introduction to lie groups and transformation groups
TRANSCRIPT
Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics
Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
7
Philippe Tondeur Department of Mathematics
University of ZfJrich
Introduction to Lie Groups
and
1965 Transformation Groups
Springer-Verlag. Berlin-Heidelberg. New York
All rights, especially that oftranalation/nto foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without
written permission from Springer Verlag. @ by Sprlnger-Verlag Berlin �9 Heidelberg 196~. Library of Congress Catalog Card Number 6~--26947. Printed in Germany. Title No. 7327
Printed by Behz, Weinhelm
PREFACE
T h e s e n o t e s w e r e w r i t t e n f o r i n t r o d u c t o r y l e c t u r e s on L i e g r o u p s
a n d t r a n s f o r m a t i o n g r o u p s , h e l d a t t h e U n i v e r s i t i e s of B u e n o s A i r e s
a n d Z u r i c h . T h e n o t i o n s of a d i f f e r e n t i a b l e m a n i f o l d , a d i f f e r e n t i a b l e
m a p a n d a v e c t o r f i e l d a r e s u p p o s e d k n o w n . T h e r e i s a n a p p e n d i x on
c a t e g o r i e s a n d f u n c t o r s .
T h e f i r s t t w o c h a p t e r s a r e i n f l u e n c e d b y a p a p e r of R . P a l a i s [ l g ] .
In s e c t i o n s 5. Z a n d 5. 3, a l o t i s t a k e n f r o m S. K o b a y a s h i a n d K. N o m i z u
[11] . In c h a p t e r 7, S. H e l g a s o n [61 w a s o f t e n u s e d . Of c o u r s e ,
C. C h e v a l l e y [ 3] w a s c o n s t a n t l y c o n s u l t e d . T h e b i b l i o g r a p h y o r i e n t s
on t h e v a r i o u s s o u r c e s . A s p e c i a l f e a t u r e of t h i s p r e s e n t a t i o n i s t h e
s y s t e m a t i c a v o i d a n c e of t h e u s e of l o c a l c o o r d i n a t e s on a m a n i f o l d . T h i s
a l l o w s t h e u s e of t h e p r e s e n t e d t h e o r y w i t h s l i g h t m o d i f i c a t i o n s f o r L i e
g r o u p s o v e r B a n a c h m a n i f o l d s . S e e e . g . B . M a i s s e n [10].
J u n e 1964 P h i l i p p e T o n d e u r
CONTENTS
.
.
.
.
G - O b j e c t s .
1.1.
1. Z.
1 .3.
" 1 . 4 .
D e f i n i t i o n a n d e x a m p l e s .
E q u i v a r i a n t m o r p h i s m s .
O r b i t s .
P a r t i c u l a r G - s e t s .
G - S p a c e s .
Z. 1. D e f i n i t i o n a n d e x a m p l e s .
Z .Z. O r b i t s p a c e .
G - M a n i f o l d s .
3.1. D e f i n i t i o n a n d e x a m p l e s of L i e g r o u p s .
3. Z. D e f i n i t i o n and e x a m p l e s of G - m a n i f o l d s .
V e c t o r f i e l d s .
4. 1. R e a l f u n c t i o n s .
4. Z. O p e r a t o r s a n d v e c t o r f i e l d s .
4 . 3 . T h e L i e a l g e b r a of a L i e g r o u p .
4 . 4 . E f f e c t of m a p s on o p e r a t o r s and v e c t o r f i e l d s .
4. 5. T h e f u n c t o r L.
4. 6. A p p l i c a t i o n s of t h e f u n c t o r a l i t y of L.
4. 7. The adjoint representation of a Lie group.
1
7
13
23
28
30
34
37
40
4Z
46
50
5Z
59
64
The * indicates a section, the lecture of which is not necessary for the
understanding of the subsequent developments.
.
.
.
V e c t o r f i e l d s a n d 1 - p a r a m e t e r ~ r o u p s of t r a n s f o r m a t i o n s .
5.1.
5. Z.
5. 3.
5 . 4 .
5 . 5 .
* 5 . 6 .
*5. 7.
1 - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s .
1 - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s a n d e q u i v a r i a n t m a p s .
T h e b r a c k e t of t w o v e c t o r f i e l d s .
1 - p a r a m e t e r s u b g r o u p s of a L i e g r o u p .
K i l l i n g v e c t o r f i e l d s .
T h e h o m o m o r p h i s m aV: RG > DX f o r a G - m a n i f o l d .
K i l l i n g v e c t o r f i e l d s a n d e q u i v a r i a n t m a p s .
T h e e x p o n e n t i a l m a p of a L i e g r o u p .
6.1.
6. Z.
6.3.
*6.4.
6.5.
D e f i n i t i o n a n d n a t u r a l i t y of e x p .
e x p is a l o c a l d i f f e o m o r p h i s m at t h e i d e n t i t y .
U n i c i t y of L i e g r o u p s t r u c t u r e .
A p p l i c a t i o n to f i x e d p o i n t s on G - m a n i f o l d s .
T a y l o r ' s f o r m u l a .
S u b g r o u p s a n d s u b a l g e b r a , s .
7 .1. L i e s u b g r o u p s .
7. Z. E x i s t e n c e of l o c a l h o m o m o r p h i s m s .
7. 3. D i s c r e t e s u b g r o u p s .
7 . 4 . O p e n s u b g r o u p s ; c o n n e c t e d n e s s .
7 . 5 . C l o s e d s u b g r o u p s .
7 . 6 . C l o s e d s u b g r o u p s of t h e f u l l l i n e a r g r o u p .
7. 7. Coset spaces and factor groups.
66
70
74
77
84
89
96
103
108
1 iZ
116
IZO
128
132
138
142
144
150
154
. G r o u p s of a u t o m o r p h i s m s .
8.1. The a u t o m o r p h i s m g roup of an a l g e b r a .
8. Z. The ad jo in t r e p r e s e n t a t i o n of a L i e a l g e b r a ,
8. 3. The a u t o m o r p h i s m g r o u p of a L i e g roup ,
160
16Z
167
Append ix : C a t e g o r i e s and f u n c t o r s . 170
B i b l i o g r a p h y 175
-1-
C h a p t e r l . G - O B J E C T S
The first two paragraphs of this chapter are essential for all that
follows, whereas paragraphs i. 3 and i. 4 are only required for the
lecture of g. g and shall not be used otherwise. For the notion of
category and functor, see appendix.
i. i Definition and examples.
If X is an object of a category ~ , we denote by Aut X the group
of equivalences of X with itself. Let G be a group.
DEFINITION I. i. I An operation of G on X is a homomorphisrn
r: G >Aut X. X is called a G-object with respect to T.
on X is a representation of G by automorphisrns An o p e r a t i o n of G
of X .
E x a m p l e 1 .1 .2
s e t X
of X .
by the s a m e l e t t e r )
A G - o b j e c t X in the c a t e g o r y of s e t s E n s is a
e q u i p p e d w i t h a h o m o m o r p h i s m 7 of G in to the g r o u p of b i j e c t i o n s
Such a homomorphism is equivalently defined by a map (denoted
G x X > X
(g, x) ~ - ~ ~ v (x) g
satisfying
-Z-
a) (x) = v ( v (x)) for gz C G xC X Tglgz gl gz gl' '
b) T (x) = x f o r e C G, x C X e
The last conditions in the example i. i. Z suggest calling an
operation in the sense of definition i. i. i more precisely a left-
o p e r a t i o n of G on X . A r i g h t - o p e r a t i o n of G on X w i l l t h e n
G ~ be a h o m o m o r p h i s m ~" : > A u t X, w h e r e G ~ i s t h e
o p p o s i t e g r o u p of G , i . e . t h e u n d e r l y i n g s e t of G w i t h t h e
m u l t i p l i c a t i o n (g lgg) o o = g z g 1. X i s t h e n a G - o b j e c t . We
s h a l l g e n e r a l l y u s e t h e w o r d o p e r a t i o n a s s y n o n y m o u s f o r
left-operation and only be more precise when right-operations
also occur.
Example 1.1. 3 Let G be a group. If to any g C G we
assign the corresponding left translation L of G defined by g
L (~/) = gv f o r V g G , g
we obtain a left-operation of G on
the underlying set of G. Similarly, the assignment of the right
translation Rg of G, Rg(V) = Vg for V C G, to any g 6 G
defines a right-operation of G on the underlying set of G.
Example 1. I. 4 Let p : G
of groups. It defines an operation
set of G' in the following way: set
> G' be a homomorphism
r of G on the underlying
= Lp(g) for g C G.
-3 -
One o b t a i n s s i m i l a r l y a r i g h t o p e r a t i o n
= Rp f o r g g G O-g (g) .
o- by the definition
E x a m p l e 1.1. 5 L e t G be a g r o u p . T o a n y g C G we
a s s i g n t h e i n n e r a u t o m o r p h i s m g
-1 induced by g, ~I (~) = g~/g
g
f o r ~ C G . T h i s d e f i n e s an o p e r a t i o n of G on i t s e l f .
E x a m p l e 1 .1 .6
c o n s i d e r the m a p G x H
m u l t i p l i c a t i o n O x O
H on the s e t O .
