Complex Semisimple Lie Algebras

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Springer

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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: [email protected] 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 <engt> 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

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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 - 5 4 3 2 1 0

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 6 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. 1. 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. 9 8 7 6 543 2 1 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

1. Lower Central Series

1 1 2 2 3 3

2. 3. 4. 5. 6. 7. 8.

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)

1. 2. 3. 4. 5. 6. 7. 8.

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 4

5 5 6 6

7 7 8 8 9

CHAPTER III Cart an Subalgebras

10

1. Definition of Cartan Subalgebras 2. Regular Elements: Rank

10 10

viii

Contents

3. The Cartan Subalgebra Associated with a Regular Element 4. Conjugacy of Cartan Subalgebras 5. The Semisimple Case 6. Real Lie Algebras

11 12 15 16

IV The Algebra

17

CHAPTER

1. 2. 3. 4. 5. 6. 7.

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 2 Applications

V Root Systems

17 17 18 19

20 21 22

CHAPTER

1. 2. 3. 4. 5. 6. 7. 8. 9. 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

1. Decomposition of 9 2. Proof of Theorem 2 3. Borel Subalgebras 4. Weyl Bases 5. Existence and Uniqueness Theorems 6. Chevalley's Normalization Appendix. Construction of Semisimple Lie Algebras by Generators and Relations

43

45 47 48 50 51 52

Contents

VII Linear Representations of Semisimple Lie Algebras

ix

CHAPTER

1. 2. 3. 4. 5. 6. 7. 8.

Weights Primitive Elements Irreducible Modules with a Highest Weight Finite-Dimensional Modules An Application to the Weyl Group Example: sl'+1 Characters H. Weyl's formula

CHAPTER VIII Complex Groups and Compact Groups

1. 2. 3. 4. 5. 6. 7.

Cartan Subgroups Characters Relations with Representations Borel Subgroups Construction of Irreducible Representations from Borel Subgroups Relations with Algebraic Groups Relations with Compact Groups

56 56 57 58

60 62 62 63

64

66

66 67

68 68 69 70 70

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 (en g)n;a. I of ideals of 9 defined by the formulae elg=g

eng = [g,en-1g]

ifn

~

2.

We have and

2. Definition of Nilpotent Lie Algebras Definition 1. A Lie algebra 9 is said to be nilpotent such that eng = O.

if there exists an integer n

More precisely, one says that 9 is nilpotent of class ~ r if C+ 1 9 = O. For r = 1, this means that [g, g] = 0; that is, 9 is abelian.

2

I. Nilpotent Lie Algebras and Solvable Lie Algebras

Proposition 1. The following conditions are equivalent: (i) 9 is nilpotent of class ~ r. (ii) For all xo, ... , x, E g, we have

[XO,[x1,[ ... ,x,] ... ]] = (adxo)(adxd ... (adx,_d(x,) = O. (iii) There is a descending series of ideals 9

= no

such that [g, n,] c ni+1 for 0

~

~

n1

~ ••• ~

a, = 0

i ~ r - 1.

Now recall that the center of a Lie algebra 9 is the set of x

E

9 such that

[x, y] = 0 for all y E g. It is an abelian ideal of g.

Proposition 2. Let 9 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 dimension n. A flag D = (Di) of descending series of vector subspaces V

= Do ~ Dl

~ '" ~ D"

v

is a

=0

of V such that codim Di = i. Let D be a flag, and let n(D) be the Lie subalgebra of End(V) = gl(V) consisting of the elements x such that x(D,) c Di + 1 • One can verify that n(D) is a nilpotent Lie algebra of class n - 1.

4. Engel's Theorems Theorem 1. For a Lie algebra 9 to be nilpotent, it is necessary and sufficient for ad x to be nilpotent for each x E g. (This condition is clearly necessary,

cr. Proposition 1.)

Theorem 2. Let V be a finite-dimensional vector space and 9 a Lie subalgebra of End( V) consisting of nilpotent endomorph isms. Then:

6. Definition of Solvable Lie Algebras

3

(a) 9 is a nilpotent Lie algebra. (b) There is a flag D of V such that 9 c n(D).

We can reformulate the above theorem in terms of g-modules. To do this, we recall that if 9 is a Lie algebra and V a vector space, then a Lie algebra homomorphism ;: 9 -+ End(V) is called a g-module structure on V; one also says that; is a linear representation of 9 on V. An element v E V is called invariant under 9 (for the given g-module structure) if ;(x)v = 0 for all x E g. (This surprising terminology arises from the fact that, if k = R or C, and if; is associated with a representation of a connected Lie group G on V, then v is invariant under 9 if and only if it is invariant-this time in the usual senseunder G.) With this terminology, Theorem 2 gives: Theorem 2'. Let;: 9 -+ End(V) be a linear representation of a Lie algebra 9 on a nonzero finite-dimensional vector space V. Suppose that ;(x) is nilpotent for all x E g. Then there exists an element v ::F 0 of V which is invariant under g.

5. Derived Series Let 9 be a Lie algebra. The derived series of 9 is the descending series (Dn g)n;!o 1 of ideals of 9 defined by the formulae

Dlg =g D/l g = [Dn-lg,D"-lg]

ifn

~

2.

One usually writes Dg for D2g = [g, g].

6. Definition of Solvable Lie Algebras Definition 2. A Lie algebra 9 is said to be solvable such that Dng = O.

if there exists an integer n

Here again, one says that 9 is solvable of derived length ~ r if Dr + l 9

= O.

1. Every nilpotent algebra is solvable. 2. Every subalgebra, every quotient, and every extension of solvable algebras is solvable. 3. Let D = (Dj ) be a flag of a vector space V, and let b(D) be the subalgebra of End(V) consisting ofthe x E End(V) such that x(D/) c D/ for all i. The algebra b(D) (a "Borel algebra") is solvable. EXAMPLES.

4

I. Nilpotent Lie Algebras and Solvable Lie Algebras

Proposition 3. The following conditions are equivalent: (i) 9 is solvable of derived length :s;; r. (ii) There is a descending series of ideals of g:

9

= a o ::J a 1

::J '"

::J

ar = 0

such that [ai' a;] c ai+ 1 for 0 :s;; i :s;; r - 1 (which amounts to saying that that the quotients adai+l are abelian). Thus one can say that solvabie Lie algebras are those obtained from abelian Lie algebras by successive "extensions" (not necessarily central).

7. Lie's Theorem We assume that k is algebraically closed (and of characteristic zero). Theorem 3. Let lP: 9 -+ End(V) be a finite-dimensional linear representation of a Lie algebra g. If 9 is solvable, there is a flag D of V such that lP(g) c b(D). This theorem can be rephrased in the following equivalent forms. Theorem 3'. If 9 is solvable, the only finite-dimensional g-modules which are simple (irreducible in the language of representation theory) are one dimensional. Theorem 3". Under the hypotheses of Theorem 3, if V =F 0 there exists an element v =F 0 of V which is an eigenvector for every lP(x), x E g. The proof of these theorems uses the following lemma. Lemma. Let 9 be a Lie algebra, ~ an ideal of g, and lP: 9 -+ End(V) a finitedimensional linear representation of g. Let v be a nonzero element of V and let A. be a linear form on ~ such that A. (h) v = lP(h)v for all h E~. Then A. vanishes on [g,q].

8. Cartan's .Criterion It is as follows:

Theorem 4. Let V be a finite-dimensional vector space and 9 a Lie subalgebra of End(V). Then: 9 is solvable- Tr(x

0

y)

=0

for all x

E

9. Y E [g, g].

(This implication => is an easy corollary of Lie's theorem.)

CHAPTER II

Semisimple Lie Algebras (General Theorems)

In this chapter, the base field k is a field of characteristic zero. The Lie algebras and vector spaces considered have finite dimension over k.

1. Radical and Semisimplicity Let 9 be a Lie algebra. If Qand b are solvable ideals of 9, the ideal Q + b is also solvable, being an extension ofb/(Q n b) by Q. Hence there is a largest solvable ideal t of 9. It is called the radical of 9. Definition 1. One says that 9 is semisimple

if its radical t

is O.

This amounts to saying that 9 has no abelian ideals other than O.

If V is a vector space, the subalgebra sI(V) of End(V) consisting of the elements of trace zero is semisimple.

EXAMPLE.

(See Sec. 7 for more examples.) Theorem 1. Let 9 be a Lie algebra and t its radical. (a) 9/t is semisimple. (b) There is a Lie subalgebra s of 9 which is a complement for t.

If s satisfies the condition in (b), the projection s -+ 9/t is an isomorphism, showing (with the aid of (a» that s is semisimple. Thus 9 is a semidirect product of a semisimple algebra and a solvable ideal (a "Levi decomposition").

II. Semisimple I ie Algebras (General Theorems)

6

2. The Cartan-Killing Criterion Let 9 be a Lie algebra. A bilinear form B: 9 x 9 -+ k on 9 is said to be invariant if we have B([x,y],z) + B(y,[x,z]) = 0 for all x, y, z E g. The Killing form B(x,y)

= Tr(adx 0 ady) is invariant and symmetric.

Lemma. Let B be an invariant bilinear form on g, and Q an ideal of g. Then the

orthogonal space

Q'

of Q with respect to B is an ideal of g.

(By definition, Q' is the set of all y E 9 such that B(x, y)

= 0 for all x E Q.)

Theorem 2 (Cartan-Killing Criterion). A Lie algebra is semisimple if and only

if its Killing form is 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 to be simple if: (a) it is not abelian, (b) its only ideals are 0 and s. EXAMPLE.

The algebra sI(V) is simple provided that dim V

~

2.

Theorem 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 then 9

= [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 D = ad x for some 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 group of G. This 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) x is said to be nilpotent if the endomorphism ad x of 9 is nilpotent. (b) x is said to be semisimple if ad x is semisimple (that is, diagonalizable after extending the ground field). Theorem 6. If 9 is semisimple, every element x of 9 can be written uniquely in the form x = s + n, with n nilpotent, s semisimple, and [s, n] = O. Moreover, every element y E 9 which commutes with x also commutes with sand n. One calls n the nilpotent component of x, and s its semisimple component. Theorem 7. Let ;: 9 -+ End(V) be a linear representation of a semisimple Lie algebra. If x is nilpotent (resp. semisimple), then so is the endomorphism ;(x).

