(Springer monographs in mathematics) jean pierre serre complex semisimple lie algebras springer

o I 0 O~O i c fa :E Springer Springer Monographs in Mathematics Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Jean -Pierre Serre Complex Semisimple lie Algebras Translated from the French by G A Jones Reprint of the 1987 Edition Springer Jean-Pierre Serre College de France 7S231 Paris Cedex os France e-mail: serre@dmi.ensJr Translated By: G A Jones University of Southampton Faculty of Mathematical Studies Southampton S09 SNH United Kingdom Library of Congress CataIoging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Serre, Jean·Pierre: Complex semisimple Lie aIgeras I Jean-Pierre Serre Transl from the French by G A Jones.• Reprin t of the 1987 ed••• Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Springer monographs in mathematics) Einheitssacht.: Algebres de Lie semi-simples complexes ISBN 3'540.67827.1 This book is a translation of the original French edition Algebres de Lie Semi-Simples Complexes, published by Benjamin, New York in 1966 Mathematics Subject Classification (2000): 17BOS,I7B20 ISSN 1439-7382 ISBN 3-540-67827-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Viola· tions are liable for prosecution under the German Copyright Law Springer.Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer.Verlag Berlin Heidelberg 2001 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publica-tion does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typeset by Asco Trade Typesetting Ltd, Hong Kong Printed on acid-free paper SPIN 10734431 41/3142LK - Jean -Pierre Serre Complex Semisimple Lie Algebras Translated from the French by G A Jones Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Jean-Pierre Serre College de France 75231 Paris Cedex 05 France Translated by: G A.Jones University of Southampton Faculty of Mathematical Studies Southampton S09 5NH United Kingdom AMS Classifications: 17B05, 17B20 With Illustrations Library of Congress Cataloging-in-Publication Data Serre, ] ean -Pierre Complex semisimple Lie algebras Translation of: Algebres de Lie semi-simples complexes Bibliography: p Includes index Lie algebras I Title 512'.55 87-13037 QA251.S4713 1987 This book is a translation of the original French edition, Alg~bres de Lie Semi-Simples Complexes :£)1966 by Benjamin, New York © 1987 by Springer-Verlag New York Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Typeset by Asco Trade Typesetting Ltd., Hong Kong Printed and bound by R R Donnelley and Sons, Harrisonburg, Virginia Printed in the United States of America 543 ISBN 0-387-96569-6 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-96569-6 Springer-Verlag Berlin Heidelberg New York Preface These notes are a record of a course given in Algiers from 10th to 21 st May, 1965 Their contents are as follows The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple Lie algebras These are well-known results, for which the reader can refer to, for example, Chapter I of Bourbaki or my Harvard notes The theory of complex semisimple algebras occupies Chapters III and IV The proofs of the main theorems are essentially complete; however, I have also found it useful to mention some complementary results without proof These are indicated by an asterisk, and the proofs can be found in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII A final chapter shows, without proof, how to pass from Lie algebras to Lie groups (complex-and also compact) It is just an introduction, aimed at guiding the reader towards the topology of Lie groups and the theory of algebraic groups I am happy to thank MM Pierre Gigord and Daniel Lehmann, who wrote up a first draft of these notes, and also Mlle Fran~oise Pecha who was responsible for the typing of the manuscript Jean-Pierre Serre Contents CHAPTER I Nilpotent Lie Algebras and Solvable Lie Algebras 1 Lower Central Series 1 2 3 Definition of Nilpotent Lie Algebras An Example of a Nilpotent Algebra Engel's Theorems Derived