L e t H be a s u b g r o u p of t h e g r o u p G and
> G d e f i n e d by r e s t r i c t i n g the
> O . It d e f i n e s a r i g h t - o p e r a t i o n of
by
l a w
E x a m p l e 1.1. 7 L e t t h e g r o u p G
T : G > Au t G ' . On the s e t G'
o p e r a t e on t h e g r o u p G'
x G the m u l t i p l i c a t i o n
(gl" g l ) (gz" gz ) = (gl ''r gl (gz ')' glg2 )
f o r g i ' g G ' , g i g G (i = ~, Z)
d e f i n e s a g r o u p s t r u c t u r e , t h e s e m i - d i r e c t p r o d u c t d e n o t e d
O ' x v G . C o n s i d e r t he h o m o m o r p h i s m s
j : G ' > G ' x~: G j (g ' ) = (g ' , e) f o r g' 6 G ' , e n e u t r a l i n G
p : G' x ~ G > G p (g ' , g) = g f o r g' g G ' , g E G
s : G > G' x G s(g) = (e' g) for e' neutral in G' g GG T J
-4 -
T h e s e q u e n c e
(*) e > G' J > P P > G > e
with P = G'x G is exact and s satisfies p oS = 1 G Conversely T
an e x a c t s e q u e n c e (*) and a h o m o m o r p h i s m s : G > P w i t h
p o s = 1G (a s p l i t t i n g of (*)) d e f i n e s an o p e r a t i o n T of G on G v :
the a u t o m o r p h i s m -r o f G ' c o r r e s p o n d i n g to g ~ G i s the g
i n n e r a u t o m o r p h i s m of P d e f i n e d b y s ( g ) , r e s t r i c t e d to the
n o r m a l s u b g r o u p G ' . T h e r e f o r e G - g r o u p s a r e in ( 1 - 1 ) - c o r r e s p o n d e n c e
w i t h s p l i t t i n g e x a c t s e q u e n c e s (*).
E x a m p l e 1.1. 8 A t y p i c a l c a s e of the s i t u a t i o n j u s t m e n t i o n e d
i s a s f o l l o w s : L e t V be a f i n i t e - d i m e n s i o n a l I R - v e c t o r s p a c e
and G L ( V ) the g r o u p of l i n e a r a u t o m o r p h i s r n s of V. T h e n G L ( V )
o p e r a t e s n a t u r a l l y on V . T h e s e m i - d i r e c t p r o d u c t V x G L ( V )
i s the g r o u p of a f f i n e m o t i o n s of V . N o t e t h a t t he m u l t i p l i c a t i o n
j u s t c o r r e s p o n d s to the n a t u r a l c o m p o s i t i o n of a f f i n e m o t i o n s .
We s h a l l on ly h a v e to c o n s i d e r c a t e g o r i e s ~ w h o s e o b j e c t s
h a v e an u n d e r l y i n g s e t an d w h o s e m o r p h i s m s a r e a p p l i c a t i o n s
of the u n d e r l y i n g s e t s . M o r e p r e c i s e l y t h i s m e a n s t h a t t h e r e
e x i s t s a f u n c t o r V : ~ > E n s w h i c h c a n be t h o u g h t of a s
f o r g e t t i n g a b o u t the a d d i t i o n a l s t r u c t u r e on X in R and t a k i n g
a m o r p h i s m j u s t a s an a p p l i c a t i o n . T o a v o i d e n d l e s s r e p e t i t i o n s
- 5 -
we m a k e t h e following c o n v e n t i o n : F r o m n o w on we s h a l l
o n l y c o n s i d e r c a t e g o r i e s of t h a t s o r t a n d s h a l l u s e t h e s a m e
n o t a t i o n X f o r a n o b j e c t X a n d i t s u n d e r l y i n g s e t V X .
A n o p e r a t i o n of t h e g r o u p G on X d e f i n e s a n o p e r a t i o n on
t h e u n d e r l y i n g s e t . M o r e g e n e r a l l y w e h a v e t h e
P R O P O S I T I O N 1 . 1 . 9
f u n c t o r f r o m t h e c a_tegor_~y
of t h e group G
FX ~ ~.'
on X C~
Let F : R --> g' be a c ovariant
to the category R'. An operation
induces a well-defined operation on
Proof: F defines a homomorphism Aut X
By composition with the given homomorphism G
> Aut FX.
> Aut X we
functor F : ~. ISO
obtain a homomorphism G > Aut FX, which is the desired
operation of G on FX.
Remark I. I. I0 If in a given category ~ we only consider
equivalences as morphisms, we obtain a new category ~. Iso
Evidently proposition I. I. 9 is still valid if we are only given a
> ~I. ISO
If F :~ > ~' is a contravariant functor, a left-operation
of G on X induces a right-operation of G on FX, and a right-
operation on X is turned into a left-operation on FX.
Example 1.1.11 Consider the covariant functor
P : Ens > Ens, making correspond to each set X the set PX
- 6 -
of i t s s u b s e t s , to e a c h m a p X
P X > P X ' of s u b s e t s . L e t X be a G - s e t .
-1 G - s e t by p r o p o s i t i o n 1 .1 .9 . T h e f u n c t o r P
h a v i n g the s a m e e f f e c t a s P
> X' t h e i n d u c e d m a p
T h e n P X is a
: E n s > E n s ,
o n o b j e c t s of E n s , but a s s i g n i n g
-1 to a m a p @ : X > X ' t h e m a p r : P X ' > P X ( i n v e r s e
images of subsets), transforms the G-set X into the G ~ -set
P X .
E x a m p l e 1.1.12
T h e c o n t r a v a r i a n t f u n c t o r
L e t R be a f i x e d o b j e c t of t h e c a t e g o r y
hR(X) = [ X , R ] , = f o
g i v e s f o r a n y l e f t - o p e r a t i o n of G
on the s e t [ X , R] . If T : G
p h i s m , we w r i t e T
i n t o t h e g r o u p of b i j e c t i o n s o f [ X , R ] .
E x a m p l e 1 . 1 . 1 3 L e t A be a r i n g ,
hR : ~ > E n s d e f i n e d by
f o r f E ; [ X t, R] , ~ : X > X ' ,
on X a r i g h t - o p e r a t i o n of G
> Au t X is the g i v e n h o m o m o r -
f o r t h e i n d u c e d h o m o m o r p h i s m of G ~
2l the c a t e g o r y of
l e f t - A - m o d u l e s . A G - m o d u l e X i s d e f i n e d by a n o p e r a t i o n of
G on X by A - l i n e a r m a p s ; i . e . a r e p r e s e n t a t i o n of G in X
in t he u s u a l s e n s e . By p r o p o s i t i o n 1 .1 .9 s u c h a r e p r e s e n t a t i o n
i n d u c e s an o p e r a t i o n of G on the s e t of s u b m o d u l e s of X .
F o l l o w i n g o u r c o n v e n t i o n on t h e c a t e g o r i e s to c o n s i d e r , i t
m a k e s s e n s e to s p e a k of an e l e m e n t of an o b j e c t X .
-7-
D E F I N I T I O N i. i . 14 A n e l e m e n t x i n t h e G - o b j e c t
c a l l e d i n v a r i a n t o r G - i n v a r i a n t i f x i s f i x e d u n d e r e v e r y
transformation m g
A subset M c
X is
: Tg(X) = x f o r a l l g ~ G .
X i s c a l l e d i n v a r i a n t i f i t i s a n i n v a r i a n t
element of PX under the induced G-operation (example i. I. Ii),
i . e . i f T (M) c M f o r a l l g C G . g
E x e r c i s e 1 . 1 . 1 5 L e t X a n d X ' b e G - o b j e c t s of ~ w i t h
respect to T: G >Aut X and I-' : G >Aut X'.
O-g(~) = T' g o ~ o ~" -I for g G G, ~: X >X' defines an g
operation of G on the set of morphisms from X to X'.
(Example i. 1. IZ is a special case of this situation, if we considert
trivial G-operation on X'.) Show that there is a suitable functor
inducing this operation according to proposition I. I. 9.
1 . 2 E q u i v a r i a n t m o r p h i s m s .
L e t G a n d G ' b e g r o u p s a n d R a c a t e g o r y . S u p p o s e X
w i t h r e s p e c t t o a h o m o m o r p h i s m
w i t h r e s p e c t t o a h o m o m o r p h i s m
a G ' - o b j e c t of ~ w i t h r e s p e c t t o a
O' > Aut X'.
A p-equivariant morphism
P:G
t o b e a G - o b j e c t of
T: G > A u t X , X '
! h o m o m o r p h i s m v :
D E F I N I T I O N I. 2. i
~: X --> X' > G '
- 8 -
i s a m o r p h i s m ~0: X > X'
f o l l o w i n g d i a g r a m c o m m u t e s
of g s u c h t h a t f o r a l l g ~ G t h e
T g
X ~o > X'
X
I
d T
>X
map.
If G =G' and P = I G, we just speak of anequivariant
E x a m p l e 1. Z. Z If X i s a G - s e t a n d X' a G ' - s e t w i t h t h e
o p e r a t i o n s g i v e n a s i n e x a m p l e 1.1. ~, t h e n a m a p ~ : X > X'
i s P - e q u i v a r i a n t i f a n d o n l y i f t h e f o l l o w i n g d i a g r a m c o m m u t e s
GxX T > X
I ' d , T
G ' x X t . > X t
E x a m p l e 1. Z. 3 L e t P : G - - > G v be a h o m o m o r p h i s m
of g r o u p s . I f G a n d G ' a r e o p e r a t i n g on i t s e l f by l e f t -
t r a n s l a t i o n a s i n e x a m p l e 1.1. 3, t h e n a m a p q~ : G > G ' i s
p - e q u i v a r i a n t i f a n d o n l y i f ~ ( g l g 2 ) = p ( g l ) ~ ( g z ) . T h e r e -
f o r e p i t s e l f i s a n e x a m p l e of a p - e q u i v a r i a n t m a p w i t h
r e s p e c t to t h e l e f t - o p e r a t i o n s .
