Recent Titles in This Series 123 M . A * Akivis and B. A. Rosenfeld, Eli e Carton (1869-1951), 199 3 122 Zhan g Guan-Hou, Theor y of entire and meromorphic functions: Deficien t an d asymptotic values and singular directions, 199 3 121 LB . Fesenko and S. V. Vostokov, Loca l fields and their extensions: A constructive approach, 199 3 120 Takeyuk i Hida and Masnyuki Hitsnda, Gaussia n processes, 199 3 119 M . V. Karasev and V. P. Maslov, Nonlinea r Poisson brackets. Geometry and quantization, 199 3 118 Kenkich i Iwasawa, Algebrai c functions, 199 3 117 Bori s Zilber, Uncountabl y categorical theories, 199 3 116 G . M. Fel'dman, Arithmeti c of probability distributions, and characterization problem s on abelian groups, 199 3 115 Nikola i V. Ivanov, Subgroup s of Teichmuller modular groups, 199 2 114 Seiz o ltd, Diffusio n equations , 199 2 113 Michai l Zhitomirskii, Typica l singularities of differential 1-form s and Pfafna n equations, 199 2 112 S . A. Lomov, Introductio n to the general theory of singular perturbations, 199 2 111 Simo n Gindikin, Tub e domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introductio n to complex analysis Part II. Functions of several variables, 1992 109 Isa o Miyadera, Nonlinea r semigroups, 199 2 108 Take o Yokonnma, Tenso r spaces and exterior algebra, 199 2 107 B . M. MakaroY, M. G. Golnzina, A. A. Lodkin, and A. N. Podkorytov, Selecte d problem s in real analysis, 199 2 106 G.-C . Wen, Conforma l mapping s and boundary value problems, 199 2 105 D . R. Yafaev, Mathematica l scattering theory: Genera l theory, 199 2 104 R . L. Dobrushin, R. Kotecky, and S. Shlosman, Wulf f construction: A global shape fro m local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and their applications, 199 2 102 A . M. II' in, Matchin g of asymptotic expansions of solutions of boundary value problems, 199 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory of differential equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbits in invariant theory, 199 2 99 Nori o Shimaknra, Partia l differential operator s of elliptic type, 199 2 98 V . A. Vassiliev, Complement s of discriminants of smooth maps: Topolog y and applications, 199 2 97 Itir o Tamura, Topolog y of foliations: A n introduction, 199 2 96 A . I. Marknshevich, Introductio n to the classical theory o f Abelian functions, 199 2 95 Guangchan g Dong, Nonlinea r partial differential equation s of second order, 199 1 94 Yu . S. Il'yashenko, Finitenes s theorems for limit cycles , 199 1 93 A . T. Fomenko and A. A. Tnzhilin, Element s of the geometry and topology of minimal surfaces i n three-dimensional space, 199 1 92 £ . M. Nikishin and V. N. Sorokin, Rationa l approximations and orthogonality, 199 1 91 Mamor u Mimnra and Hlrosi Toda, Topolog y of Lie groups, I and II, 199 1 90 S . L. Sobolev, Som e applications of functional analysis in mathematical physics, third edition, 199 1 (Continued in the back of this publication) This page intentionally left blank Elle Cartan (1869-1951) ;{V ^ A Wl^1** """'•'in&wX EU E CARTA N Apri l t, ?S69~May6 » 195 1 10.1090/mmono/123 Translations o f MATHEMATICAL MONOGRAPHS Volume 12 3 Elie Carta n (1869-1951) M. A. Akivis B. A. Rosenfel d American Mathematical Societ y I Providence , Rhode Island 9JIM KAPTA H (1869-1951 ) M. A . AKHBH C B. A . Po3eH(|)ejib,n ; Translated b y V . V . Goldber g fro m a n origina l Russia n manuscrip t Translation edite d b y Simeo n Ivano v 2000 Mathematics Subject Classification. Primar y 01A70 ; Secondary 01A60 , 01A55 . ABSTRACT. The scientific biograph y o f one of the greatest mathematicians o f the 20th century, Eli e Cartan (1869-1951) , i s presented, a s well as the development o f Cartan's idea s by mathematician s of the followin g generations . Photo credits : p . iv-Centre National de la Recherche Scientifique ; pp . 