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This page intentionally left blankCAMBRI DGE STUDI ES I N ADVANCED MATHEMATI CS 113EDITORIAL BOARDb. bollobs, w. fulton, a. katok, f. ki rwan,p. sarnak, b. si mon, b. totaroAn Introduction to Lie Groups and Lie AlgebrasWith roots in the nineteenth century, Lie theory has since found many and variedapplications in mathematics and mathematical physics, to the point where it is nowregarded as a classical branch of mathematics in its own right. This graduate textfocuses on the study of semisimple Lie algebras, developing the necessary theoryalong the way.The material covered ranges from basic denitions of Lie groups, to the theory ofroot systems, and classication of nite-dimensional representations of semisimple Liealgebras. Written in an informal style, this is a contemporary introduction to thesubject which emphasizes the main concepts of the proofs and outlines the necessarytechnical details, allowing the material to be conveyed concisely.Based on a lecture course given by the author at the State University of New York atStony Brook, the book includes numerous exercises and worked examples and is idealfor graduate courses on Lie groups and Lie algebras.CAMBRI DGE STUDI ES I N ADVANCED MATHEMATI CSAll the titles listed below can be obtained from good booksellers or from CambridgeUniversity Press. For a complete series listing visit: http://www.cambridge.org/series/sSeries.asp?code=CSAMAlready published60 M. P. Brodmann & R. Y. Sharp Local cohomology61 J. D. Dixon et al. Analytic pro-p groups62 R. Stanley Enumerative combinatorics II63 R. M. Dudley Uniform central limit theorems64 J. Jost & X. Li-Jost Calculus of variations65 A. J. Berrick & M. E. Keating An introduction to rings and modules66 S. Morosawa Holomorphic dynamics67 A. J. Berrick & M. E. Keating Categories and modules with K-theory in view68 K. Sato Levy processes and innitely divisible distributions69 H. Hida Modular forms and Galois cohomology70 R. Iorio & V. Iorio Fourier analysis and partial differential equations71 R. Blei Analysis in integer and fractional dimensions72 F. Borceaux & G. Janelidze Galois theories73 B. Bollobs Random graphs74 R. M. Dudley Real analysis and probability75 T. Sheil-Small Complex polynomials76 C. Voisin Hodge theory and complex algebraic geometry, I77 C. Voisin Hodge theory and complex algebraic geometry, II78 V. Paulsen Completely bounded maps and operator algebras79 F. Gesztesy & H. Holden Soliton Equations and Their Algebro-Geometric Solutions, I81 S. Mukai An Introduction to Invariants and Moduli82 G. Tourlakis Lectures in Logic and Set Theory, I83 G. Tourlakis Lectures in Logic and Set Theory, II84 R. A. Bailey Association Schemes85 J. Carison, S. Mller-Stach & C. Peters Period Mappings and Period Domains86 J. J. Duistermaat & J. A. C. Kolk Multidimensional Real Analysis I87 J. J. Duistermaat & J. A. C. Kolk Multidimensional Real Analysis II89 M. Golumbic &A. Trenk Tolerance Graphs90 L. Harper Global Methods for Combinatorial Isoperimetric Problems91 I. Moerdijk & J. Mrcun Introduction to Foliations and Lie Groupoids92 J. Kollar, K. E. Smith &A. Corti Rational and Nearly Rational Varieties93 D. Applebaum Levy Processes and Stochastic Calculus94 B. Conrad Modular Forms and the Ramanujan Conjecture95 M. Schechter An Introduction to Nonlinear Analysis96 R. Carter Lie Algebras of Finite and Afne Type97 H. L. Montgomery, R. C. Vaughan & M. Schechter Multiplicative Number Theory I98 I. Chavel Riemannian Geometry99 D. Goldfeld Automorphic Forms and L-Functions for the Group GL(n,R)100 M. Marcus & J. Rosen Markov Processes. Gaussian Processes, and Local Times101 P. Gille & T. Szamuely Central Simple Algebras and Galois Cohomology102 J. Bertoin Random Fragmentation and Coagulation Processes103 E. Frenkel Langlands Correspondence for Loop Groups104 A. Ambrosetti &A. Malchiodi Nonlinear Analysis and Semilinear Elliptic Problems105 T. Tao & V. H. Vu Additive Combinatorics106 E. B. Davies Linear Operators and their Spectra107 K. Kodaira Complex Analysis108 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Harmonic Analysis on Finite Groups109 H. Geiges An Introduction to Contact Topology110 J. Faraut Analysis on Lie Groups111 E. Park Complex Topological K-Theory112 D. W. Stroock Partial Differential Equations for Probabilists113 A. Kirillov An Introduction to Lie Groups and Lie AlgebrasAn Introduction to Lie Groupsand Lie AlgebrasALEXANDER KI RI LLOV, Jr.Department