II. Semisimple Lie Algebras (General Theorems)

8

6. Complete Reducibility Theorem Recall that a linear representation lP: 9 -+ End(V) is called irreducible (or simple) if V i= 0 and if V has no invariant subspaces (submodules) other than oand V. One says that lP is completely reducible (or semisimple) ifit is a direct sum of irreducible representations. This is equivalent to the condition that every invariant subspace of V has an invariant complement. Theorem 8 (H. Weyl). Every (finite-dimensional) linear representation of a semisimple algebra is completely reducible.

(The algebraic proof of this theorem, to be found in Bourbaki or Jacobson, for example, is somewhat laborious. Weyl's original proof, based on the theory of compact groups (the "unitarian 1 trick") is simpler; we shall return to it 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, SL(n + 1). For n ~ 2, BII = so(2n + 1) is the Lie algebra of the special orthogonal group in 2n + 1 variables, SO(2n + 1). For n ~ 3, CII = sp(2n) is the Lie algebra of the symplectic group in 2n variables, Sp(2n). For n ~ 4, D" = so(2n) is the Lie algebra ofthe special orthogonal group in 2n variables, SO(2n). n

+

(One can also define BII , CII , and DII for n

~

1, but then:

-There are repetitions (Al = Bl = C 1 , B2 = C2, A3 = D3)' - The algebras Dl and D2 are not simple (Dl is abelian and one dimensional, and D2 is isomorphic to Al x Ad.) In addition to these families, there are five "exceptional" simple Lie algebras, denoted by G2 , F4 , E 6 , E 7 , and E 8 • Their dimensions are, respectively, 14,52, 78, 133, and 248. The algebra G2 is the only one with a reasonably "simple" definition: it is the algebra of derivations of Cayley's octonion algebra. 1 This is often referred to as the "unitary trick"; however Weyl, introducing the idea in his book "The Classical Groups," used the more theological word "unitarian," and we will follow him.

8. The Passage from Real to Complex

9

8. The Passage from Real to Complex Let 90 be a Lie algebra over R, and 9 = 90 ® C its complexification. Theorem 9. 90 is abelian (resp. nilpotent, solvable, semisimple) if and only if 9 is. On the other hand, 90 is simple if and only if 9 is simple or of the form 5 x $, with 5 and $ simple and mutually conjugate. Moreover, 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 Let 9 be a Lie algebra, and n a subalgebra of g. Recall that the normalizer of n in 9 is defined to be the set n(n) of all x E 9 such that ad(x)(n) c n; it is the largest subalgebra of 9 which contains n and in which n is an ideal. Definition 1. A sub algebra f) of 9 is called a Cartan subalgebra of 9 satisfies the following two conditions:

if

it

(a) f) is nilpotent. (b) f) is its own normalizer (that is, f) = n(f)).

We shall see later (Sec. 3) that every Lie algebra has Cartan subalgebras.

2. Regular Elements: Rank Let 9 be a Lie algebra. 1£ x E g, we will let PAT) denote the characteristic polynomial of the endomorphism adx defined by x. We have Px(T)

= det(T -

If n = dim g, We can write PAT) in the form

ad (x».

3. The Cart an Subalgebra Associated with a Regular Element

11

;=n

PAT)

=

L

a;(x)Ti.

i=O

If X has coordinates Xl' •.. , XII (with respect to a fixed basis of g), we can view a;(x) as a function ofthe n complex variables Xl' ••• , XII; it is a homogeneous polynomial of degree n - i in Xl"'" XII' Definition 2. The rank of 9 is the least integer I such that the function a l defined abol'e is not identically zero. An element X Egis said to be regular if alx:) i= O. Remarks. Since nilpotent.

a,.

=

1, we must have I :s;; n with equality if and only if 9 is

On the other hand, if X is a nonzero element of 9 then ad(x)(x) = 0, showing that 0 is an eigenvalue of adx. It follows that if 9 i= 0 then ao = 0, so that I ~ 1. Proposition 1. Let 9 be a Lie algebra. The set gr of regular elements of 9 is a connected, dense, open subset of g. We have gr = 9 - V, where V is defined by the vanishing of the polynomial function a,. Clearly gr is open. Now if the interior of V were nonempty, the function a" vanishing on V, would be identically zero, against the definition of the rank. Finally, if x, y E g" the (complex) line D joining X and y meets V at finitely many points. We deduce that D n gr is connected, and hence that X and y belong to the same connected component of gr; thus gr is indeed connected.

3. The Cartan Subalgebra Associated with a Regular Element Let X be an element of the Lie algebra g. If ), E C, we let g~ denote the nilspace of ad (x) - ),; that is, the set of y E 9 such that (ad(x) - ,{)Py = 0 for sufficiently large p. In particular, g~ is the nilspace of ad x. Its dimension is the multiplicity of o as an eigenvalue of ad x; that is, the least integer i such that a;(x) i= o. Proposition 2. Let

X E

g. Then:

(a) 9 is the direct sum of the nilspaces g~. (b) [g~, g~J c g~+" if A, JL E C. (c) g~ is a Lie sub algebra of g.

Statement (a) is obtained by applying a standard property of vector space endomorphisms to ad x. To prove (b), we must show that, if y E g; and Z E g~,

12

III. Cart an SubaIgebras

then [y, z]

E

g:+". Now we can use induction to prove the formula

(ad x - A - Ilny,z] =

±(n)

p=o p

[(ad x - A)l'y, (ad x - Ilrpz].

If we take n sufficiently large, all terms on the right vanish, showing that [y, z] is indeed in g~+". Finally, (c) follows from (b), applied to the case A = Il = 0.

Theorem 1. If x is regular, to the rank I of g.

g~

is a Cartan subalgebra of g; its dimension is equal

First, let us show that g~ is nilpotent. By Engel's Theorem (cf. Chapter I) it is sufficient to prove that, for each y E g~, the restriction of ad y to g~ is nilpotent. Let ad 1 y denote this restriction, and ad 2 y the endomorphism induced by ady on the quotient-space g/g~. We put U = {y V

E g~ladl

y is not nilpotent}

= {y E g~lad2 y is invertible}.

The sets U and V are open in g~. The set V is nonempty: it contains the element x. Since V is the complement of an algebraic subvariety of g~, it follows that V is dense in g~. If U were nonempty, it would therefore meet V. However, let y E Un V. Since y E U, ad 1 y has as an eigenvalue with multiplicity strictly less than the dimension of g~, this dimension being visibly equal to the rank I of g. On the other hand, since y E V, is not an eigenvalue of ad 2 y. We deduce that the multiplicity of as an eigenvalue of ad y is strictly less than I, contradicting the defmition of I. Thus U is empty, and so g~ is indeed a nilpotent algebra. We now show that g~ is equal to its normalizer n(g~). Let z E n(g~). We have ad z(g~) c g~, and in particular [z, x] E g~. By the definition of g~, there is therefore an integer p such that (ad x)P[z, x] = 0, giving (ad X)p+l Z = 0, so that z E g~ as required.

°

°

°

Remark. The above process provides a construction for Cartan subalgebras; we shall see that in fact it gives all of them.

4. Conjugacy of Cartan Subalgebras Let 9 be a Lie algebra. We let G denote the inner automorphism group of g; that is, the subgroup of Aut(g) generated by the automorphisms ead(y) for y E g. Theorem 2. The group G acts transitively on the set of Cartan subalgebras of g. Combining this theorem with Theorem 1, we deduce:

4. Conjugacy of Cartan Subalgebras

13

Corollary 1. The dimension of a Cartan subalgebra of g is equal to the rank of g. Corollary 2. Erery Cartan subalgebra of g has the form g~ for some regular element x of g. FIRST PART OF THE PROOF. In this part, I) denotes a Cartan subalgebra of g. If x E I), we let ad I x (resp. adz x) denote the endomorphism of I) (resp. g/I) ind uced by x. Lemma 1. Let V = {x

E

~ I ad2 x is invertible}. The set V ist nonempty.

Let us apply Lie's Theorem (cf. Chapter I) to the I)-module g/I). This gives a flag: o = Do C DI C ... Dm = g/I) stable under I). Now I) acts on the one-dimensional space DJDi + 1 by means of a linear form :ti: if x

E

I),

::

E

Di ,

we have x . Z

== :ii(X)Z mod Di - l .

(To simplify the notation, we write X· Z instead of adz x(z).) The eigenvalues of ad 2 x are (XI (x), ... , (Xm(x). Hence it is sufficient to prove that none of the forms (Xi is identically zero. Suppose, for example, that (Xl' ... , (Xk-l i= 0 and (Xk is identically zero. Let Xo E I) be chosen so that (Xl (Xo) i= 0, ... , (Xk-l (X O) i= O. The endomorphism of Dk - l (resp. of Dk ) induced by ad 2 Xo is invertible (resp. has 0 as an eigenvalue with multiplicity 1). The nilspace D of ad 2 Xo in Dk is therefore one dimensional and is a complement for Dk- l in Dk. We shall show that the elements zED are annihilated by each ad 2 x, x E I). This is clear for Xo. Furthermore, we can use induction on n to prove the formula (z ED). x~x'Z = «adxofx)·z Since the algebra I) is nilpotent, we have (ad xof x = 0 for sufficiently large n. This shows that X· z belongs to the nilspace of ad 2 Xo in Dk, that is, X· zED. On the other hand,ad 2 x maps Dk intoDk_l ; we therefore have X· ZED n Dk- l , so x . z = 0, proving that z is indeed annihilated by each element of I). We now take z to be a nonzero element of D, and let z be a representative of z in g. The condition that X· z = 0 for all x E I) can be reinterpreted as [x, z] E I) for all x E I); thus z belongs to the normalizer n@ off). Since z is not in f) (because z i= 0), we have n(I) i= I), contradicting the definition of a Cartan subalgebra. Lemma 2. Let W = G· V be the union of the transforms of V under the action of the group G. The set W is open in g. Let x E V. It is sufficient to show that W contains a neighborhood of x. Consider the map (g, 11) ...... g' t' from G x V to g, and let () be its tangent map

III. Carlan Sub algebras

14

e

at the point (1, x). We shall see that the image of is the whole of g. Certainly this image contains the tangent space at V, namely 9. On the other hand, if y E 9 the curve tf-+e'ad(y)X = 1 + t[y,x] + ... has [y, x] as its tangent vector at the origin. We deduce from this that Im(adx) c Im(e). But since x E V, adx induces an automorphism of 9/9, and we have Im(adx)

+ 9=

g,

so that Im(e) = g. The Implicit Function Theorem now shows that the map G x V -+ 9 is open at the point (1, x), giving the lemma.