- 9 -
If w e c o n s i d e r t h e o p e r a t i o n s of G a n d G '
i n n e r a u t o m o r p h i s m s ,
on i t s e l f b y
commutes, i.e. p is
Example I. Z. 4
t h e n f o r a l l g C G t h e d i a g r a m
G P > G'
I I
G P > G'
p - e q u i v a r i a n t .
If w e c o n s i d e r t h e r i g h t - o p e r a t i o n of t h e
s u b g r o u p H of G on G a s i n e x a m p l e 1 . 1 . 6 , t h e n a h o m o m o r p h i s m
P : G - - > G s e n d i n g H i n t o H c a n b e c o n s i d e r e d a s a p / H -
p / H d e n o t e s t h e r e s t r i c t i o n of p e q u i v a r i a n t m a p , w h e r e
t o H .
E x a m p l e 1. Z. 5 A n y r i g h t - t r a n s l a t i o n of a g r o u p G i s a n
e q u i v a r i a n t m a p of t h e G - s e t G d e f i n e d b y t h e l e f t - t r a n s l a t i o n .
T h i s i s j u s t t h e a s s o c i a t i v i t y l a w i n G .
Example i. Z. 6 If
of G on X, then for any g ~ G the map
- equivariant, where g
automorphism of G defined by g.
Example i. Z. 7 Let X be a G-set.
p ( g ) = "r
T: G > A u t X d e f i n e s a n o p e r a t i o n
T :X >X i s g
[~ : G - - > G d e n o t e s t h e i n n e r g
g(X o) de fines a map p : G
F o r f i x e d x C X o
> X . I f w e c o n s i d e r t h e
- 1 0 -
o p e r a t i o n of G on G by l e f t - t r a n s l a t i o n , p is an e q u i v a r i a n t
map.
If X, X', X" are G, G,
9:G >G', P' :G'
~0: X >X', ~01 : X'
respectively, then clearly
G"-objects respectively,
> G" homornorphisms and
> X" p, p'-equivariant morphisms
! !
~0 o r is a P o p-equivariant
m o r p h i s m . F o r f i x e d G the G - o b j e c t s of a c a t e g o r y ~ t h e r e -
G f o r e f o r m a c a t e g o r y ~ w i t h t h e e q u i v a r i a n t m o r p h i s m s as
m o r p h i s m s ( D e f i n i t i o n 1 . 2 . 8 ) .
As a c o m p l e m e n t to p r o p o s i t i o n 1 .1 .9 we h a v e
P R O P O S I T ION 1. Z. 9 L e t F : K > K' b e a c o v a r i a n t
functor, X, X' respectively G, G'-objects of ~ , p : G~> G'
a h o m o m o r p h i s m and q~: X > X' a p-equivariant morphism.
!
C o n s i d e r the n a t u r a l o p e r a t i o n s i n d u c e d on F X and F X T h e n
F ( ~ ) : F X > F X ' is P - e q u i v a r i a n t w i t h r e s p e c t to t h e s e
o p e r a t i o n s . F o r a f i x e d g r o u p G th i s d e f i n e s in p a r t i c u l a r a n
extension of the map of proposition 1.1.9, sending G-objects of
into G-objects of K' , to a functor F G ~G ~,G : > .
-ii-
Proof: The commutative diagram
X ~ ,> X' r
% r [
~o X ~ X -- ' >
f
i s t r a n s f o r m e d by F in t h e c o m m u t a t i v e d i a g r a m
FX F(~) > FX'
r T
F(Tg) i i F ( ' ~
FX F(~) > FX'
p(g)
s h o w i n g t h e P - e q u i v a r i a n c e of F(~) w i t h r e s p e c t to t h e
i n d u c e d o p e r a t i o n on F X a n d F X ' . T h e r e s t i s c l e a r .
If an e q u i v a r i a n t m o r p h i s r n
!
i n v e r s e ~ : X > X in R , i . e .
t h e n ~ i s n e c e s s a r i l y e q u i v a r i a n t ,
G e q u i v a l e n c e in ~ .
T h e r e i s a c a n o n i c a l f u n c t o r V : R G
c o n s i s t s i n f o r g e t t i n g a b o u t t h e G - o p e r a t i o n .
~0 : X > X ' in ~G h a s an
% o ~0 = l x , ~0 o ~ = I x , ,
and ~0 t h e r e f o r e an
> ~ w h i c h
On t h e o t h e r h a n d ,
-12 -
w e d e f i n e a f u n c t o r I : ~ > ~G b y c o n s i d e r i n g on e v e r y
o b j e c t X of ~ t h e t r i v i a l G - o p e r a t i o n 7: G > A u t X
m a p p i n g G on 1 x . T h e r e f o r e i t m a k e s s e n s e t o s p e a k of
e q u i v a r i a n t m o r p h i s m s ~0: X > R , w h e r e X i s a G - o b j e c t
of ~ a n d R a n a r b i t r a r y o b j e c t of e . W e c a l l s u c h a m a p
a n i n v a r i a n t m o r p h i s m . M o r e p r e c i s e l y w e h a v e t h e
D E F I N I T I O N 1 .2 .11 L e t X be a G - o b j e c t of ~ , R a n
o b j e c t of ~ . A m o r p h i s m ~0 : X > R in ~ i s c a l l e d
i n v a r ! a n t i f f o r a l l g G G t h e f o l l o w i n g d i a g r a m c o m m u t e s
X
X
P R O P O S I T I O N 1. Z. 12 L e t X b e a G - s e t , X ' I
a G - s e t
a n d ~9 : X > X ' a 9 - e q u i v a r i a n t m a p w i t h r e s p e c t t o _a
h o m o r n o r p h i s m 9 : G > G ' . If x 6 X i s G - i n v a r i a n t , t h e n
(x) i s p ( G ) - i n v a r i a n t .
P r o o f : !
Tg(X) = x i m p l i e s T p ( g ) ( ~ ( x ) ) = ~ ( T g ( X ) ) = ~ ( x ) .
-13-
A s a c o n s e q u e n c e j G - i n v a r i a n t s u b s e t s of X go i n t o
p ( G ) - i n v a r i a n t s u b s e t s of X t .
E x e r c i s e 1 . 2 . 1 3 ~G c a n be c o n s i d e r e d a s a c a t e g o r y of
f u n c t o r s ( i n t e r p r e t G as a c a t e g o r y c o n s i s t i n g of a s i n g l e o b j e c t
and m o r p h i s m s g w i t h g C G w i t h n a t u r a l c o m p o s i t i o n l aw) .
E q u i v a r i a n t m o r p h i s m s a r e t h e n j u s t n a t u r a l t r a n s f o r m a t i o n s .
T h e f u n c t o r F G of p r o p o s i t i o n 1 . 2 . 9 is t he c a n o n i c a l f u n c t o r
i n d u c e d b y F b e t w e e n the c o r r e s p o n d i n g f u n c t o r c a t e g o r i e s .
E x e r c i s e 1. Z .14 If X and X' a r e G - o b j e c t s of ~ , t h e
s e t of m o r p h i s m s f r o m X to X' i s a G - s e t a c c o r d i n g to
e x e r c i s e 1 .1 .15 . T h e i n v a r i a n t e l e m e n t s u n d e r t h i s o p e r a t i o n
a r e the e q u i v a r i a n t m o r p h i s m s X - - > X ' . As a s p e c i a l c a s e ,
t he i n v a r i a n t m o r p h i s m s X > R , w h e r e R i s an o b j e c t of
, a r e t h e i n v a r i a n t e l e m e n t s u n d e r the o p e r a t i o n d e f i n e d in
e x a m p l e 1.1.1Z.
1 . 3
the g i v e n o p e r a t i o n i s the s e t
O r b i t s .
L e t X b e a G - s e t w i t h r e s p e c t t o T: G - > A u t X.
D E F I N I T I O N 1. 3.1 T h e o r b i t o r G - o r b i t of x C X u n d e r
~(x) = [ "rg(X)/g. G G } .
- 1 4 -
L E M M A I. 3.2. If
o r b i t s f o r m a p a r t i t i o n of
X i s a G - s e t , t h e d i f f e r e n t
X i n t o d i s j o i n t s e t s .
P r o o f : A s x g ~(x) , t he o r b i t s c o v e r X . We
o n l y h a v e to s h o w : i f two p o i n t s x , x ' g X h a v e i n t e r -
s e c t i n g o r b i t s [2(x), ~(x ' ) , t h e n ~(x) = ~(x ' ) . L e t
y ~ ~2(x) (] ~2(x' ): y = Vg(X), y = Vg, (x') . For z ~ ~2(x):z
= "rX(x ) we h a v e z = (-rye -rg_ 1 . - r g , ) ( x ' ) ~ ~ { x ' ) , i . e .
~(x) c ~(x' ), This shows ~(x) = ~2(x' ) .
Let X/G be the set of orbits, ~x:X -. X/G the
canonical map. An orbit is the orbit of any of its points.
This implies ~(Vg(X)) = =(x), i.e. ~ is an invariant map.
More generally we have
L E M M A 1 . 3 . 3 . L e t X be a G - s e t , m , ,
t he c a n o n i c a l m a p on to i t s o r b i t s e t X / G a n d
a r b i t r a r y , s e t . F o r a n y i n v a r i a n t m a p r X -. R
i s o n e a n d o n l y one m a p 4 : X / G -. R s u c h t h a t
~X: X -- X / G
R a n
t h e r e
= ~,Ir x
P r o o f : I f ~ , "rg = r f o r a l l g ~: G , t h e n
i s c o n s t a n t on e a c h o r b i t ~(x) , a n d t h e r e f o r e d e f i n e s a
m a p @: X / G -. R w i t h t h e d e s i r e d p r o p e r t y .