Series Definition of Solvable Lie Algebras Lie's Theorem Cartan's Criterion CHAPTER II Semisimple Lie Algebras (General Theorems) Radical and Semisimplicity The Cartan-Killing Criterion Decomposition of Semisimple Lie Algebras Derivations of Semisimple Lie Algebras Semisimple Elements and Nilpotent Elements Complete Reducibility Theorem Complex Simple Lie Algebras The Passage from Real to Complex 4 5 6 7 8 CHAPTER III Cart an Subalgebras 10 Definition of Cartan Subalgebras Regular Elements: Rank 10 10 viii Contents The Cartan Subalgebra Associated with a Regular Element Conjugacy of Cartan Subalgebras The Semisimple Case Real Lie Algebras 11 12 15 16 IV The Algebra 17 CHAPTER 512 and Its Representations The Lie Algebra 512 Modules, Weights, Primitive Elements Structure of the Submodule Generated by a Primitive Element The Modules Wm Structure of the Finite-Dimensionalg-Modules Topological Properties of the Group SL Applications V Root Systems 17 17 18 19 20 21 22 CHAPTER 10 11 12 13 14 15 16 17 Symmetries Definition of Root Systems First Examples The Weyl Group Invariant Quadratic Forms Inverse Systems Relative Position of Two Roots Bases Some Properties of Bases Relations with the Weyl Group The Cartan Matrix The Coxeter Graph Irred ucible Root Systems Classification of Connected Coxeter Graphs Dynkin Diagrams Construction ofIrreducible Root Systems Complex Root Systems 24 24 25 26 27 27 28 29 30 31 33 34 35 36 37 38 39 41 CHAPTER VI Structure of Semisimple Lie Algebras 43 Decomposition of Proof of Theorem Borel Subalgebras Weyl Bases Existence and Uniqueness Theorems Chevalley's Normalization Appendix Construction of Semisimple Lie Algebras by Generators and Relations 43 45 47 48 50 51 52 60 VII Linear Representations of Semisimple Lie Algebras (2) Theorems and give a bijection between the elements co of 1)* and the classes of irreducible g-modules with a highest weight Finite-Dimensional Modules Proposition Let V be a finite-dimensional g-module Then one has (a) (b) (c) (d) V = LV" If n is a weight of V, 7t(H~) is an integer for all ex E R If V#- 0, V contains a primitive element If V is generated by a primitive element, V is irreducible By Theorem of Chap III, the elements of 1) are semisimple; the endomorphisms of V which they define are therefore diagonalizable (Chap II, Theorem 7) Since they commute with each other, they can be diagonalized simultaneously, giving (a) Statement (c) follows from Lie's theorem (Chap I, Theorem 3) applied to the solvable algebra b Statement (d) follows from Prop (4), combined with the complete reducibility theorem (Chap II, Theorem 8) Finally, if ex E R+, V can be viewed as a module over the Lie algebra s~ generated by X~, y~, H~ (cf Chap VI) By applying Theorem of Chap IV to this module, one sees that the eigenvalues of H~ on V belong to Z Since these eigenvalues are none other than the values n(H~), one gets (b) Corollary Every finite-dimensional irreducible g-module has a highest weight This follows from (c) In view of Theorems and 2, it only remains to characterize the elements co E 1)* which are highest weights of finite-dimensional irreducible modules Theorem Let co E 1)* and let Ew be an irreducible g-module having co as highest weight For Ew to be finite dimenSional, it is necessary and sufficient that one has (*) For all ex E R +, co(H~) is an integer ~ o (Since the simple inverse roots Hi form a base for the inverse roots H~, it is sufficient that the values co(Hj) be integers ~ 0.) The necessity of condition (*) follows from the fact that, if v is a primitive element of Ew for g, it is also a primitive element for the subalgebra s~ generated by X~, Ya, H~ By Corollary to Theorem in Chap IV, co(H~) must therefore be an integer ~ o Now let us show that condition (*) is sufficient Let v be a primitive element of Ew, and let i be an integer between and n Let us put mj = CO(Hi) and Vi = Yi mj+l v Finite-Dimensional Modules 61 If j =f i, Xj and 1'; commute One then has X·v· = y•m i+ X·v = 0• J' J Moreover, Theorem of Chap IV, applied to the subalgebra 51 generated by Xj, 1';, Hi' shows that Xit'j = O If Vi were nonzero, it would then be a primitive element of E(£), of weight w - (mi + l)~j,contradicting Theorem This proves that Vi = O Theorem of Chap IV now shows that the vector subspace Fi of EO) spanned by the elements 1'; Pv, ~ P ~ mi, is a finite-dimensional 5jsubmodule of E(£) N ow let 7; be the set of finite-dimensional 5i-submodules of EO), and E; their sum If FE 7;, one checks easily that g' FE 7;; it follows that E'l is a gsubmodule of E~J' Since EO) is irreducible and E; nonzero (it contains FJ, we have E; = EO) Thus we have proved that EO) is a sum of finite-dimensional 5i-submodules Let Pw be the set of weights of Ew We shall show that Pw is invariant under the symmetry Si associated with the root (Xi To see this, let 7f E Pw , and let y be a nonzero element of E~ By Theorem 1, Pi = 7f(Hi ) is an integer Let us put x = 1';Piy if Pi ~ 0, and x = Xi-Piy if Pi ~ O By Theorem of Chap IV, applied to i and to a finite-dimensional 5i-submodule of EO) containing y, one has x =f O Since the weight of x is equal to 11: - Pi~i = 11: - W(Hi)~i = Si(1I:), this shows that Sj(1I:) is a weight of E,o, and PO) is indeed invariant under Si' N ow let us prove that PO) is finite If 11: E PO)' Theorem shows that 11: can be written as where the coefficients Pi are integers ~O All that remains is to bound these coefficients Now, because -8 is a base for R, there is an element w of the Weyl group of R sending S to -S, and this element is a product of the symmetries Si (cf Sec V.IO) It follows that w(1I:) also belongs to PO)' and can therefore be written as W(1I:) = W - L qi~i Applying ~~,-1 to this formula, one finds 11: = w- 1(W) + L ri~i Pi + rj is with rj ~ O One concludes from this that equal to the coefficient Ci of (Xi in W - W-1(W); thus Pi ~ Cj , and the coefficients Pi are indeed bounded Thus, there are only finitely many weights of E(£) Since each of them has finite multiplicity (Theorem 1), and since EO) is the sum of the corresponding eigen-subspaces, EO) is finite dimensional, as required 62 VII Linear Representations of Semisimple Lie Algebras Remarks (1) In the course ofthis proof we have seen that the set Pw of weights of Ew is invariant under the Weyl group W In fact, if n E Pw and w E W, the weights nand w(n) have the same multiplicity For it is sufficient to see this when w = s,' and in this case one easily checks that the element sends the eigen-subspace corresponding to n to that corresponding to si(n) (cf Sec IV.S, Remark 1) (2) Let (Wi) be the basis of 1)* dual to the basis (H;): ifi #:j The w, are called the fundamental weights of the root system R (with respect to the chosen base S) Condition (*) of Theorem means that the linear form W is a linear combination of the weights Wi' the coefficients being integers ~O The irreducible modules having the weights Wj as highest weights are called the fundamental modules (or fundamental representations) of g An Application to the Weyl Group Proposition The Weyl group W acts simply transitively on the set of bases ofR We know (Sec V.10) that it acts transitively Hence it is sufficient to prove that, if w(S) = S, with wE W, then w = Let P be the set of fundamental weights We have w(P) = P If W E P, we know that w(w) is a weight of the fundamental module Ew with highest weight w By Theorem 1, it follows that W - w(w) is a linear combination of the simple roots (x" with coefficients ~O This applies to every W E P But on the other hand, we have L (w - w(w» = weP L weP W - L W = O weP This is impossible unless each of the summands W - w(w) is zero Since Pis a basis for 1)*, this indeed forces w = 1, as required Example: sIn+! Let g be the algebra sln+1 of square matrices of order n + and trace zero We take 1) to be the subalgebra consisting of the diagonal matrices H = (J 1""')'n+l)' with L)" = O The roots are the linear forms (Xi,j' i #:j, given by Characters 