Lemma 3. There is a regular element x~f 9 such that

9 = g~.

Let us keep the preceding notation. Lemmas 1 and 2 show that W is open and nonempty. It therefore intersects the set gr of regular elements of 9 (cf. Prop. 1). Now if g' x is regular, it is clear that x is regular. We deduce that V contains at least one regular element x. Since ad! x is nilpotent and ad 2 x invertible, we indeed have 9 = g~. SECOND PART OF THE PROOF. We know, thanks to Lemma 3, that the Cartan subalgebras of 9 all have the form g~, with x E gr' Consider the following equivalence relation R on gr: R(x, y) ~ g~ and g~ are conjugate under G.

Lemma 4. The equivalence classes of R are open in gr' We must prove that, if x E g" every y sufficiently close to x is equivalent to x. We will apply the results of the first part of the proof to the Cartan subalgebra 9 = g~. The corresponding set V contains x. By Lemma 2, G' V is open. Hence each element y sufficiently close to x has the form g' x', with 9 E G and x' E V. We then have g~ = g' g~, = g' f) = g' g~, showing that x and yare indeed equivalent.

Since the equivalent classes of R are open, and since gr is connected (Prop. 1), there can be only one equivalence class. This shows that the Cartan subalgebras are indeed conjugate to each other, thus completing the proof of Theorem 2.

Remark. Theorem 2 remains true if one replaces the group G with the subgroup generated by the automorphisms of the form ead(Y) with ad(y) nilpotent. This form of the theorem has been extended by Chevalley to the case of an arbitrary algebraically closed base field (of characteristic zero). See expose 15 of Seminaire Sophus Lie, as well as Bourbaki, Chap. VII, Sec. 3.

5. The Semisimple Case

15

5. The Semisimple Case Theorem 3. Let g be a Cartan subalgebra of a semisimple Lie algebra g. Then: (a) 1) is abelian. (b) The centralizer of g is g. (c) Every element ofg is semisimple (cf. Sec. 11.5). (d) The restriction of the Killing form of 9 to g is nondegenerate.

h

(d) By Corollary 2 to Theorem 2, there is a regular element x such that g~. Let

=

be the canonical decomposition of 9 with respect to x (cf. Prop. 2). If B denotes the Killing form of g, then a simple calculation shows that g~ and g~ are orthogonal with respect to B provided that A + Jl i= O. We therefore have a decomposition of 9 into mutually orthogonal subspaces 9 = g~ Efj

L

(g~ Efj g;.l.).

.. #0

Since B is nondegenerate, so is its restriction to each of these subspaces, giving (d) since g = g~. (a) By applying Cartan's criterion to g and to the representation ad: g ---+ End(g), we see that Tr(ad x 0 ad y) = 0 for x E g and y E [g, f)]. In other words, [g, g] is orthogonal to g with respect to the Killing form B. Because of (d), this implies that [g, g] = O. (b) Being abelian, g is contained in its own centralizer c(g). Moreover, c(g) is clearly contained in the normalizer n(g) ofg. Since n(g) = g, we have c(g) = g. (c) Let x E g, and let s (resp. n) be its semisimple (resp. nilpotent) component (cf. Sec. 11.5). If y E g, then y commutes with x and hence also with sand n (Chapter II, Theorem 6). We therefore have s, n E c(g) = g. However, since y and n commute and ad(n) is nilpotent, ad(y) 0 ad(n) is also nilpotent and its trace B(y, n) is zero. Thus n is orthogonal to every element ofg. Since it belongs to g, n is zero by (d). Thus x = s, which shows that x is indeed semisimple. Corollary 1. g is a maximal abelian subalgebra of g. This follows from (b). Corollary 2. Every regular element of 9 is semisimple. This is because such an element is 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. The concepts of Cartan subalgebra, regular element, and rank are defined for go as in the complex case. Moreover, the rank of go is equal to that of g; a subalgebra f)o of go is a Cartan subalgebra if and only if its complexification f) is a Cartan subalgebra of g: an element of 90 is regular in go if and only if it is so in g. Theorems 1 and 3 remain true (in particular, showing the existence of Cartan subalgebras). However, this does not apply to Theorem 2: all one can say is that the Cartan subalgebras of 90 are divided into finitely many classes modulo the inner automorphisms of go. (This is because the set ofregular elements of 90 is not necessarily connected, but rather a finite union of connected open sets.) A precise description of these classes is to be found in B. Kostant, Proc. Nat. Acad. Sci. USA, 1955. For more details on Cartan subalgebras, see Bourbaki, Chapter 7.

CHAPTER IV

The Algebra 512 and Its Representations

In this chapter (apart from Sec. 6) the ground field is the field C of complex numbers.

1. The Lie Algebra sI2 This is the algebra of square matrices of order 2 and trace zero. We shall denote it by g. One can easily verify that it is a simple algebra, of rank 1. It has as a basis the three elements

X=(~ ~). H=(~

0) Y= (01 0)0·

-1 '

We have [X, Y]

= H,

[H,X]

= 2X,

[H, Y]

=

-2Y.

The endomorphism ad(H) has three eigenvalues: 2, 0, -2. It follows that H is semisimple; the line ~ = C· H spanned by H is a Cartan subalgebra of g, called the canonical Cartan subalgebra. The elements X, Yare nilpotent. The subalgebra b of 9 generated by Hand X is solvable; this is the canonical Borel subalgebra of g.

2. Modules, Weights, Primitive Elements Let V be a g-module (not necessarily finite-dimensional~ If A E C, we will let VA denote the eigenspace of H in V corresponding to A.; that is, the set of all x E V such that Hx = Ax. An element of VA is said to have weight A..

18

IV. The Algebra 1[2 and Its Representations

Proposition 1. (a) The sum L;'eC VA is direct. (b) If x has weight )., then Xx has weight ). + 2 and Yx has weight A. - 2. (a) merely expresses the well-known fact that the eigenvectors corresponding to distinct eigenvalues are linearly independent. Moreover, if Hx = ).x we have HXx = [H,X]x

and so X x has weight i.

+ XHx

= 2Xx

+ ).Xx =

().

+ 2)Xx,

+ 2. A similar argument applies to Yx.

Remark. When Vis finite dimensional, the sum L V" is equal to V (this follows, for example, from the fact that H is semisimple; cf. Chapter II, Theorem 7). This is no longer true when V is infinite dimensional. Definition 1. Let V be a g-module. and let ). E C. An element e primitive of weight ).

E

V is said to be

if it is nonzero and if we have Xe = 0,

He = i.e.

Proposition 2. For a nonzero element e of the g-module V to be primitive, it is necessary and sufficient that the line it spans should be stable under the Borel algebra b. This condition is clearly necessary. Conversely, if Ce is stable under b then we have Xe = J,Le, He = A.e, with i.., Jl E C. Using the formula [H,X] = 2X, we see that 2Jl = 0, so Jl = and e is indeed primitive.

°

Proposition 3. Every nonzero finite-dimensional g-module contains a primitive element.

This follows from Lie's Theorem (cf. Chapter I. Theorem 2'). (Alternative proof: one chooses an eigenvector x for H. and takes the last nonzero term in the sequence x, Xx, X 2 x, .... This is a primitive element.)

3. Structure of the Submodule Generated by a Primitive Element Theorem 1. Let V be a g-module and e E V a primitive element of weight i.. Let us put en = yne/n! for n ~ 0, and e_ 1 = O. Then we hat'e (i) Hen = (). - 2n)e n

+ l)en+l (A. - n + l)en-l

(ii) Yen = (n (iii) X en =

for all n ~ O.

4. The Modules W..

19

Formula (i) asserts that en has weight J. - 2n, which follows from Prop. 1. Formula (ii) is obvious. Formula (iii) is proved by induction on n (the case n = being true because of the convention that C 1 = 0); for we have

°

nXen

=

XYen-l

=

[X, Y]en - 1 + YXen- 1

= (i, =

+ 2 + (J. n + l)en -l'

- 2n

n(). -

= Hen- 1 + (J. - n + 2)Yen- 2 n + 2)(n - l))en - 1

which gives (iii) on dividing by n. Corollary 1. Only two cases arise: either (a) the elements (en), 11 ~ 0, are all linearly independent,

or (b) the weight ). of e is an integer m ~ 0, the elements eo, ... , em are linearly independent, and e; = 0 for i > m.

Since the elements ei have distinct weights, those which are nonzero are linearly independent (cf. Prop. 1). If they are all nonzero, then we have case (a). Otherwise, there is an integer m ~ 0 such that eo, .. " em are nonzero, and em + 1 = em +2 = ... = O. Applying formula (iii) with n = m + 1, we obtain

However, em+1

=

°

Xe m + 1

= (;, -

m)e m •

and em i= 0. The above formula therefore implies that

;, = m, so we are in case (b). Corollary 2. Suppose that V is finite dimensional. Then we are in case (b) of Corollary 1. The vector subspace W of V with basis eo, ... , em is stable under g; it is an irreducible g-module.

