorff formula, 114 Bracket, Lie: of vector fields, in a Lie algebra, c Canonica! coordinates: of the first kind, 88 of the second kind, 89 Cartan-Killing form, 208 Cartan matrix, 290 Cartan subalgebra ( CSA), 262 423 424 Index Casimir elţ:ment: of the universal envcloping algebra of a semisimple Lie algebra, 217 associated with a representation of a semisimple Lie algebra, 218 Casimir polynomial, 208 Center, of a Lie algebra, 50 Central series, ascending and descending, of a nilpotent Lie algebra, 193 Chamber, Weyl, 282 associated with root systems, 370, 379 Chevalley's theorem: on conjugacy of CSA's of a complex Lie algebra, 263 on invariant polynomials on a semisimple Lie algebra, 335 on polynomials invariant with respect to a finite reftection group, 383 on stabilizers in a finite reftection group, 377 Classical groups: compact, 46 complex, 46 Compact form: of a complex semisimple Lie algebra, 348 of a complex semisimple Lie group, 348 Complexification of a Lie algebra, Covering group, 62 Covering homomorphism, 62 Cyclic vector for a representation, 154 o Derivation, of an algebra, 49 Derived algebra, of a Lie algebra, 200 Diffeomorphism, 15 Differentiable functions, Differentiable manifold, dimension of, Differentiable mapping, 14 Differentiable structure, of a mapping, 14 of a homomorphism of one Lie group into another, 70 Index Differential expression, Differential operator, expression of, at a point, 1r-related to another, 14 formal adjoint of, 14 left invariant, on a Lie group, 53 Dominant linear function, 314 Dynkin diagram, 293 E Engel's theorem, 190 Enveloping algebra, of a Lie group, 55 Exponential of a matrix, 89 Exponential map, 85 differential of, 108 Exterior algebra, 166 Exterior differential form, 11 Exterior differentiation, 11 F Filtered algebra, 176 Filting decomposition, 152 Free action of a group, 81 Free associative algebra, 164 Free Lie algebra, 174 Frobenius theorem: local, 28 global, 30 Fundamental group, 61 G G-space, 74 transitive, 74 Galois extension, 149 Germ of a function, Graded algebra, 176 associated with a filtered algebra, 177 425 426 Index H Harish-Chandra's theorem, on the center of the universal enveloping algebra of a semisimple Lie algebra, 339 Hilbert's fifth problem, 42 Homogeneous space, 74 Homomorphism, of Lie algebras, Ideal, of a Lie algebra, 47 lmbedding, 16 quasiregular, 17 regular, 16 Immersion, 16 Infinitesimal character, 184, 339 Integral linear function, 314 Involutive analytic system, of tangent spaces, 25 canonical, 26 coordinates adapted to, 28 integrable, 25 integral manifold of, 25 J Jacobi identity, 6, 47 Jordan decomposition: of an endomorphism of a vector space, 151 of an element of a semisimple Lie algebra, 215 L Levi decomposition: of a Lie algebra, 224 of an analytic group, 245 Levi subalgebra, of a Lie algebra, 224 Levi subgroup, of an analytic group, 245 Lexicographic ordering, of a real vector space, 285 Index Lie algebra, 46 abelian, 48 of a Lie group, 51 quotient of, 48 structure constants of, subalgebra of, Lie group, 41 analytic action of, on a manifold, 75 quotient of, 80 Lie subalgebra defined by a Lie subgroup, 57 Lie subgroup, 42 Local homomorphism, 63 Locally isorriorphic groups, 63 M Maurer-Cartan forms, 138 Maxima! torus, of a compact semisimple analytic group, 353 Measure, corresponding to an n-form, 12 Monodromy, 63 N Nil ideal, of a Lie algebra, 206 Nil radical: of a Lie algebra, 206 of an analytic group, 244 Nil representation, of a Lie algebra, 189 Nilpotent analytic group, 195, 243 Nilpotent component, of an element of a semisimple Lie algebra, 215 Nilpotent element, of a semisimple Lie algebra, 215 Nilpotent Lie algebra, 189 o Orbit, 80 Orientable manifold, 12 Orientation, of a manifold, 12 427 428 Index p Poincare-Birkhoff-Witt theorem, 168 Poincare series, of a graded vector space, 381 Positive system of roots, 280, 370, 379 Product: of Lie algebras, 48 of Lie groups, 41 Product manifolds, 19 Q Quotient, of a manifold relative to a map, 16 R Radical: of a Lie algebra, 204 of an