-15 -
On t h e o t h e r h a n d , a m a p x~ : X / G
Tr x : X > X / G g i v e s a n i n v a r i a n t m a p
h a v e p r o v e d
> R c o m p o s e d w i t h
~o= ~ o ~ W e X"
P R O P O S I T I O N 1. 3 . 4 L e t X be a G - s e t , WX : X > X / G
t h e c a n o n i c a l m a p o n t o i t s s e t of o r b i t s a n d R a n a r b i t r a r y s e t .
T h e c o r r e s p o n d e n c e % ~ - ~ - > % o ~ , s e n d i n ~ m a p s f r o m X / G
t o R i n t o i n v a r i a n t m a p s f r o m X t_~o R i s b i j e c t i v e .
Remark. X/G is characterized by this universal
property up to a canonical bijection by a standard argument.
This property allows therefore the definition of X/G in an
arbitrary category. Of course, there remains to show the
existence of such an orbit-object in a given category.
PROPOSITION I. 3.5 Let X be a G-set, X' a G'-set,
p : G >G' a homomorphism and q~ : X >X' a
9 - e q u i v a r i a n t m a p . T h e n t h e r e e x i s t s o n e a n d o n l y one m a p
~: X/G > X'/G' , such that the following diagram commutes
Q0 X >X
I I
~ x ~ X / G ' > / G t
-16 -
P r o o f : By t h e u n i v e r s a l p r o p e r t y s t a t e d in p r o p o s i t i o n 1. 3 . 4 ,
it is sufficient to show that w x, o ~ : X > X t/G t is an invariant
map. But
( ~ x ' ~ ~ ) ~ Tg = = x ' ~ ( ~ ~ *g) = ~ x ' o(T;(g) - 0 )
!
= (~r x, o Tp(g ) ) o ~ = w x, o ~2 WXV b e i n g
an i n v a r i a n t m a p . ~ is n o w d e f i n e d as t h e f a c t o r i z a t i o n of
,r x, o ~ through X / G .
E x a m p l e 1. 3. 6 L e t G be a g r o u p a n d H a s u b g r o u p ,
o p e r a t i n g by r i g h t t r a n s l a t i o n s on G ( e x a m p l e 1. 1 .6) . T h e n G / H
d e n o t e s t h e s e t of o r b i t s , t he s e t of l e f t c o s e t s m o d u l o H . L e t
G v be a n o t h e r g r o u p and H v a s u b g r o u p of G v L e t f u r t h e r
: G > G v be a m a p s u c h t h a t ~ ( H ) C H v a n d ~ ( g h )
= ~0(g}~0(h) f o r g ~ G , h ~ H . T h e n ~o/H : H > H' i s a
h o m o m o r p h i s m and ~ is ~ / H - e q u i v a r i a n t . B y p r o p o s i t i o n
1. 3 .5 t h e r e e x i s t s one a n d o n l y one m a p ~0 : G / H > GV/H t
s u c h t h a t t h e d i a g r a m
G ~ > G I
r l Gf f - G ,
G / H > G /H '
-17-
c o m m u t e s . In t h e c a s e w h e r e H and H' a r e n o r m a l s u b g r o u p s
of G and G' r e s p e c t i v e l y and ~0 i s a h o m o m o r p h i s m , (p is t he
i n d u c e d h o m o m o r p h i s m of the q u o t i e n t g r o u p s .
C o n s i d e r n o w a f i x e d g r o u p G . F o r a n y G - s e t X we h a v e
d e f i n e d the o r b i t s e t X / G . M o r e o v e r by p r o p o s i t i o n 1. 3 .5 a n y
e q u i v a r i a n t m a p (p: X - > X' i n d u c e s one and o n l y one m a p
r X / G > X ' / G . In t h i s w a y we o b t a i n a c o v a r i a n t f u n c t o r
Ens G "~ B : ~ Ens from G-sets to sets: B(X) = X/G, B(~0) : (p.
A standard consequence is that an equivalence ~0: X > X'
in Ens G induces a bijection ~0: X/G .... > X'/G.
Remark. If we consider the "forget-functor" V: Ens G
defined as forgetting about the G-set structure, we see that
w : V >B is a natural transformation of V into B.
To the beginning of this paragraphjfor a G-set X 2 we have
introduced the map
by the map w x : X
can be extended to a map PX --> PX and we interpret now
-i as the map w x , w x : PX > PX. For M c X Q(M) is
just the orbit of M under the induced G-operation on PX.
Explicitly
> Ens
: X > P X , w h i c h c a n a l s o be d e s c r i b e d
-1 X / G as ~ = ~ X ~ T h e r i g h t s i d e
- (-,- (x) l g c G , x e ; M } . g
-18-
2 ( M ) i s t h e r e f o r e t h e s a t u r a t i o n of M w i t h r e s p e c t t o G ,
i . e . t h e u n i o n of a l l G - o r b i t s of X i n t e r s e c t i n g M .
T h e i n v a r i a n c e of M c X c a n n o w be e x p r e s s e d by
(M) = M. For an arbitrary M c X the set
intersection of all invariant sets containing M.
are the minimal invariant sets.
~ ( M ) i s t h e
T h e o r b i t s
CfK-"
Isotropy groups.
{g e G / = x }.
Let X be a G-set and x C X.
G is a subgroup of G. x
Consider
DEFINITION i. 3.7. G is called the isotropy group of x. x
= Gxg-I PROPOSITI ON I. 3.8. Gg x g
Proof:
h6G x
g Gxg -1 c G gx
g-1 c G G g x g x
For simplicity we write Vg(X) = gx. Then for
we have ghg -I . gx = ghx = gx, which implies
A s g - 1 . g x = x , we h a v e b y t h e s a m e a r g u m e n t
o r Gg x c g G x g - 1 , w h i c h p r o v e s t h e p r o p o s i t i o n .
This can also be expressed in the following way.
the map ~0 : X
by qg(x) = G x .
orbit being a c o n j u g a c y class
Consider
> SG into the set of subgroupsof G, defined
G operates by inner automorphisms on SG, .an
of subgroups. By proposition .
I. 3.8 the diagram
-19 -
X r > SG
. l [
X ~ > SG
g
c o m m u t e s f o r a l l g ~ G , i . e . ~0 i s a n e q u i v a r i a n t m a p . T h e r e -
f o r e ~0 i n d u c e s f o l l o w i n g p r o p o s i t i o n 1 . 3 . 5 a m a p
N
~0: X / G > S G / G , i . e . t o e v e r y o r b i t of X t h e r e c o r r e s p o n d s
a w e l l - d e f i n e d c o n j u g a c y c l a s s of s u b g r o u p s of G , c a l l e d t h e
o r b i t - t y p e o f t h e o r b i t .
P a r t i c u l a r o r b i t - t y p e s a r e t h e c o n j u g a c y c l a s s e s {e} a n d
G . L e t x be in ~ ( X o ) of o r b i t - t y p e { e } . T h e n f o r x 6 ~ ( x ) 0
t h e r e i s one a n d o n l y one g C G s u c h t h a t gx ~ = x . B e c a u s e
= l g l x = i m p l i e s g~ lg 1 = e a n d gl = g z " glXo gzXo o r g~ o Xo
If Xo i s i n ~] (x o) of o r b i t - t y p e G , t h e n Xo i s G - i n v a r i a n t ,
a n d (Xo) - x o . T h e r e f o r e t h e f i x e d p o i n t s a r e e x a c t l y t h e
o r b i t s of o r b i t - t y p e G .
E x a m p l e 1. 3 .9 . C o n s i d e r t h e f u l l l i n e a r g r o u p G L ( n , JR),
c o n s i s t i n g of t h e r e a l q u a d r a t i c m a t r i c e s w i t h n e n t r i e s h a v i n g
a d e t e r m i n a n t d i f f e r e n t f r o m z e r o , w i t h t h e n a t u r a l o p e r a t i o n
on n~ n . T h e o r i g i n O and i t s c o m p l e m e n t n~ n - { o } a r e t h e
o r b i t s of t h i s o p e r a t i o n . T h e o r b i t - t y p e of O i s G L ( n , ]1%).
- 2 0 -
E x a m p l e 1. 3 .10. C o n s i d e r IR n w i t h t h e s t a n d a r d e u c l i d e a n
m e t r i c a n d the c o r r e s p o n d i n g o r t h o g o n a l g r o u p O ( n , JR). T h e
o r b i t s of t h e n a t u r a l o p e r a t i o n a r e t he s p h e r e s w i t h t he o r i g i n
as c e n t e r . T h e i s o t r o p y g r o u p of a po in t d i f f e r e n t f r o m the
o r i g i n i s i s o m o r p h i c to t he o r t h o g o n a l g r o u p O(n-1 , JR).
E x a m p l e 1. 3.11. L e t X d e n o t e t h e c o m p l e x p l a n e w i t h a
p o i n t a t i n f i n i t y a d j o i n e d . T h e g r o u p of t r a n s f o r m a t i o n s of t he
az +b t y p e z " " > w i t h a, b, c d 6 ~; a n d ad - bc ~ O
cz + d
o p e r a t e s on X . X is the o r b i t of a n y po in t x ~ X .