Clearly case (a) of Corollary 1 is impossible. On the other hand, formulae (i), (ii1 and (iii) show that W is a g-submodule of V (it is the g-submodule generated bye). By (i), the eigenvalues of H on Ware equal to In, m - 2, m - 4, ... , -m, and have multiplicity 1. If W' is a nonzero subspace of W stable under H, then it contains one of the eigenvectors ei (0 ~ i ~ m); however, if W' is stable under g. formulae (iii) show that W' contains e;-l"'" eo = e, and formuTae (ii) show that it contains e;, e;+l' ., .. We therefore have W' = W, proving the irreducibility of W

4. The Modules Wm Let m be an integer ~ 0, and let Wm be a vector space of dimension m + 1, with basis eo, ... , em. Let us define endomorphisms X, Y, H of Wm by the following formulae (with the convention that e-l = em + 1 = 0):

20

IV. The Algebra 5[2 and Its Representations

(i) Hen

= (m - 2n)en

(ii) Yen = (n + l)en+1 (iii) X en = (m - n + l)e n- l • A direct computation shows that

HXe n - XHe n = 2Xen, HYe n - YHe n = -2Yen, XYe n - YXe n = Hen, in other words the endomorphisms X, Y, H make Wm into a g-module. Theorem 2. (a) Wm is an irreducible g-module. (b) Every irreducible g-module of dimension m + 1 is isomorphic to Wm • (a) follows from Corollary 2 to Theorem 1, and the fact that Wm is generated by the images of the primitive element eo, which has weight m. Let V be an irreducible g-module of dimension m + 1. By Prop. 3, V con tains a primitive element e. Corollary 2 to Theorem 1 shows that the weight of e is an integer m' ~ 0, and that the g-submodule Wof V generated by e has dimension m' + 1. Since Vis irreducible, we must have W = V, so that m' = m, and the formulae of Theorem 1 show that Vis isomorphic to Wm , as required. EXAMPLES. The module Wo is the trivial g-module of dimension 1. The space C 2 with its natural g-module structure is isomorphic to WI. The algebra g,

regarded as a g-module by means of the adjoint representation, is isomorphic to W2 • Remark. One can show that Wm is isomorphic to the moth symmetric power of the module WI = C 2 •

5. Structure of the Finite-Dimensional g-Modules Theorem 3. Each finite-dimensional g-module is isomorphic to a direct sum of modules Wm • Indeed, by H. Weyl's theorem (Chapter II, Theorem 8), such a module is a direct sum of irreducible modules, and we have just seen that each finitedimensional irreducible g-module is isomorphic to some Wm • Theorem 4. Let V be a finite-dimensional g-module. Then: (a) The endomorphism of V induced by His diagonalizable. Its eigenvalues are integers. If ± n (with n ~ 0) is an eigenvalue of H, then so are n - 2, n - 4,

... , -no

6. Topological Properties of the Group SL 2

21

(b) If n is an integer ~ 0, the linear maps

yn: vn --+ v- nand xn: v- n--+ vn are isomorphisms. In particular, vn and v- nhave the same dimension. (Recall that vn denotes the set of elements of V of weight n.) By Theorem 3, we may assume that Vis one of the g-modules Wm , in which case (a) and (b) are clear. Remarks. (1) The fact that vn and V- n have the same dimension can also be seen by using the endomorphism = eXe-Ye X of V (notice that X and Yare nilpotent on V, so that their exponentials are just polynomials). Now one checks that:

e

eo X = - Yo e, eo Y =

-X

0

e, eo H = -H 0 e,

e

and the last identity shows that maps vn to v-no (2) Here is an example of an application of Theorems 3 and 4, independent of the interpretation of sI2 as the Lie algebra of SL 2 : Let U be a compact Kahler variety of complex dimension n, and let V be the cohomology algebra H*(U, C). Hodge theory associates endomorphisms A and L of V with the kahlerian structure on U (cf. A. Weil, Varietes kiihleriennes, Chap. IV); let us take X and Y to be these endomorphisms, and define H by the relation Hx = (n - p)x if x E HP(U, C). Then one can check (Weil, loc. cit.) that V becomes a g-module. By applying Theorems 3 and 4 to this module, one retrieves Hodge's theorem's on "primitive" cohomology classes.

6. Topological Properties of the Group SL 2 This is the group of complex matrices of order 2 and determinant equal to 1. It is a complex Lie group, with Lie algebra s1 2 . The elements X, Y, H of sI2 generate the following one-parameter subgroups: e tX

=

(1o

t) 1 '

etY

=

(1t 0) , e (e e-0) 1

tH

=

t

0

t

•

Similarly we will consider the subgroup SU 2 of SL 2 formed by the unitary matrices; its Lie algebra will be denoted by SU2' Theorem 5. (a) SL 2 is isomorphic (as a real analytic variety) to SU 2 X R3. (b) SU 2 is isomorphic (as a Lie group) to the group of quaternions of norm 1, which is itself homeomorphic to the sphere S3' (c) SU 2 and SL 2 are connected and simply connected. (d) The algebra sI2 can be identified with the complexification of the real Lie algebra SU2: we have sI2 = SU 2 EB i· SU2'

22

IV. The Algebra 5[2 and Its Representations

The algebra SU 2 consists of the skew hermitian matrices of order 2 and trace zero; if P denotes the set of hermitian matrices of trace zero, then clearly P = i· su 2 and sI 2 = sU 2 Ei3 P, giving (d). Moreover, it is straightforward to check that the map (u,p) ...... u·e P

is an isomorphism (ofreal analytic varieties) from SU 2 x Ponto SL 2. Since P is isomorphic to R 3 , this proves (a). Statement (b) is well-known, and (c) follows from (b) and the fact that S3 is connected and simply connected. Now Weyl's "unitarian trick" takes the following form: Theorem 6. For each complex Lie group G, with Lie algebra g, the following canonical maps are bijections:

(Notation: HomdSL 2, G) denotes the set of complex analytic homomorphisms from SL 2 to G, HomR(su 2, g) denotes the set ofR-homomorphisms from the Lie algebra su 2 to the Lie algebra g, etc. The maps a and d are the restriction maps; the maps band c arise from the functor "Lie group" ...... "Lie algebra". PROOF. The maps band c are bijective because SL 2 and SU 2 are connected and simply connected; the map d is bijective because sI2 is the complexification of su 2; the bijectivity of a (which is not a priori obvious) follows from the commutativity of the diagram. 0

Corollary. The finite-dimensional linear representations of SU 2, SL 2, SU 2, and sI2 correspond bijectively with each other. It is sufficient to apply the theorem to the group G = GLn(C) for n

= 0,

1, ....

7. Applications The global results ofthe preceding section provide alternative proofs of certain properties of sI 2-modules. Thus, for example: (i) The complete reducibility of finite-dimensional sI2 -modules follows from the fact that, by the corollary to Theorem 6, these modules correspond to the linear representations of the compact group SU 2'

7. Applications

23

(ii) The fact that the eigenvalues of H are integers can be seen in the following way: let V be a finite-dimensional 512 -module, and let x E V be an eigenvector of H, with eigenvalue }.. By the corollary to Theorem 6, the group SL 2 acts on Vj in particular, the element e tH of SL 2 sends x to et.h:; but, if t = 2in, we have e tH = 1 in SL 2 , so e tH x = x. We must therefore have etl = 1 for t = 2in, implying that A. is an integer. (iii) The automorphism () introduced at the end of Sec. 5 corresponds to the action of the element ( _

~ ~) of SL

2•

CHAPTER V

Root Systems

In this chapter (apart from Sec. 17) the ground field is the field R of real numbers. The vector spaces considered are all fmite dimensional.

1. Symmetries Let V be a vector space and :x a nonzero element of V. One defines a symmetry with vector ex to be any automorphism s of V satisfying the following two conditions: (i) s(ex) = - ex. (ii) The set H of elements of V fixed by s is a hyperplane of V.

It is clear that H is then a complement for the line Rex spanned by ex, and that s has order 2. The symmetry s is completely determined by the choice of Rex and of H. Let V* be the dual space of V, and let ex* be the unique element of V* which vanishes on H and takes the value 2 on ex. We have s(x)

= x - (a*, x)a

for all x

which we can write as s = 1 - ex* ® ex,

on identifying End(V) and V* ® V. Conversely, if ex E V and a* E V* satisfy ( a* , a)

= 2,

the element 1 - a* ® a is a symmetry with vector a.

E

V,

2. Definition of Root Systems

25

Lemma. Let 0: be a nonzero element of V, and let R be a finite subset of V which spans V There is at most one symmetry with vector 0: which leaves R invariant. Let sand s' be two such symmetries, and let u be their product. The automorphism u has the following properties: u(R)

=

R,

u(o:) = ct, II

induces the identity on V/Ro:.

The last two properties show that the eigenvalues of u are equal to 1. Moreover, because R is finite there is an integer n ~ 1 such that un(x) = x for all x E R, so that un = 1 since R spans V This implies that u is diagonalizable. Since its eigenvalues are equal to I, we therefore have u = 1, so that s = s'.

2. Definition of Root Systems Definition 1. A subset R of a vector space V is said to be a root system in V the following conditions are satisfied:

if

(1) R is finite, spans V, and does not contain O. (2) For each 0: E R, there is a symmetry S2' with vector 7, leat'ing R invariant. (This symmetry is unique, by Lemma 1.) (3) For each 0:, PER, s,,(P) - P is an integer multiple of 7.

The dimension of V is called the rank of R. The elements of R are called the roots of V (relative to R). By Sec. 1, the symmetry s" associated with the root 0: can be written uniquely as s"

= 1 - 0:* ® 0:

with (ct*, 7) = 2.

The element 0:* of V* is called the inverse root of 0:. Condition (3) is equivalent to the following: (3') For all

Let 0:

E

0:,

PER, we hare (o:*,P)

R. By (2) and (3), we have - ct

E

R, since - ct

E

=

Z. sAct).

Definition 2. A root system R is said to be reduced if, for each ct E R,7 and are the only roots proportional to 0:.

0:

If a root system R is not reduced, it contains two proportional roots 0: and to:, with 0 < t < 1. Applying (3) to P = to:, we see that 2t E Z, which implies that t = l Then the roots proportional to 0: are simply - ct,

- ct/2,

0:/2,

ct.

26

V. Root Systems

Remark. The reduced roots systems are those which arise in the theory of

semisimple Lie algebras (or algebraic groups) Over an algebraically closed field; they are the only ones we shall need. Nonreduced systems occur when one no longer assumes that the base field is algebraically closed.

3. First Examples (We shall see others in Sec. 16.) Clearly, the only reduced system of rank 1 is the system

•

-:x

•o.

•

(denoted by

Ad.

IX

There is one nonreduced system ofrank 1:

•

•

-2:x

-IX

• 0

•

•

IX

2:x

One can show (cf. Secs. 8, 15) that every reduced system of rank 2 is isomorphic to one of the following four:

-a+'

(type Al x

Ad

0

-Il

---4~-""-a

-Il Il +Cll

tl +2a

.......

O----~--

-J3-2a

-Il-a

a

-13

(type A 2 )

27

5. Invariant Quadratic Forms

2ft +3a

Il

--Il-3a

-1l-1a

--Il-a

-Il

Complete the root system B2 so as to obtain a nonreduced system. Can one do the same with A2 and G2 ?