of Mathematics, SUNY at Stony BrookCAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So PauloCambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UKFirst published in print formatISBN-13 978-0-521-88969-8ISBN-13 978-0-511-42319-2 A. Kirillov Jr. 20082008Information on this title: www.cambridge.org/9780521889698This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgeBook (EBL)hardbackDedicated to my teachersContentsPreface page xi1 Introduction 12 Lie groups: basic denitions 42.1. Reminders from differential geometry 42.2. Lie groups, subgroups, and cosets 52.3. Lie subgroups and homomorphism theorem 102.4. Action of Lie groups on manifolds andrepresentations 102.5. Orbits and homogeneous spaces 122.6. Left, right, and adjoint action 142.7. Classical groups 162.8. Exercises 213 Lie groups and Lie algebras 253.1. Exponential map 253.2. The commutator 283.3. Jacobi identity and the denition of a Lie algebra 303.4. Subalgebras, ideals, and center 323.5. Lie algebra of vector elds 333.6. Stabilizers and the center 363.7. CampbellHausdorff formula 383.8. Fundamental theorems of Lie theory 403.9. Complex and real forms 443.10. Example: so(3, R), su(2), and sl(2, C) 463.11. Exercises 48viiviii Contents4 Representations of Lie groups and Lie algebras 524.1. Basic denitions 524.2. Operations on representations 544.3. Irreducible representations 574.4. Intertwining operators and Schurs lemma 594.5. Complete reducibility of unitary representations:representations of nite groups 614.6. Haar measure on compact Lie groups 624.7. Orthogonality of characters and PeterWeyl theorem 654.8. Representations of sl(2, C) 704.9. Spherical Laplace operator and the hydrogen atom 754.10. Exercises 805 Structure theory of Lie algebras 845.1. Universal enveloping algebra 845.2. PoincareBirkhoffWitt theorem 875.3. Ideals and commutant 905.4. Solvable and nilpotent Lie algebras 915.5. Lies and Engels theorems 945.6. The radical. Semisimple and reductive algebras 965.7. Invariant bilinear forms and semisimplicity of classical Liealgebras 995.8. Killing form and Cartans criterion 1015.9. Jordan decomposition 1045.10. Exercises 1066 Complex semisimple Lie algebras 1086.1. Properties of semisimple Lie algebras 1086.2. Relation with compact groups 1106.3. Complete reducibility of representations 1126.4. Semisimple elements and toral subalgebras 1166.5. Cartan subalgebra 1196.6. Root decomposition and root systems 1206.7. Regular elements and conjugacy of Cartansubalgebras 1266.8. Exercises 1307 Root systems 1327.1. Abstract root systems 1327.2. Automorphisms and the Weyl group 1347.3. Pairs of roots and rank two root systems 135Contents ix7.4. Positive roots and simple roots 1377.5. Weight and root lattices 1407.6. Weyl chambers 1427.7. Simple reections 1467.8. Dynkin diagrams and classication of root systems 1497.9. Serre relations and classication of semisimpleLie algebras 1547.10. Proof of the classication theorem insimply-laced case 1577.11. Exercises 1608 Representations of semisimple Lie algebras 1638.1. Weight decomposition and characters 1638.2. Highest weight representations and Verma modules 1678.3. Classication of irreducible nite-dimensionalrepresentations 1718.4. BernsteinGelfandGelfand resolution 1748.5. Weyl character formula 1778.6. Multiplicities 1828.7. Representations of sl(n, C) 1838.8. HarishChandra isomorphism 1878.9. Proof of Theorem 8.25 1928.10. Exercises 194Overview of the literature 197Basic textbooks 197Monographs 198Further reading 198Appendix A Root systems and simple Lie algebras 202A.1. An = sl(n 1, C), n 1 202A.2. Bn = so(2n 1, C), n 1 204A.3. Cn = sp(n, C), n 1 206A.4. Dn = so(2n, C), n 2 207AppendixB Sample syllabus 210List of notation 213Bibliography 216Index 220PrefaceThis book is an introduction to the theory of Lie groups and Lie algebras, withemphasis on the theory of semisimple Lie algebras. It can serve as a basis fora two-semester graduate course or omitting some material as a basis fora rather intensive one-semester course. The book includes a large number ofexercises.The material covered in the book ranges from basic denitions of Lie groupsto the theory of root systems and highest weight representations of semisim-ple Lie algebras; however, to keep book size small, the structure theory ofsemisimple and compact Lie groups is not covered.Exposition follows the style of famous Serres textbook on Lie algebras[47]: we tried to

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