E x a m p l e 1. 3.1Z. C o n s i d e r t h e o p e r a t i o n of a g r o u p G on
i t s e l f by i n n e r a u t o m o r p h i s m s . T h e f i x p o i n t s a r e t he e l e m e n t s
of t he c e n t e r C G . We h a v e a l r e a d y c o n s i d e r e d t h e i n d u c e d
G - o p e r a t i o n on the s e t SG of s u b g r o u p s of G . T h e o r b i t of a
s u b g r o u p i s i t s c o n j u g a c y c l a s s . T h e r e f o r e t he i n v a r i a n t s u b -
g r o u p s of G a r e e x a c t l y t h e f i x p o i n t s u n d e r t h i s o p e r a t i o n .
M o r e o v e r i t f o l l o w s t h a t t h e d i f f e r e n t c o n j u g a c 7 c l a s s e s f o r m
a p a r t i t i o n of S G .
T h e e f f e c t of an e q u i v a r i a n t m a p on t h e i s o t r o p y g r o u p s
i s d e s c r i b e d by the
-21-
PROPOSITION i. 3.13. L e t X be a G - s e t , X v G' a -set,
P: G >G' a homomorphism and ~0 : X >X' a
p-equivariant m a p . T h e n p (G x) c G ~ ( x ) .
Proof: Let g ~ G , i.e. gx= x. Then x
p (g)~0(x) = ~0 (gx) = ~(x) i.e. p(g) ~ G' ' ( p ( x ) "
G ! E x e r c i s e 1. 3 . 1 4 . L e t X b e a G - s e t , X v a - s e t ,
p: G > G' "~ a h o m o m o r p h i s m a n d ~o : X / G > X ' / G ' a m a p .
S t u d y t h e c o n d i t i o n s u n d e r w h i c h ~ i s i n d u c e d by a p - e q u i v a r i a n t
m a p ~0 : X > X t i n t h e s e n s e of p r o p o s i t i o n 1 . 3 . 5 .
Exercise I. 3. 15. For a G-set X consider the map
= Tr X o ir X : PX > PX defined above. Show that
h a s t h e f o l l o w i n g p r o p e r t i e s :
a) ~ (~)) = (~ fo~.the empty set r of X
b) M c ~ (M) f o r M c X
f o r a f a m i l y ( M x ) x 6 A of M X c X .
-ZZ -
T h e r e f o r e Q i s a " K u r a t o w s k i - o p e r a t o r " on X and d e f i n e s
a t o p o l o g y on X a c c o r d i n g to the d e f i n i t i o n : M c X is c l o s e d
if and o n l y if ~ ( M ) = M . T h i s r e m a i n s t r u e i f we c o n s i d e r
an a r b i t r a r y e q u i v a l e n c e r e l a t i o n R on X (no t n e c e s s a r i l y
d e f i n e d by a g r o u p G) and the map ~= ~-I o ~r : PX x X
>PX,
w h e r e lr : X x
> X / R is the c a n o n i c a l m a p onto t h e q u o t i e n t
s e t X / R . Show t h a t m o r e g e n e r a l l y f o r an a r b i t r a r y r e l a t i o n
R on a set X the "saturation-operator " ~: PX > PX
d e f i n e d by
(M) = [ y ~ X / x R y f o r s o m e x ~ M ]
f o r M c X s a t i s f i e s t he p r o p e r t i e s 1) to 4) if a n d o n l y i f R
is a r e f l e x i v e and t r a n s i t i v e r e l a t i o n on X .
E x e r c i s e 1 . 3 . 1 6 . C o n s i d e r t he t o p o l o g y d e f i n e d in e x e r c i s e
1. 3.15 on a s e t X e q u i p p e d w i t h an e q u i v a l e n c e r e l a t i o n R .
Show t h e f o l l o w i n g p r o p e r t i e s :
1) M c X is c l o s e d i f a n d o n l y if M is a u n i o n of
e q u i v a l e n c e c l a s s e s ;
2) M c X is closed if and only if M is open.
- Z 3 -
Wha t a r e t h e c o n d i t i o n s on X / R f o r t h e t o p o l o g y in q u e s t i o n to s a t i s f y
i)
z)
3)
Exercise i. 3.17. Let
t h e s e c o n d c o u n t a b i l i t y a x i o m ,
t o be c o m p a c t ,
t o be c o n n e c t e d ?
X be a G - s e t , R a n a r b i t r a r y s e t a n d ~ : X ' > R
a map. Suppose X equipped with the topology defined in exercise I. 3.15 and
R topologized by the discrete topology. Then (~ is invariant if and only if
(~ is continuous.
I. 4 Particular G-sets.
Let X be a G-set, defined by a homomorphism 7 : G
define some particular properties an operation can have.
7 is an effective operation if 7 D E F I N I T I O N 1 .4 . 1
Ker T = {e}.
> B i j X . We
is injective, i.e.,
W e o b s e r v e t h a t K e r T = N G x , an e l e m e n t of K e r T x C X
exactly an element of G contained in every isotropygroup. If
e f f e c t i v e , t h e n t h e r e e x i s t s a f a c t o r i z a t i o n T t h r o u g h G / K e r
i Bij x
G/Ker T
a n d G / K e r T o p e r a t e s e f f e c t i v e l y on X .
Example i. 4. Z The operation [~ of a group O
has the center CG = Ker ~ as kernel.
being
T is n o t
T
by inner automorphisms
- 2 4 -
D E F I N I T I O N 1 . 4 . 3
x 6 X i m p l i e s g = e .
T h i s m e a n s t h a t a t r a n s f o r m a t i o n
T i s a f r e e o p e r a t i o n , i f Tg(X) = x f o r s o m e
T f o r g ~ e h a s n o f i x p o i n t . g
F r e e m e a n s " f r e e of f i x p o i n t s " . T h e i s o t r o p y g r o u p i s r e d u c e d t o t h e
n e u t r a l e l e m e n t : G = { e } f o r e v e r y x ~ X . X i s a l s o c a l l e d a x
" G - p r i n c i p a l s e t " . N o t e t h a t a f r e e o p e r a t i o n i s e f f e c t i v e .
E x a m p l e 1 . 4 . 4 T h e o p e r a t i o n of G on G b y l e f t - t r a n s l a t i o n s i s f r e e .
L e t H b e a s u b g r o u p of G ,
o p e r a t i o n i s f r e e .
D E F I N I T I O N 1 . 4 . 5 T
t h e r e e x i s t s a g ~ G s u c h t h a t
t h e e l e m e n t g i s u n i q u e .
o p e r a t i n g on G b y r i g h t - t r a n s l a t i o n s . T h i s
i s a t r a n s i t i v e o p e r a t i o n , i f f o r x 1 , x z ~ X
7g(Xl) = x z , s i m p l 7 t r a n s i t i v e , i f , m o r e o v e r ,
A s i m p l y t r a n s i t i v e o p e r a t i o n i s f r e e . C o n v e r s e l y , a f r e e o p e r a t i o n
i s s i m p l y t r a n s i t i v e on e a c h o r b i t . B e c a u s e i f x = g iXo ( i = 1, Z) , t h e n
-I -i -i ={e}, i.e. gl = gz" Xo = gz x = gz glXo and therefore gz gl C Gxo
The definition of a transitive operation can also be put in the following
form: there exists an element x C X such that ~(Xo) = X. X is then O
the orbit of each point x C X. This shows that the set of orbits X/G is
a point. This property allows us to define the transitivity of a G-operation
in an arbitrary category, as so~n as the notion of point is defined.
DEFINITION i. 4. 6 A G-set X is called homogeneous, if G
transitively on X.
o p e r a t e s
- 2 5 -
E x a m p l e 1 .4 . 7 T h e o r t h o g o n a l g r o u p O(n, JR) o p e r a t e s t r a n s i t i v e l y
on the un i t s p h e r e S n-1 in IR n .
M o r e g e n e r a l l y , an o p e r a t i o n of G on X d e f i n e s a t r a n s i t i v e o p e r a t i o n
on e a c h G - o r b i t .
E x a m p l e 1 .4 . 8 T h e g r o u p of h o l o m o r p h i s m s of t h e un i t d i s k in t he
c o m p l e x p l a n e o p e r a t e s t r a n s i t i v e l y .
A f u n d a m e n t a l e x a m p l e of a h o m o g e n e o u s G - s e t is o b t a i n e d in t he
f o l l o w i n g w a y . C o n s i d e r a g r o u p G and a s u b g r o u p H o p e r a t i n g on G by
r i g h t - t r a n s l a t i o n s . T h e n we c a n d e f i n e an o p e r a t i o n of G on t h e o r b i t s e t
G / H . T h e l e f t t r a n s l a t i o n L : G > G s a t i s f i e s L g ( v H ) = g v H and g
t h e r e f o r e d e f i n e s by r ( v H ) = g v H a m a p �9 : G / H > G / H . r is g g
t he d e s i r e d o p e r a t i o n , m a k i n g G / H a G - s e t , w h i c h i s e v i d e n t l y h o m o g e n e o u s .
R e m a r k . T h e i s o t r o p y g r o u p of H i s H .
We s h a l l s h o w t h a t f o r an a r b i t r a r y h o m o g e n e o u s G - s e t X t h e r e
e x i s t s a s u b g r o u p H of G and an e q u i v a l e n c e ~ : G / H > X of G - s e t s ,
w h e r e G / H is c o n s i d e r e d as a G - s e t in t he s e n s e i n d i c a t e d .
F i r s t l e t X be an a r b i t r a r y G - s e t and x ~ X . We put H = G and O x o
d e f i n e ~ : G / H > ~ ( X o ) X by r = g x o .