EXERCISE.

4. The Weyl Group Definition 3. Let R be a root system in a vector space V. The Weyl group of R is the subgroup W of GL(V) generated by the symmetries s,,' ex E R. The group W is a normal subgroup of the group Aut(R) of automorphisms of V leaving R invariant. Since R spans V, these two groups can be identified with subgroups of the group of all permutations of R; they are finite groups. When R is a reduced system of rank 2, the group W is isomorphic to the dihedral group of order 2n, with n = 2 (type Ai x Ad, n = 3 (type A 2 ), n = 4 (type B 2 ), or n = 6 (type G2 ). We have Aut(R) = W when R is of type B2 or G2 , and IAut(R): WI = 2 when R is of type Ai x Ai or A 2 • EXAMPLE.

5. Invariant Quadratic Forms Proposition 1. Let R be a root system in V. There is a positive definite symmetric bilinear form (,) on V which is invariant under the Weyl group Wof R. This follows simply from the fact that W is finite. For if B(x, y) is any positive definite symmetric bilinear form on V, the form (x, y) =

L weW

is invariant, and (x, x) > 0 for all x "# O.

B(wx, wy)

28

V. Root Systems

From now onwards, we let (,) denote such a form. The choice of (,) gives V the structure of a Euclidean space, with respect to which the elements of W are orthogonal transformations. In particular this applies to the symmetries Sa; we deduce from this that we have

=x -

s:z(x)

V

(x, ex) 2--ex (ex, ex)

for all x

E

V.

Let ex' be the element of V corresponding to ex* under the isomorphism V* detennined by the chosen bilinear form. By definition, we have

--+

s:z(x)

=x -

for all x

(ex', x)ex

E

V.

Comparing this with the preceding fonnula, we get

, 2ex ex = - - . (ex, ex) (Thus we pass from ex to ex' by an "inversion in a sphere of radius the sense of elementary geometry.) Condition (3) for root systems can be written as

2(ex,{J) (ex, ex)

E

Z

-/2," in

for ex, {J E R.

Th us one can retrieve the traditional definition of root systems, cf. J aco bson or Seminaire S. Lie. (The definition in Sec. 2 is that of Bourbaki, Systemes de Racines-it has the advantage of separating the roles of V and of V*.)

6. Inverse Systems Let R be a root system in V. Proposition 2. The set R* of inverse roots ex*, ex E R, is a root system in V*. Moreover, ex** = ex for all ex E R. Clearly R* is finite and does not contain O. To prove that it spans V* it is sufficient (by the isomorphism V -+ V*) to show that the elements ex' = 2ex/(ex, ex) span V, which is obvious. If ex* E R* we take the corresponding symmetry to be the transpose 'SIJl = 1 - ex ® ex* of SIJl' Since slJl(R) = R, we have sa.(R*) = R*. Similarly, we see that ex** = ex. Finally, if ex*, {J* E R*, we have

(ex**,{J*) = ({J*, ex)

E

Z,

as required. The system R* is called the inverse (or dual) system of the system R. Its Weyl group can be identified with that of R by means of the map

29

7. Relative Position of Two Roots

7. Relative Position of Two Roots Let us keep the notation of the preceding sections. If oc, Pare two roots, we put

(oc,P) (oc, oc)

n(p,oc) = (oc*,P) = 2--. We have n(p, rx) E Z. Now if we let loci denote the length of oc (that is, (rx, rx)1/2), and ,p the angle between oc and P(with respect to the Euclidean structure on V), then we have (oc, P) = locilPI cos,p, so that

n(p,oc) =

2~cos,p. loci

From this we deduce the formula

n(p, rx)n(oc, p) = 4 cos 2 ,p. Since n(p, rx) is an integer, 4cos 2 ,p can take only the values 0,1,2,3,4; the last case being that in which oc and Pare proportional. Returning to the case of nonproportional roots, we see that there are 7 possibilities (up to transposition of oc and p):

2

n(oc,p) = 0,

n(p,oc) = 0,

,p = n12.

n(oc,p) = 1,

n(p,oc) = 1,

,p = n13,

IPI = loci.

n(p,oc) = -1,

,p = 2n13,

IPI = loci·

n(p,oc) = 2,

,p = n14,

IPI=j2l oc l.

n(p,oc) = -2,

,p = 3n14,

IPI=j2l oc l.

n(p,oc) = 3,

,p = n16,

IPI = y"3l oc l.

3 n(oc,p) = -1, 4

n(oc,p) = 1,

5 n(oc,p) = -1, 6

n(oc,p) = 1,

7 n(oc,p) = -1, n(p,oc) = -3, ,p = 5n16,

IPI = y"3l oc l.

Notice that knowledge of the angle,p determines the set {n(oc,p), n(p,oc)}, or, what amounts to the same thing, the set of ratios of lengths

loci IPI} { ijI'l;f , provided that we have ,p i= n12. Proposition 3. Let oc and oc - Pis a root.

P be two

nonproportional roots. If n(p, oc) > 0, then

°

(N otice that n(p, oc) > is equivalent to (oc, P) > 0: the two roots form an acute angle.) The above list shows that we have either n(p, oc) = 1 or n(oc, p) = 1. In the first case,

oc - P = -(P - n(p,oc)oc) = -s,,(P), so that oc - PER. In the second case, oc - P = sfl(oc)

E

R.

30

V. Root Systems

8. Bases Let R be a root system in V. Definition 4. A subset S of R is called a base for R are satisfied:

if the following two conditions

(i) S is a basis for the vector space V. (ii) Each PER can be written as a linear combination

P= L

lZeS

mlZex,

where the coefficients mlZ are integers with the same sign (that is, all all ~ 0).

~

0 or

Instead of "base," the terms "simple root system" or "fundamental root system" are also used; the elements of S are then called the "simple roots." Theorem 1. There exists a base. We shall prove a more precise result. Let t E V* be an element such that (t, ex) "# 0 for all ex E R. Let R( be the set of all ex E R such that (t, ex) is > 0; we have R = R,+ u (- Rn. An element ex of R( is called decomposable if there exist p, y E R,+ such that ex = P + y; otherwise, ex is called indecomposable. Let S, be the set of indecomposable elements of R(. Proposition 4. S, is a base for R. Conversely, if S is a base for R, and is such that (t,ex) is >0 for all ex E S, then S = St.

if t E V*

Let us show that St is a base for R. We will do this in stages. Lemma 2. Each element of R( is a linear combination, with non-negative integer coefficients, of elements of St. Let I be the set of ex E Ri which do not have the property in question. If I were nonempty, there would be an element ex E I with (t, ex) minimal. The element ex is decomposable (otherwise it would belong to St); if we write ex = P+ y, with p, y E R(, we have (t,ex) = (t,P)

+ (t,y),

and since (t,P) and (t,y) are >0, they are strictly smaller than (t,ex). We therefore have P~ I and y ~ I, clearly giving ex ~ I, a contradiction. Lemma 3. We have (ex, P) ~ 0 if ex, PESt.

9. Some Properties of Bases

31

Otherwise, Prop. 3 would show that y = ex - P is a root. We would then have either t' E R,+, in which case ex = P+ Y would be decomposable, or else -y E R,+, in which case P= ex + (-y) would be decomposable. Lemma 4. Let t E V* and A c V be such that: (a)

(t, ex) > 0

for all ex E A,

(b)

(ex, P)

for all ex, pEA.

~

0

Then the elements of A are linearly independent. (In other words, vectors which subtend obtuse angles with each other, and lie in the same half-space, are linearly independent.) Each relation between the elements of A can be written in the fonn

LY(JP = LZyY, where the coefficients Y(J and Zy are ~O, and where P and y range over disjoint finite subsets of A. Let J. E V be the element LY(JP, We have

(J., ).) = L }'pZy(P, y), so that (J., ).) ~ 0, by (b). We deduce that). = O. But then we have

0= (t,J.) = LY(J(t,P), so that Y(J = 0 for all P, and similarly Z1 = 0 for all y, as required. Lemmas 2, 3,4 show that Sr is a base for R. Conversely, let S be a base for R, and let t E V * be such that (t, ex) is > 0 for all ex E S. If we let R + denote the set of linear combinations with non-negative integer coefficients of elements of S, then we have R+ c R,+ and (-R+) c (-Rt), so that R+ = R,+ since R is the union of R+ and -R+. We deduce that the elements of S are indecomposable in Rt, that is, that S c Sr. Since Sand Sr have the same number of elements (namely the dimension of V), we have S = Sr. EXAMPLE. Suppose that dim V = 2, and let {ex, P} be a base for R. Since the angle between ex and Pis obtuse (Lemma 31 only cases 1/, 3/,5/, 7/ of Sec. 7 are possi ble (allowing for a possible transposition of ex and Pl. They correspond to the systems of types Ai x Ai' A 2 , B2 , and G2 respectively (cf. Sec. 3).

9. Some Properties of Bases In the following sections, S denotes a base for the root system R. We denote by R + the set of roots which are linear combinations with non-negative integer coefficients, of elements of S. An element of R+ is called a positive root (with respect to S).

