Ergebnisse der Mathematik und ihrer Grenzgebiete Band 40 IIerausgegeben von P. R. IIalmos . P.]. FIilton . R. Remmert· B. Sz6kefalvi-Nagy Unter Mitwirkung von L. V. Ahlfors . R. Baer . F. L. Bauer· R. Courant· A. Dold ]. L. Doob . S. Eilenberg . M. Kneser . M. M. Postnikov II. Rademacher· B. Segre . E. Sperner Geschaftsfiihrender IIerausgeber: P.]. IIilton G. B. Seligman Modular Lie Algebras Springer-Verlag Berlin Heidelberg New York 1967 Prof. Dr. G. B. Seligman Yale University Department of Mathematics New Haven, Conn. 06520/ USA ISBN-13: 978-3-642-94987-6 e-ISBN-13: 978-3-642-94985-2 DOl: 10.1007/978-3-642-94985-2 All rights reserved, especially that of translation into foreign languages It is also forbidden to reproduce this book, either whole or in part, by photomecbanical means (photostat, microfilm and/or microcard) or any other means without written pennission from the publishers ® by Springer-Verlag Berlin Heidelberg 1967 Softcover reprint of the hardcover 1s t edition 1967 Library of Congress Catalog Card Number 67-28452 Titel-Nr. 4584 Foreword The study of the structure of Lie algebras over arbitrary fields is now a little more than thirty years old. The first papers, to my know ledge, which undertook this study as an end in itself were those of JACOBSON (" Rational methods in the theory of Lie algebras ") in the Annals, and of LANDHERR ("Uber einfache Liesche Ringe") in the Hamburg Abhandlungen, both in 1935. Over fields of characteristic zero, these thirty years have seen the ideas and results inherited from LIE, KILLING, E. CARTAN and WEYL developed and given new depth, meaning and elegance by many contributors. Much of this work is presented in [47, 64, 128 and 234] of the bibliography. For those who find the rationalization for the study of Lie algebras in their connections with Lie groups, satisfying counterparts to these connections have been found over general non-modular fields, with the substitution of the formal groups of BOCHNER [40] (see also DIEUDONNE [108]), or that of the algebraic linear groups of CHEVALLEY [71], for the usual Lie group. In particular, the relation with algebraic linear groups has stimulated the study of Lie algebras of linear transformations. When one admits to consideration Lie algebras over a base field of positive characteristic (such are the algebras to which the title of this monograph refers), he encounters a new and initially confusing scene. It is not simply the case that new methods must be found to establish analogues of the theorems for characteristic zero,but rather that almost the only analogues which remain true (with the same degree of generality) are those whose traditional proofs tum out to have been independent of the characteristic anyway. Chapter V of this report deals with a number of analogues of fundamental classical theorems, and attempts in particular (Chap. V, § 4) to organize the rather awkward array of simple modular Lie algebras which would be totally unexpected to one acquainted only with the non-modular case. Chapter VI is an indication of some ways in which Lie algebras, especially those of prime characteristic, have arisen in other areas of mathematics; indeed, §§ 1 and 4 would be meaningless except for modular Lie algebras. Present indications seem to be that the Lie algebra is assured a lasting and prominent place in the theories of VI Foreword fonnal groups and algebraic groups of arbitrary characteristic, even though it does not serve as handily for general fields as for non-modular ones. An attempt has been made in §§ 2 and 3 to sketch its status in these theories at present. It is quite likely that progress, especially concerning group schemes, has already made my comments obsolete at this writing; such obsolescence is probably inevitable by the time this monograph reaches public view. I beg the pati(;nce of my better informed readers with my efforts to give a hint of these theories to readers who may be totally uninitiated. One setting in which a rather full modular analogue of the classical theory exists, without constituting a word-for~word translation of that theory, is that of Lie algebras with non-singular Killing forms. I have been influenced by the fact of my own participation in the develop ment of this analogue to give it a considerable amount of space (Chap. II, III, IV and part of Chap. I). I have tried to make the exposition of this material nearly complete and self-contained, whereas the rest of the book consists more often than not of summary comments with references to the appropriate literature. It has been my intention to include in the bibliography all papers known to me to be relevant, with the occasional exception of short research announcements (as in Comptes Rendus or Doklady) whose results have since been published in more complete fonn. I have leaned heavily on