(Springer monographs in mathematics) jean pierre serre complex semisimple lie algebras 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:
Translated By:
G. A. Jones
University of Southampton
Faculty of Mathematical Studies
Southampton S09 SNH
United Kingdom

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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

ISBN 3-540-67827-1 Springer-Verlag Berlin Heidelberg New York
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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.
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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.