32

V. Root Systems

Proposition 5. Every positive root fJ can be written as

+ ... + exk

fJ = exl

with exi E S,

in such a way that the partial sums

are all roots. Let t E V* satisfy (t, ex) = 1 for all ex E S. Since fJ is a positive root, (t, fJ) is a non-negative integer. We shall prove the proposition by induction on k = (t, fJ). First note that the values of (ex, fJ), fJ E S, cannot all be ~ O. If they were, then Lemma 4 would show that fJ and the elements of S were linearly independent, which is absurd. Hence there exists some :x E S such that (ex, fJ) > O. If ex and fJ are proportional, we have fJ = ex or fJ = 2ex, and Prop. 5 is true. Otherwise, Prop. 3 shows that y = fJ - ex is a root. If Y E - R + , ex = fJ + (- (') would be decomposable, which is absurd. Hence we have y E R + and (t, "I) = k - 1. The induction hypothesis can be applied to y, so we obtain the result by taking exk = ex. Proposition 6. Suppose that R is reduced, and let :x E S. The symmetry Sa associated with ex leaves R+ - {ex} int'ariant. Let fJ

E

R+ - {ex}. We have fJ =

L

my}'

1eS

Since R is reduced, and fJ "# ex, fJ is not proportional to ex, and there exists some y "# ex such that my "# O. Since sa(fJ) = fJ - n(fJ, ex):x, we see that the coefficient of y in sa(fJ) is equal to m," This gives sa(fJ) E R+, proving the proposition. Corollary. Let P be half the sum of the positit'e roots. We hat'e

sa(P) = P - ex for all ex E s. Let Pa be half the sum of the elements of R+ - {ex}. Clearly we have sa(Pa) = Pa' On the other hand, P = Pa + ex/2. SincesAex) = -ex we deduce from this that sa(P) = P - :x. Proposition 7. Suppose that R is reduced. The set S* of inverse roots of the

elements of S is a base for R*. Let R' be the root system consisting of the vectors ex' = 2ex/(ex, ex) for ex E R. By the isomorphism V -+ V* (cf. Sec. 5) it is sufficient to prove that the vectors ex', ex E S, form a base for R'. If t E V* is such that (t,:x) > 0 for all ex E S, then

10. Relations with the Weyl Group

33

(R')t consists of the vectors IX' with IX E R+. The convex cone C generated by (R')t is therefore the same as that generated by R +. Let S; be the corresponding base of R'. The half-lines generated by the elements of S; are the extremal generators of C; hence they are the half-lines R + IX, with IX E S. Since R is reduced, such a half-line contains a unique root of R', which must be IX'. Thus S; = S, as required. Remark. In the general case, let Sl (resp. S2) be the subset of S consisting of the roots IX such that 2IX is not a root (resp. 2IX is a root). We obtain a base for R* by taking the elements IX*, IX E Sl' and the elements IX*/2, IX E S2'

10. Relations with the Weyl Group We assume that R is reduced.. Theorem 2. Let W be the Weyl group of R. (a) (b) (c) (d)

For each t E V*, there exists w E W such that (w(t), IX) ~ 0 for all IX If S' is a base for R, there exists W E W such that w(S') = S. For each fJ E R, there exists WE W such that w(fJ) E S. The group W is generated by the symmetries S2' IX E S.

E

S.

Let HiS be the subgroup of W generated by the symmetries SO' IX E S. We first prove (a) for the group HiS. Let t E V*, and let p be half the sum of the positive roots (cf. Prop. 6, Corollary). Let us choose an element w of Ws so that

(w(t),p) is maximal. In particular, we have

(w(t),p)

~

(s",w(t),p)

if IX E S.

But we have

(S2W(t),P) = (w(t), sAp) = (w(t),p - IX) (cf. the corollary to Prop. 6). Hence we conclude that (w(t), IX) ~ 0, which proves our assertion. We now prove (b) for the group HiS. Let t' be an element of V* such that (t', IX') > 0 for all IX' E S'. By (a), there exists WE HiS such that, if we put t = w(t'), then (t,IX) ~ 0 for all IX E S. Since (t,IX) = (t', W-1(IX), and since t' is not orthogonal to any root, we in fact have (t, IX) > 0 for all IX E S. By Prop. 4, we have

S

= S, and S' = S, ..

Since w sends t' to t, it also sends S' to S.

34

V. Root Systems

We now prove (c) for the group Ws. Let PER, and let L be the hyperplane of V* orthogonal to p. The hyperplanes associated with the roots other than ± P are distinct from L, and there are only finitely many of them. Hence there is an element to of L not contained in any of these hyperplanes. We have (to,P) = 0

(to,y)

and

=1=

0

fory

R, y =1=

E

±p.

One can find an element t sufficiently close to to that (t, P) = e, with e > 0, the absolute value of each (t, y), y =1= ±p, being strictly greater than e. Let S, be the base of R associated with t as described in Sec. 8; clearly Pbelongs to S,. By (b), there exists WE W such that w(St) = S. We then have w(P) E S. We finally show that Ws = W, which will prove (d). Since W is generated by the symmetries s{J' with PER, it is sufficient to show that s{J E Ws. By (c), there exists W E Ws such that oc = w(P> belongs to S. We have S2

= sW({Jl = W'S{J'

W

-1

so that s{J = w- 1 • S2' w, which indeed shows that s{J

E

Ws.

Remarks. (1) The element W given in (b) is unique (cf. Sec. VII.5). (2) The set of elements t E V* such that (t, oc) > for all oc E S is called the Weyl chamber associated with S. By (a) and (b), the Weyl chambers are the connected components of the complement in V* of the hyperplanes orthogonal to the roots; the group W permutes them transitively. (3) One can refine (d) by showing that the relations between the generators sa(oc E S) of Ware all consequences of the following:

°

S2 2

=

1,

(s::z: S fJ )m(a.{Jl = 1'

where m(oc, P) is equal to 2, 3,4, or 6 as the angle between oc and Pis n/2, 2n/3, 3n/4, or 5n/6. See, for example, Seminaire Chevalley, 1956-58, expose 14, or Bourbaki, Chaps. IV-V.

11. The Cartan Matrix Definition 5. The Cartan matrix of R (with respect to the chosen base S) is the matrix (n(oc, P»a.{JES' We recall (cf. Sec. 7) that n(oc, P) = (p*, oc) is an integer. We have n(oc, oc) = 2; if oc =1= p, we know (cf. Lemma 3) that n(oc, p> :s;; 0. We have n(oc, P) = 0, -1, - 2, or -3. EXAMPLE.

The Cartan matrix of G2 is

(-32 -21).

Proposition 8. A reduced root system is determined, up to isomorphism, by its Cart an matrix.

12. The Coxeter Graph

35

More precisely: Proposition 8'. Let R' be a reduced root system in a vector space V', let S' be a base for R', and let lP: S ---+ S' be a bijection such that n(lP(oc~ lP(P) = n(a, [3) for all oc, [3 E S. If R is reduced, then there is a unique isomorphism f: V ---+ V' which is an extension of lP and maps R onto R'. To define f, we extend lP by linearity from S to V. If oc, [3 E S, we have s;("l

0

f([3)

=

s;("M([3))

=

lP([3) - n(lP(oc), lP([3))lP(oc)

and f

0

s,,([3)

=

f([3 - n([3, oc)oc)

=

lP([3) - n([3, oc)lP(oc).

Comparing these, we see that s;("l 0 f = f 0 s" for all oc E S. If W (resp. W') denotes the Weyl group of R (resp. R'), we see that W' = fWf- i . Since R = W(S) and R' = W(S'~ we deduce that f(R) = R', as required. In particular, let E be the group of permutations of S which lea ve the Cartan matrix invariant. By the above argument, E can be identified with the group of automorphisms of R which leave the base S invariant. Proposition 9. The group Aut(R) is the semidirect product of E and W If w E W n E, we have w(S) = S, so that w = 1 by a result which will be proved later (Sec. VII.S). Moreover, if u E Aut(R), u(S) is a base for R, hence there exists WE W such that w(u(S» = S (cf. Theorem 2). We therefore have wu E E, showing that Aut(R) = W· E.

Corollary. The group Aut(R)/W is isomorphic to E.

12. The Coxeter Graph Definition 6. A Coxeter graph is a finite graph, each pair of distinct vertices being joined by 0, 1, 2, or 3 edges. Let R be a root system, and let S be a base for R. The Coxeter graph of R (with respect to S) is defined as follows: the vertices are the elements of S, two distinct vertices oc and [3 being joined by 0,1,2, or 3 edges as n(oc, [3)' n([3, oc) is equal to 0, 1, 2, or 3. (Recall that if lP denotes the angle between oc and [3, then n(oc, [3). n([3, oc) = 4 COS 2 lP,

cf. Sec. 7.) Of course, the transitivity theorem in Sec. 10 shows that the graphs associated with different bases of R are isomorphic.

v.

36

EXAMPLES.

Root Systems

The Coxeter graphs of the root systems in Sec. 3 are the following: o

type Ai

ootype A 1 0--0

typeA2

0=======0

type B2

~

typeG2·

X

Ai

13. Irreducible Root Systems Proposition 10. Suppose that V is the direct sum of two subspaces Vi and V2, and that R is contained in Vi U V2. Let Ri = R n V;, i = 1,2. Then: (a) Vi and V2 are orthogonal. (b) R j is a root system in V;.

If ex E Ri and p E R 2, ex - p is not contained in Vi U J-2, so it is not a root. By Prop. 3, we therefore have (ex, fJ) " O. Since this also applies to ex and - p, we see that (ex, fJ) = O. Since Ri spans V;, (a) follows. For (b), it is sufficient to notice that, by (a), the symmetry associated with an element of Ri preserves J-2, and hence also Vi'

One says that the system R is the sum of the subsystems R i . If this can happen only trivially (that is, with Vi or V2 equal to 0), and if V #- 0, then R is said to be irreducible. Proposition 11. Every root system is a sum of irreducible systems. This is obvious. One can show that such a decomposition is unique. Proposition 12. For R to be irreducible, it is necessary and sufficient that its Coxeter graph should be connected and nonempty.

If R is the sum of two nontrivial subsystems Ri and R 2, we can take the union of two bases Si and S2 for Ri and R2 to be a base S for R. If ex E Sl and p E S2' then ex and p are orthogonal and are therefore not joined by any edge in the Coxeter graph of S. We deduce that the latter is the disjoint sum of the Coxeter graphs of the bases Sj, and is therefore not connected. Conversely, if S has a nontrivial partition

S = Si US2

14. Classification of Connected Coxeter Graphs

37

such that every element of Si is orthogonal to every element of S2' then the vector subspaces Vi and V2 spanned by Si and S2 are orthogonal, and are therefore invariant under the symmetries s,,' !X E S. Hence R is contained in Vi u V2 , and is therefore reducible.