Mathematical Reviews for guidance to these papers, and may therefore be less than complete as to quite recent work. My unfamiliarity with Chinese and Japanese, and my inadequacy in Russian, may have caused me to give insufficient notice to work published in these languages. Joint papers are listed under the name of the (lexicographically) first author only; I hope that my colleagues who share with me the tail of the alphabet will not feel neglected thereby. No attempt has been made to set or to follow fixed procedures as to notation. Rather, I have chosen notation and tenninology on the basis of my own previous conditioning to the matter at hand and on the basis of the usage in original sources. Whenever a conflict between these guiding principles has arisen, I have not hesitated to follow my own preferences. As the reader will soon notice (perhaps to his annoy ance) these include a choice of what I regard as local clarity over global consistency and a quite conservative attitude toward terminology. I regret that Professor NAKAYAMA, who solicited this work for the Ergebnisse series, has been taken from us while it was in progress. He will be remembered with gratitude and deepest respect. It seems certain that I should not be authoring this volume, and quite likely that much of the material presented here would not yet have been developed, if it were not for the continuing contributions, Foreword VII both to mathematics and to my education, of Professor NATHAN JACOB SON. His methods and ideas are for me models of elegance and imagina tion. As teacher, colleague and friend, he has every right to such honor as I may do him. A multitude of colleagues and students have given valuable advice and assistance in the preparation of this work. Among the former, I cite P. CARTIER, WALTER FEIT, J. PETER MAY, T. TAMAGAWA, MAGUERITE FRANK, RICHARD BLOCK and JOHN WALTER. The last three have generous ly supplied me with reports of their work prior to publication. My students, RICHARD POLLACK and JAMES HUMPHREYS, have read portions of the manuscript and have taken me to task for some obscurities and errors. HARRY ALLEN and JOSEPH FERRAR have kept me abreast of developments concerning Lie algebras related to the exceptional Jordan algebras. As student and later as colleague, DAVID J. WINTER has been a frequent and very helpful critic and contributor. The secretarial staff of the Yale mathematics department and the editorial staff of the Springer-Verlag have been most generous and gra cious in the typing and further preparation of the manuscript for pub lication. Financial support during several summers from grants AFOSR-402-63, from the Air Force Office of Scientific Research, and NSF-GP-4017, from the National Science Foundation, as well as a Senior Faculty Fellowship from Yale University, assisted by National Science Foundation grant NSF- G P-6 558, during the academic year 1966-67, has contributed to the completion of the work while the author is still rela tively young. New Haven (Conn.), August 1967 G. B. SELIGMAN Contents Chapter I. Fundamentals . . . . . . . . 1. Definitions ... . . . . . . . . 1 2. The Poincan~-Birkhoff-Witt theorem 5 3. Free Lie algebras. Restricted Lie algebras 6 4. Iwasawa's theorem. . . . . . . . . . 10 5. Nilpotent Lie algebras. Engel's theorem 11 6. Cartan subalgebras . . . . . . . . . . 14 7. Semisimplicity. The Killing form 15 8. Trace forms, derivations, and restrictedness 17 9. Extension of the base ring . . . . . . 18 Chapter II. Classical Semisimple Lie Algebras. 21 1. The Cartan decomposition ..... 21 2. Split 3-dimensional algebras and applications 24 3. Classical Lie algebras. . . . . . . 28 4. Strings of roots and Cartan integers 30 5. Fundamental root systems 31 6. Semisimplicity and simplicity . . . 36 7. Determination of the fundamental systems 39 8. Existence of isomorphisms 42 9. The Weyl group ....... . 44 10. Existence of the classical algebras 45 11. Generalizations of the theory . . 48 Chapter III. Automorphisms of the Classical Algebras 50 1. The Chevalley groups .. . . . . . . . . . 50 2. The fundamental decomposition of G. Consequences 55 3. Structure of the Chevalley group . . 60 4. Conjugacy of Cartan subalgebras 63 5. Structure of the automorphism group 65 6. Realizations . . . . . . . . . . . . 66 Chapter IV. Forms of the Classical Lie Algebras 73 1. Forms and splitting fields . . . . . . . 73 2. Galois semi-automorphisms and i-cohomology 77 3. Simple involutorial algebras and the types A-D 79 4. Derivation algebras of alternative and Jordan algebras 84 5. Other types . . . . . . 86 6. Finite fields . . . . . . 88 7. On automorphism groups 93 Contents IX Chapter V. Comparison of the Modular and Non-modular Cases. 96 1. Solvable and nilpotent algebras 96 2. Representations . . . . . 98 3. Cohomology ........ . 101 4. Known simple Lie algebras . . 