L E M M A 1 . 4 . 9 @
P r o o f : F o r V ~ G
( q~ o - r y ) ( g H ) = ~0 ( "vgH) =
t he e q u i v a r i a n c e of
i s e q u i v a r i a n t and i n j e c t i v e .
one has(T~o~)(gH) =v v(gx o) = Vgx o and
�9 ~ = ~D o 0 " i . e . Vgx o , and t h e r e f o r e v V V '
~ . T o s h o w t h e i n j e c t i v i t y , c o n s i d e r g l ' gz ~ G
- Z 6 -
s u c h t h a t q~ (g l H) = O ( g z H ) .
-1 a n d t h e r e f o r e gz gl E; H . B u t gl
I f X i s h o m o g e n e o u s , t h e n
We h a v e p r o v e d
This means glXo = gZXo - i
o r gz glXo = Xo
E; gz H implies gl H = gz H , q . e . d .
Q ( x o) = X a n d cp i s a n e q u i v a l e n c e .
P R O P O S I T I O N 1 . 4 . 1 0 L e t X be a h o m o g e n e o u s G - s e t a n d s e l e c t
x 0
on G / H i n d u c e d by t h e l e f t - t r a n s l a t i o n of G . T h e n t h e m a p @ : G / H
d e f i n e d by ~ ( g H ) = g x o i s a n e q u i v a l e n c e of G - s e t s .
T h e g r o u p H d e p e n d s on t h e c h o i c e of x o 6 X , b u t t h e c o n j u g a c y
c l a s s of H i s w e l l - d e f i n e d by t h e o p e r a t i o n i n v i e w of t h e t r a n s i t i v i t y .
6 X . Le___tt H be t h e i s o t r o p y g r o u p of x o a n d c o n s i d e r t h e G - o p e r a t i o n
> X
We c o n c l u d e t h i s c h a p t e r b y s o m e r e m a r k s on e f f e c t i v e a n d t r a n s i t i v e
o p e r a t i o n s on s e t s . I n v i e w of t h e p r e c e d i n g p r o p o s i t i o n , w e c a n c o n s i d e r
w i t h o u t l o s s of g e n e r a l i t y G - s e t s of t h e t y p e G / H , w h e r e H i s a s u b g r o u p
of G . T h e k e r n e l K of t h e h o m o m o r p h i s m d e f i n i n g t h e o p e r a t i o n of G
on G / H i s t h e i n t e r s e c t i o n of t h e i s o t r o p y g r o u p s , t h e r e f o r e
K = N gHg I .
g C G
K is an invariant subgroup of G
an invariant subgroup of G with
= i l i gH g l ' H f o r s o m e ~ L i n v i e w of
s i g n i f i e s L c K . T h e r e f o r e w e h a v e
c o n t a i n e d in H . C o n v e r s e l y , i f L i s
L c H , t h e n L c K , b e c a u s e
L g = g L a n d l g H = g H w h i c h
-27 -
PROPOSITION I. 4. Ii Let G be a sroup, H a subgroup and_consider
the G-operation on G/H induced by the left-translations of G. The kernel
K of the homomorphism ~: G > Bij {G/H) definin~ this operation is
the g r e a t e s t i n v a r i a n t s u b g r o u p of G c o n t a i n e d in H and c a n be d e s c r i b e d
a s
-I K = fl gHg
g~G
C O R O L L A R Y 1.4. 12 G o p e r a t e s e f f e c t i v e l y on G / H if and only if
H c o n t a i n s no i n v a r i a n t s u b g r o u p of G d i f f e r e n t f r o m { e } .
Exercise I. 4.13 Study the effect of the choice of the point x o G X
in proposition I. 4. i0.
- 2 8 -
C h a p t e r Z. G - S P A C E S
Z. 1 D e f i n i t i o n and e x a m p l e s .
D E F I N I T I O N Z, 1.1 A t o p o l o g i c a l g r o u p G is a g r o u p w h i c h is a
t o p o l o g i c a l s p a c e s u c h t h a t t he m a p s
G x G > G
( g l ' gz ) ~ g lg2
, G > G
-1
a r e c o n t i n u o u s .
D E F I N I T I O N Z. 1. Z L e t G b e a t o p o l o g i c a l g r o u p . A G - s p a c e X
is a t o p o l o g i c a l s p a c e w h i c h i s a G - s e t w i t h r e s p e c t to a m a p G x X > X .
M o r e o v e r t h i s m a p i s s u p p o s e d to be c o n t i n u o u s . T h e p a i r (G,X) i s a l s o
c a l l e d a t o p o l o g i c a l t r a n s f o r m a t i o n g r o u p .
I t i s c l e a r t h a t t he g r o u p G is a c t i n g by h o m e o m o r p h i s m s on X ,
so t h a t X i s a G - o b j e c t in t he c a t e g o r y of t o p o l o g i c a l s p a c e s . We
r e q u i r e , m o r e o v e r , the c o n t i n u i t y of t h e m a p G x X > X .
L e t G a n d G' be t o p o l o g i c a l g r o u p s .
D E F I N I T I O N Z. 1. 3 A h o r n o m o r p h i s m 9 : G > G w of t o p o l o g i c a l
g r o u p s i s a h o m o m o r p h i s m of g r o u p s , w h i c h i s c o n t i n u o u s .
L e t X be a G - s p a c e , X' G' a a - s p a c e a n d P: G > G '
h o m o m o r p h i s m .
-Z9 -
DEFINITION Z. i. 4 A 9-equivariant map ~: X > X' is a
p -equivariant map in the sense of definition I. 2.1 which is continuous.
The map r makes the following diagram commutative
G x X > X
[ [ px ] $
X' X' G' x >
~0 i s c o n t i n u o u s and t h e r e f o r e a l s o P
the u n i v e r s a l p r o p e r t y of t he p r o d u c t t o p o l o g y .
An e q u i v a l e n c e of G - s p a c e s X, X t is an e q u i v a l e n c e
of G - s e t s w h i c h i s a h o m e o m o r p h i s m .
E x a m p l e Z. 1 .5 L e t G be a t o p o l o g i c a l g r o u p .
x @, as follows immediately by
~:X >X'
T h e o p e r a t i o n of
G on G by l e f t o r r i g h t - t r a n s l a t i o n s m a k e s t he s p a c e G a G - s p a c e .
T h e o p e r a t i o n of G on G by i n n e r a u t o m o r p h i s m s a l s o m a k e s G a
G - s p a c e .
R e m a r k . L e t X be a t o p o l o g i c a l s p a c e and G the g r o u p of
h o m e o m o r p h i s m s of X . T h e d i s c r e t e t o p o l o g y on G c e r t a i n l y m a k e s
X a G - s p a c e .
L e t X be a c o m p a c t G - s p a c e . C o n s i d e r t h e g r o u p Aut X
of h o m e o m o r p h i s m s w i t h t he c o m p a c t - o p e n t o p o l o g y . I t c a n be p r o v e d
t h a t Au t X i s a t o p o l o g i c a l g r o u p , a n d t h a t t h e m a p G x X > X is
continuous .
- 3 0 -
2. Z Orbitspace.
Let G be a topological group and X a G-space. Consider the set
of orbits X/G and the canonical map w : X > X/G. The quotient x
topology on X/G is the strongest topology on X/G making w x continuous.
T h e o p e n s e t s of X / G a r e t h e s e t s h a v i n g a n o p e n s a t u r a t i o n in X .
D E F I N I T I O N Z. Z. 1 T h e o r b i t s p a c e X / G of t h e G - s p a c e X i s
t h e s e t of o r b i t s w i t h t h e q u o t i e n t t o p o l o g y .
P R O P O S I T I O N Z. Z. Z w : X : > X / G i s a n o p e n m a p . T h e X ....
t o p o l o g y on X / G i s c h a r a c t e r i z e d a s b e i n 8 t h e u n i q u e t o p o l o g y m a k i n g
t h e m a p w c o n t i n u o u s a n d o p e n . X
P r o o f : L e t M C X be o p e n . 7g (M) i s o p e n a n d t h e r e f o r e a l s o
(~rxl o Wx)(lVi ) b e i n g t h e u n i o n of a l l s e t s T ( M ) . B u t t h i s ~(M) ' g
m e a n s t h a t ~rx(M) i s o p e n by d e f i n i t i o n of t h e q u o t i e n t t o p o l o g y . T o
p r o v e t h e s e c o n d s t a t e m e n t , c o n s i d e r m o r e g e n e r a l l y a m a p ~ : X > Y
f r o m X to a s e t Y . T w o t o p o l o g i e s on Y m a k i n g b o t h ~ c o n t i n u o u s
a n d o p e n n e c e s s a r i l y c o i n c i d e . B e c a u s e i f O i s a n o p e n s e t of Y in t h e
i s o p e n i n X a n d ~ ( ~ - 1 ( O ) ) = O i s a l s o o p e n i n one t o p o l o g y , ~ - 1 ( 0 )
t h e o t h e r t o p o l o g y .
Example Z. Z. 3 L e t G be a t o p o l o g i c a l g r o u p a n d H a s u b g r o u p
of G w i t h t h e r e l a t i v e t o p o l o g y . T h e o p e r a t i o n of H on G b y r i g h t
translations makes G an H-space. The canonical map w G : G > G/H
onto the orbitspace is continuous and open.