14. Classification of Connected Coxeter Graphs Theorem* 3. Every connected nonempty Coxeter graph which is attached to a root system is isomorphic to one of the following: An:

0-------0---

(n vertices, n

~

1)

Bn:

0-------0---

(n vertices, n

~

2)

Dn:

0-------0---

(n vertices, n

~

4)

The principle of the proof is as follows. One takes a nonempty connected Coxeter graph G, with vertex-set S. One associates with G a symmetric bilinear form (,) on the space RS with basis (e')"E s, by defining (e",e,,) = 1

= cos(n/2), cos(2n/3), cos(3n/4), cos(5n/6) as ()( and p are joined by 0, 1, 2, or 3 edges. For G to be the Coxeter graph of a root system, it is necessary that this form should be positive definite (for it can be realized by one of the invariant forms of Sec. 5). One then shows, by a series of ingenious reductions, that this positivity condition is sufficient to force an isomorphism between G and An, Bn , •• , or E 8 • For further details, see Seminaire S. Lie, expose 13; Jacobson, pp. 128-134; or Bourbaki, Chap. 6, Sec. 4.

(e", ep )

38

V. Root Systems

15. Dynkin Diagrams (For simplicity, we restrict our attention to root systems which are both reduced and irreducible.) The Coxeter graph is not sufficient to determine the Cartan matrix (and hence the root system); indeed it gives only the angles between the pairs of roots in the base, without indicating which is the longer. Two mutually inverse systems (like Bn and Cn; cf. Sec. 16) have the same Coxeter graph. However, the Cartan matrix is determined if we specify the ratios of lengths of the roots. This leads us to attach, to the vertices of the Coxeter graph, coefficients proportional to the square (ex, ex) of the length of the relevant root ex. The Coxeter graph, thus labelled, is called the Dynkin diagram of R. If we agree to identify two Dynkin diagrams which differ only by a coefficient of proportionality, we have: Proposition 13. Specifying a Dynkin diagram is equivalent to specifying a Cartan matrix. They determine the root system up to isomorphism. Let us explain how to determine the Cartan matrix from the Dynkin diagram: if ex = {3, we have n(ex, {3) = 2; if ex :F {3, and if ex and {3 are not joined by an edge, we have n(ex, {3) = 0; if ex :F {3, if ex and {3 are joined by at least one edge, and if the coefficient of ex is less than or equal to that of {3, we have n(ex, {3) = -1; if ex :F {3, if ex and {3 are joined by i edges (1 t;; i t;; 3), and if the coefficient of ex is greater than or equal to that of {3, we have n(ex, {3) = - i. (In this last case, the coefficient of ex is i times that of {3; hence there is no need to draw mUltiple edges.) Theorem* 4. Each nonempty connected Dynkin diagram is isomorphic to one of the following: 1

1

1

An:

0----0--

Bn:

0----0--

Cn:

0----0--

Dn:

0----0---

G2 :

~

F4 :

Q---------O====

2 1 1 1 1

2

2

1

--0===0

1

1

2

--0===0

1

---0

Theorem 4. (a) One has g = n- Ei3g Ei3 n = n- Ei3 b. (b) nand n- are subalgebras of g consisting of nilpotent elements; they are nilpotent. (c) b is a solvable subalgebra of g; its derived algebra is n. (a) is trivial. (b): Let x En; for each integer k > 0, and each PE g*, one has ad(x)"(gP)

C

L

gPH'+"·Hk.

:1,>0

If k is sufficiently large, p and p + a 1 + ... + a" cannot both be in R u {O}. Hence one has ad(x)" = 0, which shows that x is nilpotent. The fact that n is nilpotent follows from Engel's Theorem (Sec. 1.4) or from a direct argument. The case of n- is similar. (c) follows from the equation [g, g] = n.

The algebra b is called the Borel subalgebra corresponding to

g and S.

48

VI. Structure of SemisimpIe Lie Algebras

Theorem* 5 (Borel-Morozov). Eeery solvable subalgebra of g can be mapped to a subalgebra of f) by an inner automorphism of g. In particular, b is a maximal solvable subalgebra of g. For the proof see A. Borel, Ann. of Maths. 64 (1956), pp. 66-67; Bourbaki, Chap. 8, Sec. 10; or Humphreys, Sec. 16.3. Corollary. Every subalgebra of g consisting of nilpotent elements can be mapped to a subalgebra of n by an inner automorphism of g. This follows from Theorem 5, and from the fact that each nilpotent element of g contained in b belongs to n.

4. Weyl Bases We will keep the notation of the preceding section. We let (CX 1, ... , cx n) denote the chosen base S; n = dim f) is the rank of g (cf. Chap. III). For each i, we put Hi = Hai , and we choose elements Xi Ega" Yj E g-a, such that [Xi' Yj] = Hi (cf. Theorem 2). Finally, we put n(i,j) = cxj(H;).

The matrix formed by the numbers n(i,j) is the Cartan matrix of the given system; recall (cf. Sec. V.11) that n(i,j) is an integer for all i. One has u 2 = 1. Let us put H; = - Hi' X; = - Yi, Yi' = - Xi' Clearly the elements Xi, Yi/, H; satisfy the Weyl relations and the relations (Jij, (Jij. Hence by (ii) there is a homomorphism u: g --+ g mapping Xi' Yi, Hi to X;, Yi/, H;. Since u 2 fixes Xi> Yi, Hi' it is the identity, as required. Remark. Theorem 7 gives an explicit description of g and of n in terms of the Cartan matrix (n(i,j)).

VI. Structure of Semisimple Lie Algebras

50

5. Existence and Uniqueness Theorems The conjugacy theorem for Cartan subalgebras (Sec. 111.4) shows that the root system of a semisimple Lie algebra is independent (up to isomorphism) of the chosen Cartan subalgebra. Furthermore: Theorem 8. Two semisimple Lie algebras corresponding to isomorphic root systems are isomorphic. More precisely: Theorem 8'. Let 9 (resp. g') be a semisimple Lie algebra, 9 (resp. 9') a Cartan subalgebra of 9 (resp. g'), S (resp. S') a base for the corresponding root system, and r: S ---+ S' a bijection sending the Cartan matrix of S to that of S'. For each i E S (resp. j E S'), let Xi (resp. Xl) be a nonzero element of gi (resp. g,j). Then there is a unique isomorphism f: 9 -> g' sending Hi to H;(i) and Xi to X;(i) for all i E S. Let Yi (resp. lj') be the element of g-i (resp. of g,-j) such that [Xi' YiJ = Hi (resp. [X;, lj'J = H;). Then Theorem 7 shows that there is a unique homomorphism f: 9 ---+ g' sending Xi' Yi, Hi to X:(i)' Y,.(i) , H;(i), and clearly this is an isomorphism. Remark. By applying this result in the case g' = g, obtains another proof of the corollary to Theorem 7.

9' = 9, s' = - S, one

Finally, here is the existence theorem: Theorem 9. Let R be a reduced root system. There exists a semisimple Lie algebra 9 whose root system is isomorphic to R. Let S = {a b ... , an} be a base for R, with (n(i,j)) the corresponding Cartan matrix. Let 9 be the Lie algebra defined by 3n generators Xi> Yi, Hi and by the relations in Theorem 6 (i.e. the Weyl relations and the relations ()ij, ()ij). One shows (cf. the Appendix) that this Lie algebra is finite dimensional, semisimple, and has a root system isomorphic to R. Hence the theorem. Corollary. For 9 to be simple, it is necessary and sufficient that R should be irreducible. This is obvious.

6. Chevalley's Normalization

51

6. Chevalley's Normalization For each oc E R, choose a nonzero element Xa if OC if OC

Ega.

Then we have

+ 13 E R, + 13 ¢ R,

OC

+ 13 :F 0,

where Na • p is a nonzero scalar. The coefficients N2 .(J determine the "multiplication table" of g. However, they depend on the choice of the elements X~. Theorem 10. One can choose the elements X 2 so that

for all oc E R, for oc, 13, oc

+ 13 E R.

Let R+ be the set of positive roots relative to a base S of R, and let u be an automorphism of 9 equal to -Ion g, and such that u 2 = 1 (cf. the Corollary to Theorem 7). We have u(ga) = 9- 2 • Let oe E R+, and let us choose a nonzero element x~ of g2. We have [.1: 2, u(x 2 ) ] E Ega, g-2], so there is a nonzero scalar t2 such that [x 2 • u(x2 )] = t 2 H 2 • Let U 2 be a square root of - t 2 , and let us put X2

We now have [X 2 ,X- 2 ] =

=

U;l X2 ,

H2

and X 2 N2 .(J

=

X- 2

=

-u(X2 )·

+ u(X = O. The identity 2 )

-N_ 2 . -P

is then obtained by writing [uX2 , uXp] = u[X2 ,X(JJ. Theorem* 11 (Chevalley). Suppose that the conditions of Theorem 10 are satisfied. Let oe, 13 E R be such that oe + 13 E R, and let p be the greatest integer such that 13 - poe E R (cf. VI.2.12). Then one has N2 • P = ±(p

+

1).

For the proof, see Chevalley, T6koku Math. J., 7 (1955), pp. 22-23, or Bourbaki, Chap. 8, Sec. 2, No.4. Remarks. (1) Let g(Z) be the abelian subgroup of g generated by the elements

Ha and by a family of elements Xa satisfying the conditions of Theorem 10. It follows from Theorem 11 that g(Z) is a Lie algebra over Z. For each field K, one can therefore define the Lie algebra g(K) = g(Z) ® K. This is the starting point for the construction of the "Chevalley groups" (cf. Chevalley, lOhoku, loco cit.; see also Carter's survey, J. London Math. Soc., 40 (1965), and his book Simple Groups of Lie Type, Wiley (1972)).

52

VI. Structure of Semisimple Lie Algebras

(2) Tits (Publ. Math. I.H.E.S., 31 (1966), pp. 21-58) has determined the ± signs in Theorem 11 (however, it is necessary to index the elements X. differently). From this he has deduced a new proof of the existence theorem (Theorem 9). (3) Let f be the real vector subspace of g spanned by the elements iH., the elements X. - X-at, and the elements i(Xat + X-at)' One easily checks that f is a real Lie subalgebra of g, and that the Killing form off is negative. Moreover, g can be identified with the complexification f ® C of f. One says that f is a compact form of g. The existence of such a form is the basis ofWeyl's "unitarian trick." When g = 51 2 , we have f = sU 2 (cf. Secs. IV.6, IV.7).

Appendix. Construction of Semisimple Lie Algebras by Generators and Relations Let R be a root system in a complex vector space V. For consistency with the notation of the preceding sections, the dual of V will be denoted by 9, so that V = 9*. Let S = {OC1"'" cxn } be a base for R, let Hlo ... , Hn E 9 be the inverse roots of OC 1, ••• , OCn , and let n(i,j)

= (ocj,Hi ).