105 A. The Jacobson-Witt algebras }ill" 105 B. Some simple subalgebras of }ill" 106 a) The class 6" 107 b) The class j8" 107 c) The class ffi" 108 d) The class :r". 108 C. Algebras defined by finite groups of functions. 109 a) Generalized Witt algebras. . . . . . . . 109 b) Another generalization of the Witt algebra 109 c) The algebras of BLOCK . . . . . . . . . 110 D. Isomorphisms among known simple algebras. 110 5. Derivations . . . . . . . 112 6. Extension of the base field . . . . . . . 115 7. Cartan subalgebras. . . . . . . . . . . 116 8. Nilpotent elements and special subalgebras 121 Chapter VI. Related Topics ......... . 126 1. Nilpotent groups and Lie algebras. The restricted Burnside problem 126 2. Linear algebraic groups and Lie algebras. . . . . . . . 129 3. Formal groups, hyperalgebras and Lie algebras . . . . . . 133 4. Lie derivation algebras and purely inseparable extensions . 139 5. Infinite-dimensional analogues of the classical Lie algebras. 143 Bibliography. 146 Index . ... 163 Chapter I Fundamentals § 1. Definitions Let tr be a commutative ring with unit; by a Lie algebra over tr we understand a unitary tr-module B, together with a mapping (x, y) -+ [x y] from B X B into B which is a homomorphism in each of its variables when the other is fixed, and which satisfies in addition the following conditions: A nticommutativity : [x x] = 0; ] acobi identity: [[x y] z] + [[y z] x] + [[ z x] y] = 0; for all x, y, z E B. If tr = Z, the integers, B is called a Lie ring; clearly every Lie algebra may be regarded as a Lie ring. + + Consideration of the quantity [(x y) (x y)J - [x x] - [y y], which is zero in any Lie ring B, shows that + [xy] [yx] = 0 holds for all x, y E B, and conversely if the additive group of B is without 2-torsion, the condition (1) applied to y = x implies anti commutativity. If B and ID1 are Lie algebras over the commutative ring tr, we understand by a homomorphism of B into ID1 a mapping rJ: B -+ ID1 which is a homomorphism of tr-modules and which satisfies [x y] rJ = [x 'fJ, y'fJ] for all x, y E B. (We follow here the convention of writ ing mappings on the right of elements of their domains, as well as the concomitant convention that the product written rp 'fJ of two mapp ings rp, rJ represents the result of applying first rp, then rJ. Thus x (rp 'fJ) = (x rp) 'fJ, if x is an element of the domain of rp such that x rp is in sr the domain of 'fJ.) By an ideal in B is meant a submodule such that sr sr; [x y] E for all x E B, y E by (1), all ideals are two-sided. In this case, the quotient module Bjsr carries the structure of a Lie algebra over tr, the product being specified by requiring that the canonical mapping of B onto Bjsr be a homomorphism of Lie algebras. The fundamental homomorphism theorems of group and ring theory have their counterparts for Lie algebras. We cite: 2 1. Fundamentals (A) If 'YJ: £ -+ m is a homomorphism of Lie algebras, then the kernel ~ of 'YJ is an ideal in £, the image £ 'YJ is a subalgebra of m, and there is a unique isomorphism 'YJ' of £/ ~ onto £ 'YJ such that the diagram '1 £-£'YJ canon.~ /,/' £/~ is commutative. If cp is any homomorphism of £ into a Lie algebra m (over the same ring) such that the kernel of cp contains ~, there is a unique homomorphism 11' of £ 'YJ onto £ cp such that the diagram is commutative. The mapping ~ -+ ~ 'YJ is a bijection of the set of sub algebras of £ containing ~ onto the set of sub algebras of £ 'YJ, under which ideals in £ and ideals in £ 'YJ correspond. + (B) If m is a subalgebra of £ and if ~ is an ideal in £, then m ~ is a sub algebra of £, m r. ~ is an ideal in m, and there is a unique m/(m + isomorphism cp of r.~) onto (m ~)/~ making the diagram m .m+~ canon.~ ~canon. + commutative. Here the mapping of minto m ~ is the inclusion mapping. If ffi is an associative algebra over is, one verifies at once that the definition [x y] = x y - y x gives ffi the structure of a Lie algebra. If £ is a Lie algebra over is, and if ~ is an is-module, a representation of £ in ~ is a homomorphism of £ into the set of endomorphisms ~ (~) of ~, where ~(~) has the Lie algebra structure resulting as above from its structure as associative algebra (the associative product of endo morphisms being their composite). An associative embedding of £ is an isomorphism of £ onto a Lie subalgebra of an associative algebra ffi over is. Since every associative is-algebra ffi (1 E ffi is assumed) is mapped isomorphically onto a sub algebra of ~(ffi) by the map x -+ Rx, where y Rx = y x for all y E ffi, we see that if £ has an associative embedding, then £ is isomorphic to a Lie sub algebra of some ~ (~) , ~ an is-module; i.e., £ has a faithful representation. Certain Lie algebras and their subalgebras will be especially import ant in this exposition; they should also serve as examples to illustrate