- 3 1 -
T h e q u o t i e n t t o p o l o g y on X / G c a n a l s o b e c h a r a c t e r i z e d b y t h e
f o l l o w i n g p r o p e r t y . L e t R b e a n a r b i t r a r y t o p o l o g i c a l s p a c e . T h e
m a p x~--------> ~ o lr , s e n d i n g c o n t i n u o u s m a p s x~ : X / G - - > R i n t o
c o n t i n u o u s m a p s X > R i s i n j e c t i v e .
t h e r e f o r e n o w b e c o m p l e t e d b y
P R O P O S I T I O N Z. Z. 4 L e t G
~r : X x
s p a c e .
from X/G t o R
bijective.
PROPOSITION Z. Z. 5
a homomorphism and X, X'
T h e p r o p o s i t i o n 1. 3. 4 c a n
b e a t o p o l o g i c a l ~ r o u p , X a G - s p a c e ,
> X / G t h e c a n o n i c a l m a p o n t o t h e o r b i t s p a c e a n d R a n a r b i t r a r y
T h e c o r r e s p o n d e n c e $ ~ qt o ~ , s e n d i n 8 c o n t i n u o u s m a p s
o n t o i n v a r i a n t c o n t i n u o u s m a p s f r o m X t o R i s
map 4: X > X' induces one and only one continuous map
> X'/G' such that the loll owing diagram commutes
Let G, G' be topological groups, p: G-->G'
respectively G, G'-spaces. A p-equ/variant
~: X/G
X 0 > X'
I i ~x [ 7fX! I
X/G - ~ > X'/G'
Proof: There is only t o show t h e continuity of ~ . But t h i s is a
consequence of the continuity of ~r , o (~ in view of proposition Z. Z. 4. x
Exercise 2.2.6 Consider a G-space X with a transitive operation
of G on X S e l e c t x 6 X a n d l e t H b e t h e i s o t r o p y g r o u p of x �9 0 0
-32-
D e f i n e , as in p r o p o s i t i o n 1 .4 . 10, a m a p ~ : G / H > X . T h i s m a p i s an
e q u i v a l e n c e of G - s e t s and c o n t i n u o u s , bu t no t n e c e s s a r i l y a h o m e o m o r p h i s m .
T h e f o l l o w i n g c o u n t e r - e x a m p l e i s t a k e n f r o m B o u r b a k i . L e t ]R o p e r a t e on
TZ = IR~-/~Z by 7x(x I, x 2) = (x I + a(X), x 2 + a(8•)) where X C IR,
a:]R > ]R/Z the canonical homomorphism and Z
(x 1, x 2) ~ 3" ,
i r r a t i o n a l n u m b e r . F i x i n g (x 1, xz) 6 T z we d e f i n e
1 e m i ~ a 1 .4 . 9, o b t a i n i n g a c o n t i n u o u s i n j e c t i o n .
X = ~ (x 1, x Z) w i t h t h e r e l a t i v e t o p o l o g y . ~ : ]R
b i j e c t i o n , bu t n o t a h o m e o m o r p h i s m . B e c a u s e X i s d e n s e in
c a n n o t be h o m e o m o r p h i c to t he c o m p l e t e s p a c e ]R.
0 an
~:]R > T z as in
C o n s i d e r t he i m a g e
> X is a c o n t i n u o u s
T Z and
E x e r c i s e Z. Z. 7 L e t G be a t o p o l o g i c a l g r o u p a n d H an o p e n s u b -
g r o u p . T h e n H is c l o s e d in G . ( C o n s i d e r t h e p a r t i t i o n of G d e f i n e d
by t h e e l e m e n t s of G / H . )
E x e r c i s e Z. Z. 8 L e t G be a c o n n e c t e d t o p o l o g i c a l g r o u p a n d U a
-1 n e i g h b o r h o o d of e . T h e n e i g h b o r h o o d V = U N U h a s t he p r o p e r t i e s :
Vc U, V "I V n = V. Consider the sets ={gl .... gn/gi CV, i = l,...,n}.
The union V ~176 = UV n is a group, the group generated by V. e is an
inner point of V O~ as e ~ V cV c~ Any point of V ~176 is therefore an
inner point, the left-translations being homeomorphisms leaving V ~176
invariant. V ~176 is an open subgroup of G and therefore closed. As G
is connected, this shows V ~176 = G. This proves that G is generated by
an arbitrary neighborhood U of e.
- 3 3 -
E x e r c i s e 2 . 2 . 9 Let G be a topological group and G the connected o
component of the neutral element e C G, the identity component of G.
Show that G o is a closed invariant subgroup of G
- 3 4 -
C h a p t e r 3. G - M A N I F O L D S
This chapter introduces the fundamental notions of these lectures.
In the following chapters, we proceed to a detailed study of G-manifolds
and Lie groups.
3. i Definition and examples of Lie ~roups.
Manifold will mean a Hausdorff, but not necessarily connected
manifold.
DEFINITION 3. I. I A Li e group is a group G which is an analytic
manifold such that the maps
G x G > G G > G
(gl ' gz ) ~ glgz g ~ g-1
are analytic.
C ~ Differentiable shall always mean . If one replaces analycity
by differentiability in the definition above, it doesn't change anything;
i. e. , analycity is then automatically satisfied (Pontrjagin, [14] ,
p. 191). For a great part of the theory, we shall only make explicit use
of d i f f e r e n t i a b i l i t y .
In the d e f i n i t i o n a b o v e ,
manifold.
analytic manifold means real analytic
Replacing it by complex analytic manifold, one obtains the
notion of a complex Lie group.
- 3 5 -
T w o a r b i t r a r y c o n n e c t e d n e s s c o m p o n e n t s G 1, 0 2 of a L i e g r o u p O
a r e a n a l y t i c a l l y d i f f e o m o r p h i c . F o r gl ~ O l ' g2 6 O 2 t h e m a p
~-~-~> g 2 g l l g i s a n e x a m p l e of s u c h a d i f f e o m o r p h i s m . A l l t h e g
c o n n e c t e d n e s s c o m p o n e n t s t h e r e f o r e h a v e t h e s a m e d i m e n s i o n a n d i t
m a k e s s e n s e t o s p e a k of t h e d i m e n s i o n of a L i e g r o u p .
E x a m p l e 3. I. 2 T h e a d d i t i v e g r o u p ]R n o r (~ n . 3rn ] R n / Z n , = ;
G L ( n , IR ) - t h e g r o u p of q u a d r a t i c m a t r i c e s w i t h n r o w s a n d d e t e r m i n a n t
d i f f e r e n t f r o m z e r o .
E x a m p l e B. 1. 3 L e t G b e a L i e g r o u p a n d T G t h e t a n g e n t b u n d l e .
T h e n T G i s a L i e g r o u p . T h i s f o l l o w s f r o m t h e f a c t t h a t T i s a f u n c t o r
c o n s e r v i n g d i r e c t p r o d u c t s .
E x a m p l e S. 1 . 4 L e t G 1 a n d G 2 b e L i e g r o u p s . T h e n t h e d i r e c t
p r o d u c t G 1 x G 2 i s a L i e g r o u p .
D E F I N I T I O N 3. 1 . 5 L e t G a n d G ' b e L i e g r o u p s . A h o r n o m o r p h i s m
p : G > G of Lie groups is a homomorphism of groups which is analytic.
Remark. It is to be noted that in the literature the term homomorphism
is often reserved for analytic homomorphisms of groups such that the mEp
p: G > 9 (G) is open.
Example 3. i. 6 Let V be an n-dimensional vector space over JR.
T h e c h o i c e of a b a s e e 1 . . . . , e n of V d e f i n e s a n i s o m o r p h i s m
G L ( V ) > G L ( n , IR) of g r o u p s , p e r m i t t i n g u s t o d e f i n e a L i e g r o u p
s t r u c t u r e on t h e g r o u p of l i n e a r a u t o m o r p h i s m s G L ( V ) of V . T h i s
- 3 6 -
s t r u c t u r e i s i n d e p e n d e n t of the c h o i c e of the b a s e . B e c a u s e t w o c h o i c e s
of t he b a s e of V c o r r e s p o n d t o t w o i s o m o r p h i s m s G L ( V ) > G L ( n , JR)
w h i c h d i f f e r b y an i n n e r a u t o m o r p h i s m of G L ( n , ]1%).
E x a m p l e 3. 1. 7 L e t G b e a L i e g r o u p and TG the t a n g e n t b u n d l e
w i t h i t s L i e g r o u p s t r u c t u r e ( e x a m p l e 3. 1. 3). C o n s i d e r t he t a n g e n t s p a c e
G e of G a t t he i d e n t i t y e of G and i t s n a t u r a l i n j e c t i o n j : G > T G . e
If G e i s e q u i p p e d w i t h the L i e g r o u p s t r u c t u r e d e f i n e d b y a d d i t i o n , j i s
a h o m o m o r p h i s m of L i e g r o u p s . T h e n a t u r a l p r o j e c t i o n p : T G > G ,
a s s i g n i n g to e a c h t a n g e n t v e c t o r i t s o r i g i n , i s a l s o a h o m o m o r p h i s m of
L i e g r o u p s . T h e s e q u e n c e
P O >G ~ > TG >G > e
e
i s e x a c t . M o r e o v e r , t h e r e e x i s t s a s p l i t t i n g , t he n a t u r a l i n j e c t i o n
s : G > T G , s a t i s f y i n g p o s = 1G.
E x e r c i s e 3 . 1 . 8 L e t G be a l o c a l l y E u c l i d e a n t o p o l o g i c a l g r o u p ,
i. e. , h a v i n g a n e i g h b o r h o o d of the i d e n t i t y e h o m e o m o r p h i c to an o p e n
s u b s e t of an E u c l i d e a n s p a c e . T h e i d e n t i t y c o m p o n e n t G O of G h a s a
c o u n t a b l e b a s e . T h e r e f o r e G i s p a r a c o m p a c t .