2, 3, 9 , 10 , 17 , 19 , 25, 27 , 28, 29-Henri Cartan ; p . 31-Department o f Geometry, Kaza n University , Tatarstan , Russia Library o f Congres s Cataloging-in-Publicatio n Dat a Akivis, M . A . (Mak s Aizikovich ) [Elie Kartan (1869-1951) . English ] Elie Carta n (1869-1951)/M . A . Akivis , B . A . Rosenfeld ; [translate d fro m th e Russia n b y V. V . Goldberg ; translatio n edite d b y Simeo n Ivanov] . p. cm.—(Translation s o f mathematical monographs , ISS N 0065-9282 ; v. 123 ) Includes bibliographica l references . ISBN 0-8218-4587- X (acid-free ) 1. Cartan , Elie , 1869-1951 . 2 . Mathematicians-Prance-Biography . 3 . Li e groups . 4. Geometry , Differential . I . Rozenfel'd , B . A . (Bori s Abramovich ) II . Title. III . Series . QA29.C355A6613 199 3 93-693 2 516.3'76'092-dc20 CI P Copyright © 199 3 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e granted t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Information o n copyin g an d reprintin g ca n b e foun d i n the bac k o f this volume . This volum e wa s typeset usin g ^v^vS-TgX , the America n Mathematica l Society' s T^ X macr o system . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 6 1 5 1 4 1 3 1 2 1 1 Contents Preface x i Chapter 1 . Th e Life and Work of E . Cartan 1 §1.1. Parents'hom e 1 §1.2. Studen t at a school and a lycee 2 §1.3. Universit y student 4 § 1.4. Docto r of Scienc e 6 §1.5. Professo r 8 §1.6. Academicia n 1 7 §1.7. Th e Cartan famil y 2 4 §1.8. Carta n and the mathematicians o f the world 2 7 Chapter 2. Li e Groups and Algebras 3 3 §2.1. Group s 3 3 §2.2. Li e groups and Lie algebras 3 7 §2.3. Killing' s paper 4 2 §2.4. Cartan' s thesis 4 5 §2.5. Root s of the classical simple Lie groups 4 6 §2.6. Isomorphism s o f complex simpl e Lie groups 5 1 §2.7. Root s of exceptional complex simple Lie groups 5 1 §2.8. Th e Cartan matrices 5 3 §2.9. Th e Weyl groups 5 5 §2.10. Th e Weyl affine group s 6 0 §2.11. Associativ e and alternative algebra s 6 3 §2.12. Cartan' s works on algebras 6 7 §2.13. Linea r representations of simpl e Lie groups 6 9 §2.14. Rea l simple Lie groups 7 3 §2.15. Isomorphism s o f real simple Lie groups 7 8 §2.16. Reductiv e and quasireductive Li e groups 8 2 §2.17. Simpl e Chevalley groups 8 4 §2.18. Quasigroup s and loop s 8 5 viii CONTENT S Chapter 3 . Projectiv e Space s and Projective Metric s 8 7 §3.1. Rea l spaces 8 7 §3.2. Comple x spaces 9 3 §3.3. Quaternio n space s 9 5 §3.4. Octav e planes 9 6 §3.5. Degenerat e geometries 9 7 §3.6. Equivalen t geometries 10 1 §3.7. Multidimensiona l generalization s o f the Hesse transfe r principle 10 7 §3.8. Fundamenta l element s 10 9 §3.9. Th e duality and triality principles 11 3 §3.10. Space s over algebras with zero divisors 11 6 §3.11. Space s over tensor products of algebras 11 8 §3.12. Degenerat e geometries over algebras 12 1 §3.13. Finit e geometries 12 3 Chapter 4. Li e Pseudogroups and Pfaffia n Equation s 12 5 §4.1. Li e pseudogroups 12 5 §4.2. Th e Kac-Moody algebra s 12 7 §4.3. Pfaffia n equation s 12 9 §4.4. Completel y integrable Pfaffian system s 13 0 §4.5. Pfaffia n system s in involution 13 2 §4.6. Th e algebra of exterior forms 13 4 §4.7. Applicatio n of the theory of systems in involution 13 5 §4.8. Multipl e integrals, integral invariants, and integra l geometry 13 6 §4.9. Differentia l form s an d the Betti numbers 13 9 §4.10. Ne w methods in the theory of partial differential equation s 14 2 Chapter 5 . Th e Method o f Moving Frames and Differentia l Geometry 14 5 §5.1. Movin g trihedra of Frenet and Darboux 14 5 §5.2. Movin g tetrahedra and pentaspheres of Demoulin 14 7 §5.3. Cartan' s moving frames 14 8 §5.4. Th e derivational formula s 15 0 §5.5. Th e structure equations 1 5 2 §5.6. Application s o f the method o f moving frames 15 3 §5.7.