The numbers n(i,j) form the Cartan matrix of R with respect to S. We aim to prove the following theorem. Theorem. Let g be the Lie algebra defined by 3n generators Xi' the relations

y;, HI

and by

=0

(W.l)

[Hi,Hj]

(W.2)

[Xit Y;] = Hi>

(W.3)

[Hi' Xj]

(e ij )

ad(Xirn(l.j)+l(Xj ) = 0

iji#j

(eij)

ad(Y;r n(i. J1+l(Yj) = 0

iji #j.

= n(i,j)Xj'

lJ] = [Hi' lJ] =

[Xit

0 iji#j -n(i,j)Yj

Then g is a semisimple Lie algebra, with the subalgebra elements Hi as a Cart an subalgebra; its root system is R.

9 generated by the

Let us first consider the algebra Q defined by the 3n generators XI> y;, Hi and by the relations (W.l), (W.2), (W.3). The structure of Q is known; it has been determined by Chevalley, Harish-Chandra, and Jacobson. We just state the result (for the proof, see, e.g. Bourbaki, Chap. 8, Sec. 4, No.2): One has

Appendix. Construction of Semisimple Lie Algebras by Generators and Relations

53

where 1) (resp. x) is the Lie algebra generated by the elements Y; (resp. XI), and where I) has the elements Hi as a basis. Moreover, 1) (resp. x) can be identified with the free Lie algebra generated by the elements Y; (resp. X;). (By abuse of notation, we identify the original vector space I) with the vector subspace of a spanned by the elements Hi') Now let

and

(}ij = ad(Y;r n (I.J1+ 1 (lj). We have Olj E x, Oil E 1). We denote by u (resp. u-) the ideal of x (resp. of 1)) generated by the elements (}ij (resp. Oil) for i :F j. Let r = u EB U-. (a) u, u-, and r are ideals of a. Let Ua be the universal enveloping algebr.l of a. The adjoint representation ad: a -+ End(a) defines a Ua-module structure on a. The ideal ulJ of a generated by OiJ is equal to the submodule (Ua)' Oij' By the Birkhoff- Witt theorem, uij is spanned (as a vector space) by the elements XYHOlj , with X E Ux, YE U1), HE UI). Clearly HOij is proportional to Olj; moreover, a straightforward calculation (cf. Jacobson, Lemma 1, p. 216) shows that ad(Yk)(Olj) = 0 for all k. It follows that YOIj is proportional to Olj' Thus the ideal ulJ is generated by the elements XOij' and is therefore contained in u. We then have u = Ulj' showing that U is indeed an ideal of a. A similar argument may be applied to u-; the result for r then follows. Th us, r is the smallest ideal generated by the elements Oij and (}ij. Hence the algebra 9 which we wish to study is simply t~e quotient a/r.

L

(b) One has 9

= n-

EB I) EB n, where n = x/u, n- = 1)/u-. This is obvious.

(c) The endomorphisms ad(Xi) and ad(Y;) of 9 are locally nilpotent. Let l'; be the set of all z E 9 such that ad(Xlz = 0 for some integer k. We must prove that l'; = g. Now a simple computation shows that V; is a Lie subalgebra of g. Since V; contains the elements X" (by the relations (}ij = 0) and Yk (by the relations W), it contains the elements H" = [X"' Yk], so we indeed have l'; = g. The same argument can be applied to ad(Y;). We now introduce some notation. If A. is a linear form on I), we denote by aA (resp. gA) the set of all z E a (resp. z E g) such that ad(H)z = J.(H)z for all HE I); such an element z will be said to have weight A.. It follows from the decomposition of a given above that a is the direct sum of the subspaces a A; hence 9 is the direct sum ofits subspaces gA. If a A :F 0,). is a linear combination of the simple roots ai' with integer coefficients, all of the same sign. We have I) = aO, x = LA>O aA, 1) = LA ••• , an} of R, and we denote by R + the set of positive roots (with respect to S). For each a E R+, we choose X", E g"',}',. E g-"'sothat [X",,}',.] = H",(cf. Chap. VI). When a is one of the simple roots ai' we write Xi' fi, Hi instead of X"", Y"", H"". We put n = L",>og"', n- = L", = E~ EB E~. Since we have just seen that dim EO> = 1, we must have E'" = E~ or E'" = E~. In the first case, we have v E Eb and since v generates E this forces E = Eland E2 = O. We apply a similar argument in the second case; thus E is indecomposable.

3. Irreducible Modules with a Highest Weight Theorem 1. Let V be an irreducible g-module containing a primitive element v of weight w. Then: (a) v is the only primitive element of V (up to scalar multiplication); its weight w is called the "highest weight" of V. (b) The weights 11: of V have the form 11: =

w -

"I mj!Xj

with

mj E

N.

They have finite multiplicity; in particular, w has multiplicity 1. One has V=IV". (c) For two irreducible g-modules Vi and V2 with highest weights Wi and W2 to be isomorphic, it is necessary and sufficient that Wi = w 2 • (Statement (b) shows that the weights of V are dominated by w, in an 0 bvious sense; this justifies the terminology "highest weight".)

The g-submodule E of V generated by v is nonzero, and hence equal to V since V is irreducible. By applying Prop. 2 to it, one obtains (b). Let us now prove (a): let v' be a primitive element of V, of weight w'. By (b), w' can be written as w' = w - "~ m·!X· • I'

Similarly, exchanging the roles of t' and v', we see that w = w, -

"

~

' mj!Xj,

mi ~O.

m;

These two eq uations are possible only if mj = = 0 for all i, that is, w = w'. By (b), v and v' are then proportional giving (a). For (c), it is sufficient to prove that, if Wi = w 2 , the modules Vi and V2 are isomorphic. Let Vj (i = 1, 2) be a primitive element of V;, of weight w = Wi = w 2· Clearly the g-module V = Vi EB V2 has v = Vi + V2 as a primitive element of weight w. Let E be the g-submodule of V generated by v. The second projection pr2: V ---+ V2 induces a g-module homomorphism f2: E ---+ V2. One has f(v) = v 2 ; since v 2 generates J-;, it follows that f is surjective. Moreover, the kernel N2 = Vi nE of f2 is a submodule of Vi' This submodule does not

3. Irreducible Modules with a Highest Weight

59

contain Vi (because, by Prop. 2, the only elements of E of weight ware the mUltiples of V, and Vi is not a multiple of v). It is therefore distinct from Vb and since Vi is irreducible, one has N z = O. Thusfz: E -+ Vz is an isomorphism. Similarly, one proves that E is isomorphic to Vi; therefore Vi and V2 are isomorphic. Remark. One can give examples of irreducible g-modules which have no highest weight (in other words, which do not contain a primitive element); these modules are necessarily infinite dimensional (cf. Sec. 4). Theorem 2. For each w equal to w.

E

g*, there is an irreducible g-module with highest weight

(Theorem 1 shows that such a module is unique up to isomorphism.) (i) Our first step will be to construct a g-module V"" containing a primitive element V of weight w, and generated by v. First let LO) be a one-dimensional b-module, having as basis an element v such that Hv

= w(H) if H

E

g, X V = 0

if X

E

n.

We can view LO) as a Vb-module, where Vb denotes the universal enveloping algebra of b. By taking a tensor product with Vg, we obtain from this a Vg-module V", = Vg ®ubL",.

It is clear that the module VO) is generated by the element 1 ® v (which we shall write simply as v); this element is nonzero since, by the Birkhoff-Witt Theorem, V g is a free Vb-module having a basis containing the unit element 1. Moreover, the formulae written above clearly show that v is primitive of weight w. (In fact, VO) has as a basis the family of elements Yp~I ... Yp:-v, but we do not need this .) (ii) Let VO) be the module constructed above, and let us put

If V' is a g-submodule of VO) distinct from V"" then V c V",-. Indeed since V' is sta ble under g, one has V' = L V''', and if V'O) were nonzero, it would contain v and one would have V' = VO)' One therefore has V' = V''', that is, V' c V;. This being so, let NO) be the g-submodule of VO) generated by all the g-submodules of VO) distinct from VO)' By the above, one has NO) c V;, so that NO) =f. VO)' The quotient module EO) = VO)/N is obviously irreducible with highest weight w.

L,.",O)

0)

Remarks. (1) Depending on the given linear form w, it can happen that NO) or NO) =f. 0; both cases arise even for g = s12'

=0

60

VII. Linear Representations of Semisimple Lie Algebras

(2) Theorems 1 and 2 give a bijection between the elements co of 1)* and the classes of irreducible g-modules with a highest weight.

4. Finite-Dimensional Modules Proposition 3. 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 3 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. 2 (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 4 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 1 and 2, it only remains to characterize the elements co E 1)* which are highest weights of finite-dimensional irreducible modules.

Theorem 3. 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 2 to Theorem 1 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 1 and n. Let us put

mj

= CO(Hi)

and

Vi

= Yi mj+l v.

4. Finite-Dimensional Modules

61

If j =f. i, Xj and 1'; commute. One then has X·v· = y•m i+ 1 X·v = 0• J' J Moreover, Theorem 1 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 1. This proves that Vi = O. Theorem 1 of Chap. IV now shows that the vector subspace Fi of EO) spanned by the elements 1'; Pv, 0 ~ 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 4 of Chap. IV, applied to 5 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 1 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 3 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.

5. An Application to the Weyl Group Proposition 4. 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 = 1. 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.

6. Example: sIn+! Let g be the algebra sln+1 of square matrices of order n + 1 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

7. Characters

63

For a base, we take the roots O(i = O(i.i+1> 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 + 1.

- 1 , Aj

°

= Ai + '" + Ai'

The fundamental weight Wi is the highest weight of the natural representation of SIn+l on the vector .space E = e+ 1 . 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.)

7. 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 ) = 1 for all i, so that pEP. (iii) We put D = (e lZ/ 2 - 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.

8. H. WeyI's Formula

65

Theorem* 4. Let E be a finite-dimensional irreducible g-module, and m its highest weight. One has 1

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 1. The dimension of E is given by the formula dimE

=

TI

TI

a>O

<m + p,Ha> = (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 2. 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 9 = 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.

1. 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 1. (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 1 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.

2. Characters

67

Theorem 2.