63 For a base, we take the roots O(i = O(i.i+1> ~ i corresponding to 0(; has components Ai = 1, Ai+l = The fundamental weights Wi are given by wi(H) ~ n The element Hi E 1) = ifj #- i, i + - , Aj ° = Ai + '" + Ai' The fundamental weight Wi is the highest weight of the natural representation of SIn+l on the vector space E = e+ More generally, Wi is the highest weight of the i-th exterior power of E (In fact, all the finite-dimensional irreducible representations of SIn+l can be obtained by decomposing the tensor powers of E; for more details, see H Weyl, The Classical Groups.) Characters Let P be the subgroup of 1)* consisting of the elements 11: such that 1I:(Ha) E Z for all 0( E R (or equivalently, for all 0( E S) The group P is a free abelian group, having a basis consisting of the fundamental weights Wi' ••• , W n• We will denote by A the group-algebra Z[PJ of the group P with coefficients in Z By definition, A has a basis (e")"EP such that e'" elf' = e"+lo oc; one can show that p(Hj ) = for all i, so that pEP (iii) We put D = (e lZ/ - e- IZ/2), t n IZ>O the product being evaluated in the algebra Z[tp] In fact, we have DE Z[P], since one can show that D= L "'·EW e(w)eW(Pl H WeyI's Formula 65 Theorem* Let E be a finite-dimensional irreducible g-module, and m its highest weight One has L = -' ch(E) D e(w)ew(w+P) weW The original proof of this theorem (Weyl, 1926) used the theory of compact groups (cf Seminaire S Lie, expose 21) A "purely algebraic" (but less natural) proof was found in 1954 by Freudenthal; it is reproduced in Jacobson's book (see also Bourbaki, Chap 8, Sec 9) Corollary The dimension of E is given by the formula dimE = TI TI a>O = (w + p,IY.) a>O (p,lY.) One deduces this from the theorem by computing the sum of the coefficients of ch(E) (cf Bourbaki, Chap 8, Sec 9) Corollary Let V be a finite-dimensional g-module, and let n(V, m) be the multiplicity with which E appears in a decomposition of V as a direct sum of irreducible modules Then n(V, w) is equal to the coefficient of e W +P in the product D·ch(V) This is a simple consequence of the theorem For = sl2' there is a unique positive root IY equal to 2p The group P consists of the integer multiples of p A highest weight m can be written as m = mp, with m ~ O Weyl's formula gives EXAMPLE ch(E) = e(M+1)P _ e-(M+l)p eP - e _ = e MP + e(M-2)P + + e- MP , P which is indeed consistent with the results in Chap IV CHAPTER VIII Complex Groups and Compact Groups This chapter contains no proofs All the Lie groups considered (except in Sec 7) are complex groups Cartan Subgroups From now on, G denotes a connected Lie group whose Lie algebra g is semisimple Such a group is called a complex semisimple group Let 1) be a Cartan subalgebra of g, and let H be the Lie subgroup of G corresponding to 1) The conjugates of H are called the Cartan subgroups of G Theorem (a) H is a closed group subvariety of G (b) H is a group of multiplicative type (i.e isomorphic to a product of groups C*) Let us describe the structure of H more precisely Let R be the root system of g with respect to fJ, let R* C fJ be the inverse system, let r be the subgroup of fJ generated by the elements Ha of R*, and let r be the subgroup of fJ consisting of those x E IJ such that a(x) E Z for all a E R One has Furthermore, let e: 1) -+ H be the map xf-+exp(2inx) This is a homomorphism, since 1) is abelian Characters 67 Theorem [...]... later.) 