analytic group, 244 Rank, of a Lie algebra, 260 Real form, of a complex Lie algebra, 133 Reductive Lie algebra, 232 Reflection, in a vector space, 369 Reflection group, 369 Regular element, of a Lie algebra, 261 Regular point of an algebraic set, 23 Replica, of an endomorphism of a vector space, 157 Representation: of associative algebra, 152 of Lie algebra, 49 of Lie group, 102 complex analytic, of a complex Lie group, 102 irreducible, 51 quotient representation of, 51, 153 subrepresentation of, 51, 153 scmisimple, 153 Representation with weights, 316 Representations, of a Lie algebra: direct sum of, 50 equivalence of, 51 outer tensor product of, 50 tensor product of, 50 Index Root, 273, 370 Root space decomposition, of a semisimple Lie algebra, 274 Root subspace, 273 Root system, 370 integral, 377 extended integral, 378 s Schur's lemma, 153 Semidirect product: of analytic groups, 228 of Lie algebras, 224 Semisimple analytic group, 207 Semisimple component, of an element of a semisimple Lie algebra, 215 Semisimple element, of a semisimple Lie algebra, 215 Semisimple Lie algebra, 204 Simple Lie algebra, 211 Simple system of roots, 280, 370, 379 Solvable analytic group, 204, 243 Solvable Lie algebra, 201 Spin representation, 329 Stability subgroup, 74 Submanifold of a differentiable manifold, 18 quasi-regular, 18 regular, 18 Submersion, 15 Symmetric algebra, 165 Symmetrizer map, 180 Symplectic group, 46 T Tangent space, holomorphic, 22 Tangent vector, holomorphic, 22 Taylor expansions, on a Lie group, 94 429 430 Index Tensor: degree of, 163 skew-symmetric, 165 symmetric, 165 Tensor algebra, 163 Torus, 43 Transformation group: global, 123 infinitesimal, 123 local, 122 u Unipotent representation, of an analytic group, 245 Unipotent subgroup of GL(V), 196 Unitarian trick, 349 Universal complexification, of a real analytic group, 256 Universal covering group, 62 Universal enveloping algebra, of a Lie algebra, 167 V Vector field, w Weight: of a representation, 193, 316 subspace, 193, 316 highest, of a representation, 317 Weyl: hasis, 280 character formula of, 366 dimension formula of, 327, 368 group, 279 integral formula of, 363 Weyl's theorem: on semisimplicity of representations, 222, 348 on finiteness of fundamental groups of compact semisimple groups, 345 Graduate Texts in Mathematics ( continued from page ii) 62 KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol l 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 W ATERHOUSE lntroduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields Il 70 MASSEY Singular Homology Theory 71 FARKAS/KRA Riemann Surfaces 2nd ed 72 STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed 73 HUNGERFORD Algebra 74 DAVENPORT Multiplicative Number Theory 2nd ed 75 HOCHSCHILD Ba~ic Theory of Algebraic Groups and Lie Algebras 76 liTAKA Algebraic Geometry 77 HECKE Lectures on the Theory of Algebraic Numbers 78 BuRRJs/SANKAPPANAVAR A Course in Universal Algebra 79 W ALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 2nd ed 81 FORSTER Lectures on Riemann Surfaces 82 Borr/Tu Differential Forms in Algebraic Topology 83 WASHINGTON lntroduction to Cyclotomic Fields 2nd ed 84 lRELAND/RosEN A Cla~sical lntroduction to Modern Number Theory 2nd ed 85 EDWARDS Fourier Series Voi Il 2nd ed 86 VAN LJNT Introduction to Coding Theory 2nd ed 87 BROWN Cohomo1ogy of Groups 88 PIERCE Associative Algebras 89 LANG Introduction to Algebraic and Abelian Functions 2nd ed 90 BR0NDSTED An Introduction to Convex Polytopes 91 BEARDON On the Geometry of Discrete Groups 92 DIESTEL Sequences and Series in Banach Spaces 93 DUBROVIN/FOMENKO/NOVIKOV Modem Geometry-Methods and Applications Part I 2nd ed 94 WARNER Foundations of Differentiable Manifolds and Lie Groups 95 SHIRYAEV Probability 2nd ed 96 CoNw AY A Course in Functional Analysis 2nd ed 97 KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed 98 BROCKERITOM D!ECK Representations of Compact Lie Groups 99 GROVEIBENSON Finite Reflection Groups 2nd ed 100 BERG/CHRISTENSEN/RESSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions 101 EDW ARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 3rd ed 104 DUBROVIN/FOMENKO/NOVIKOV Modern Geometry-Methods and Applications Part Il 105 LANG SLlR) 106 SILVERMAN The Arithmetic of Elliptic Curves 107 OLVER Applications of Lie Groups to Differential Equations 2nd ed 108 RANGE Ho1omorphic Functions and Integral Representations in Severa! Complex Variables 109 LEHTO Univalent Functions and Teichmiiller Spaces 110 LANG Algebraic Number Theory 111 HUSEMOLLER Elliptic Curves 112 LANG Elliptic Functions 113 KARATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed 114 KoBLITZ A Course in Number Theory and Cryptography 2nd ed 115 BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces 116 KELLEY/SRINIVASAN Mea~ure and Integral Voi I 117 SERRE Algebraic Groups and Class Fields 118 PEDERSEN Analysis Now 119 RoTMAN An Introduction to Algebraic Topo1ogy 120 ZIEMER Weak1y Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAUS/HERMES et al Numbers Readings in Mathematics 124 DUBROVIN/FOMENKo/NOVIKOV Modem Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Ba.~ic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FuLTON/HARRIS Representation Theory: A First Course Readings in Mathematics 130 DoosoN/PoSTON Tensor Geornetry 131 LAM A First Course in Noncommutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRis Algebraic Geornetry: A First Course 134 RoMAN Coding and lnforrnation Theory 135 RoMAN Advanced Linear Algebra 136 ADKINs/WEINTRAUB Algebra: An Approach via Module Theory 137 AxLERIBouRDoNIRAMEY Harrnonic Function Theory 138 CoHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geornetry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERfWEISPFENNING/KREDEL Grobner Ba.~es A Cornputational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 Doos Measure Theory 144 DENNIS/FARB Noncommutative Algebra 145 VICK Hornology Theory An lntroduction to Algebraic Topology 2nd ed 146 BRIDGES Cornputability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWNIPEARCY An lntroduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN lntegration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable Il 160 LANG Differential and Riemannian Manifolds 161 BORWEINIERDELYl Polynomials and Polynomial lnequalities 162 ALPERINIBELL Groups and Representations 163 DIXON/MORTIMER Perrnutation Groups 164 NATHANSON Additive Number Theory: The Classical Ba.~es 165 NATHANSON Additive Number Theory: Inverse Problerns and the Geornetry of Sumsets 166 SHARPE Differential Geornetry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Cornbinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riernannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 OIESTEL Graph Theory 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An lntroduction to Knot Theory 176 LEE Riernannian Manifolds 177 NEWMAN Analytic Nurnber Theory 178 CLARKEILEDYAEV/STERN/WOLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KREss Numerical Analysis 182 WALTER Ordinary Differential 183 184 185 186 Equations MEGGINSON An lntroduction to Banach Space Theory BOLLOBAS Modem Graph Theory COxiLIITLEIO'SHEA Using Algebraic Geometry RAMAKRISHNANNALENZA Fourier Analysis on Number Fields 187 HARRIS/MORRISON Moduli of Curves 188 GOLDBLATI Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 BSMONDEIMURTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCHILACOMBE Blements of Fuctional Analysis ... 2.14 25 Lie Groups and Lie Algebras Definition and Examples of Lie Groups 41 Lie Algebras 46 The Lie Algebra of a Lie Group 51 The Enveloping Algebra of a Lie Group 55 Subgroups and Subalgebras... Lie groups, Lie algebras, and tbeir representations (Graduate texts in mathematics; 102) Bibliography : p Includes index Lie groups Lie algebras Representations of groups Representations of algebras. .. fourth and final chapter is the heart of the book and is a fairly complete treatment of the fine structure and representation theory of semisimple Lie algebras and Lie groups The root structure and