7 Complex Simple Lie Algebras The next few sections are devoted to the classification of these algebras We will state the result straight away: There are four series (the "four infinite families") All, BII , CII , and DII • the index n denoting the "rank" (defined in Chapter III) Here are their definitions: For n ~ 1, All = s[(n + 1) is the Lie algebra of the special linear group in 1 variables,... each complex simple Lie algebra 9 is the complexification of several nonisomorphic real simple Lie algebras; these are called the "real forms" of 9 For their classification, see Seminaire S Lie or Helgason CHAPTER III Cartan Subalgebras In this chapter (apart from Sec 6) the ground field is the field C of complex numbers The Lie algebras considered are finite dimensional 1 Definition of Cartan Subalgebras... Bibliography 72 Index 73 CHAPTER I Nilpotent Lie Algebras and Solvable Lie Algebras The Lie algebras considered in this chapter are finite-dimensional algebras over a field k In Sees 7 and 8 we assume that k has characteristic O The Lie bracket of x and y is denoted by [x, y], and the map y 1 + [x, y] by ad x 1 Lower Central Series Let 9 be a Lie algebra The lower central series of 9 is the descending series... nondegenerate 3 Decomposition of Semisimple Lie Algebras Theorem 3 Let 9 be a semisimple Lie algebra, and Q an ideal of g The orthogonal space Q' of Q, with respect to the Killing form of g, is a complement for Q in g; the Lie algebra 9 is canonically isomorphic to the product Q x Q' Corollary Every ideal, every quotient, and every product of semisimple algebras is semisimple Definition 2 A Lie algebra s is said... [90 g] 5 Semisimple Elements and Nilpotent Elements 7 4 Derivations of Semisimple Lie Algebras First recall that if A is an algebra, a derivation of A is a linear mapping + A satisfying the identity D: A D(x' y) = Dx' y + x' Dy The derivations form a Lie subalgebra Der(A) of End(A) In particular, this applies to the case where we take A to be a Lie algebra g A derivation D of 9 is called inner if... x E g, or in other words if D belongs to the image of the homomorphism ad: 9 -+ Der(g) Theorem 5 Every derivation of a semisimple Lie algebra is inner Thus the mapping ad: 9 -+ Der(g) is an isomorphism Corollary Let G be a connected Lie group (real or complex) whose Lie algebra 9 is semisimple Then the component AutO G of the identity in the automorphism group Aut G of G coincides with the inner automorphism... contained in a Cartan subalgebra of g Remark One can show that every maximal abelian subalgebra of 9 consisting of semisimple elements is a Cartan subalgebra of g However, if 9 i= 0 there are 16 III Carlan Subalgebras maximal abelian subalgebras of ~ which contain nonzero nilpotent elements, and which are therefore not Cartan subalgebras 6 Real Lie Algebras Let 90 be a Lie algebra over R, and g its complexification... follows from the fact that the Lie algebra of AutO G coincides with Der(g) Remark The automorphisms of 9 induced by the inner automorphisms of G are (by abuse of language) called the inner automorphisms of g When 9 is semisimple, they form the component of the identity in the group Aut(g) 5 Semisimple Elements and Nilpotent Elements Definition 3 Let 9 be a semisimple Lie algebra, and let x E g (a)... 4 A Lie algebra 9 is semisimple if and only if it is isomorphic to a product of simple algebras In fact, this decomposition is unique More precisely: Theorem 4' Let 9 be a semisimple Lie algebra, and (Q,) its minimal nonzero ideals The ideals product Qi are simple Lie algebras, and 9 can be identified with their Clearly, if s is simple we have s = [s,s] Thus Theorem 4 implies: Corollary, If 9 is semisimple. .. be a Lie algebra and let n be an ideal contained in the center of g Then: 9 is nilpotent.-g/n is nilpotent The above two propositions show that the nilpotent Lie algebras are those one can form from abelian algebras by successive "central extensions." (Warning: an extension of nilpotent Lie algebras is not in general nilpotent.) 3 An Example of a Nilpotent Algebra Let V be a vector space of finite ... Jones.• Reprin t of the 1987 ed••• Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Springer monographs in mathematics) Einheitssacht.:... SNH United Kingdom Library of Congress CataIoging -in- Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Serre, Jean·Pierre: Complex semisimple Lie aIgeras I Jean-Pierre. ..Springer Monographs in Mathematics Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Jean -Pierre Serre Complex Semisimple lie Algebras Translated