Exercise 3.1.9
Exercise 3.1. I0
open subgroup of G.
A L i e g r o u p i s l o c a l l y c o n n e c t e d .
T h e i d e n t i t y c o m p o n e n t G o of a L i e g r o u p i s an
- 3 7 -
3 . 2 D e f i n i t i o n a n d e x a m p l e s of G - m a n i f o l d s .
D E F I N I T I O N 3. 2 . 1 L e t G b e a L i e g r o u p . A G - m a n i f o l d X i s a
d i f f e r e n t i a b l e m a n i f o l d X w h i c h i s a G - s e t w i t h r e s p e c t t o a m a p
G x X > X . M o r e o v e r t h i s m a p i s s u p p o s e d t o b e d i f f e r e n t i a b l e . T h e
p a i r (G , X) i s a l s o c a l l e d a L i e t r a n s f o r m a t i o n g r o u p .
T h e g r o u p G i s a c t i n g b y d i f f e o m o r p h i s m s on X , s o t h a t X i s a
G - o b j e c t i n t h e c a t e g o r y of d i f f e r e n t i a b l e m a n i f o l d s . M o r e o v e r t h e
d i f f e r e n t i a b i l i t y of t h e m a p G x X
!
L e t X b e a G - m a n i f o l d , X a
h o m o m o r p h i s m of L i e g r o u p s .
> X i s r e q u i r e d .
v GI G -manifold, and p : G > a
D E F I N I T I O N 3. 2. ~ A 9 - e q u / v a r i a n t m a p ~ : X > X ~ i s a
9 - e q u i v a r i a n t m a p i n t h e s e n s e of d e f i n i t i o n 1. Z. 1 w h i c h i s d i f f e r e n t i a b l e .
E x a m p l e B. 2. 3 I R - m a n i f o l d s a r e of f u n d a m e n t a l i m p o r t a n c e f o r
t h e t h e o r y of G - m a n i f o l d s . T h e y h a v e r e c e i v e d a s p e c i a l n a m e : o n e -
p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s . W e s h a l l t a k e u p t h e s t u d y of
o n e - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s in c h a p t e r 5.
E x a m p l e 3 . 2 . 4 T h e o p e r a t i o n of a L i e g r o u p G on t h e u n d e r l y i n g
m a n i f o l d b y l e f t - t r a n s l a t i o n s d e f i n e s G a s a G - m a n i f o l d . T h e o p e r a t i o n
of G on i t s e l f b y i n n e r a u t o m o r p h i s m s a l s o d e f i n e s G a s a G - m a n i f o l d .
E x a m p l e 3. Z. 5 L e t V b e a f i n i t e - d i m e n s i o n a l ] R - v e c t o r s p a c e .
GL(V) is then a Lie group. Let G be a Lie group and 7: G >GL(V)
a homomorphism. We call 7 a representation of the Lie group G in
V,
-38-
Remark.
compact G-space X the continuity of the map G x X
expressed by the continuity of the homomorphism G
As observed at the end of section 2. I, for a locally
>X canbe
> Aut X defining
the operation, if Aut X is equipped with the compact-open topology.
One would like to describe similarly the differentiability of the map
G x X--> X for a G-space X. But for this the group Aut X of diffeo-
morphisms of X should first be turned into a manifold (modeled over
a suffi'ciently general topological vectorspace), which presents serious
difficulties. Nevertheless we shall use this viewpoint for heuristical
remarks.
Example 3.2.6 Let X be a G-manifold and T the functor assigning
to each differentiable manifold its tangent bundle. Then TX is a
TG-manifold, because T conserves direct products. G being a subgroup
of TG (example 3. i. 7), TX is also a G-manifold. This justifies many
classical notations in the theory of transformation groups, which at
first sight seem abusively short.
Example 3. Z. 7 Let G and G' be Lie groups and G' a G-manifold
w i t h r e s p e c t t o an o p e r a t i o n T : G > Au t G ' . T h e n t h e s e m i - d i r e c t
p r o d u c t G" xwG d e f i n e d in e x a m p l e 1 .1 .7 i s a L i e g r o u p w i t h t he a n a l y t i c
s t r u c t u r e of t he p r o d u c t - m a n i f o l d . T h i s g e n e r a l i z e s e x a m p l e 3. 1.4~
w h i c h c o r r e s p o n d s to t he t r i v i a l o p e r a t i o n of G on G ' .
L e t V be a f i n i t e d i m e n s i o n a l ] R - v e c t o r s p a c e . T h e g r o u p of a f f i n e
m o t i o n s of V , w h i c h i s t h e s e m i - d i r e c t p r o d u c t V x G L ( V ) w i t h r e s p e c t
- 3 9 -
to t h e n a t u r a l o p e r a t i o n of GL(V)
E x a m p l e 3. 2 . 8
s e q u e n c e
on V, i s a L i e g r o u p by t h e p r e c e d i n g .
L e t G be a L i e g r o u p a n d c o n s i d e r t he e x a c t
O > G e
J >TG P > G >e
of e x a m p l e 3. 1. 7.
i n j e c t i o n of G
G e d e f i n e d by
T h e s p l i t t i n g s : G > TG d e f i n e d by the n a t u r a l
g i v e s r i s e to an o p e r a t i o n of G on t h e a d d i t i v e g r o u p
Wg = g s ( g ) / G e ( e x a m p l e 1 .1 .7 ) . T h i s r e p r e s e n t a t i o n
of G in G e p l a y s an i m p o r t a n t r o l e in t he t h e o r y of L i e g r o u p s ( a d j o i n t
r e p r e s e n t a t i o n ) . , TG is i s o m o r p h i c to t he s e m i - d i r e c t p r o d u c t C, e xTG
w i t h r e s p e c t to t h i s o p e r a t i o n r .
-40 -
C h a p t e r 4. VECTORFIEI.nS
In this chapter we begin with the detailed theory of G-manifolds and
Lie groups. The Lie algebra of a Lie group is defined and the formal
properties of this correspondence are studied.
4. i. Realfunctions.
The adjective "differentiable" shall be omitted from now on, it being
understood that all manifolds and maps are differentiable.
Let X be a manifold and denote by CX the set of real-valued functions
on X. CX is a commutative ring with identity, the operations on functions
being defined pointwise. It can also be considered as an algebra over the
reals ]R, identifying the set of constant functions on X -with IR.
Let X' be another manifold. A map ~0 : X > X' induces a map
g)*: CX' > CX defined by g)*(f')= f' o ~ for f' 6 CX'. ~0" is a ring
homomorphism respecting identities.
structure on CX and CX', then ~* is
If we consider the ]R-algebra
a homomorphism of ]R -algebras
r e s p e c t i n g i d e n t i t i e s . T h i s s h o w s t h a t the c o r r e s p o n d e n c e X ~ C X ,
~ ~ * d e f i n e s a c o n t r a v a r i a n t s C : ~ > e f r o m the c a t e g o r y
~/ of m a n i f o l d s to the c a t e g o r y i~ of c o m m u t a t i v e r i n g s w i t h i d e n t i t y ,
r e s p e c t i v e l y c o m m u t a t i v e R - a l g e b r a s w i t h i d e n t i t y .
Now l e t X be a G - m a n i f o l d . A c c o r d i n g to p r o p o s i t i o n 1 .1 .9 and the
r e m a r k 1.1.10, CX is a G O - r i n g , i . e . a r i n g on w h i c h G o p e r a t e s f r o m the
r i g h t . If T : G > Aut X is the g i v e n o p e r a t i o n , "r* : G > Aut CX s h a l l
-41-
denote the induced operation. We r e p e a t the d e f i n i t i o n : I" f = f o T g g
f o r f ~ C X .
Exercise 4.1. i. Let X and X' be manifolds, CX and CX' the corres-
p o n d i n g s e t s of r e a l - v a l u e d f u n c t i o n s . Show t h a t an a r b i t r a r y r i n g h o m o -
morphism CX' > CX is a homomorphism of ]R-algebras.
E x e r c i s e 4.1. Z. L e t the s i t u a t i o n be a s in e x e r c i s e 4. 1.1 a n d
q~i:X > X (i = 1, 2) be m a p s s u c h t h a t ~1 = ~ Z " Show t h a t t h e n ~1 = Og"
Exercise 4. i. 3. Let the situation be as in exercise 4. I. i and consider
t he m a p
f r o m m a p s X
@ ~ - . > ~
[x, x'] > [cx' , cx]
> X' to r i n g h o m o m o r p h i s m s CX' > CX d e f i n e d by
E x e r c i s e 4. 1. g s h o w s t h a t t h i s m a p is i n j e c t i v e . Show t h a t
for paracompact manifolds X, X' this map is bijective. (Hint: Try to
imitate the theory of duality for A-modules over a ring A , considering
CX as the dual space of X. The study of the bidual space will then give the
desired result. ) This result should allow on principle a cornplete algebrai-
sation of the theory of differentiable manifolds.
Exercise 4. i. 4. A manifold X is connected if and only if the ring CX
is not decomposable in a direct product of non-trivial rings.
- 4 Z -
4. 2. O p e r a t o r s and v e c t o r f i e l d s .
L e t X be a m a n i f o l d and CX the s e t of r e a l - v a l u e d f u n c t i o n s , c o n -
s i d e r e d as an ]R - v e c t o r s p a c e .
D E F I N I T I O N 4. Z. 1. An o p e r a t o r A
A : CX > C X .
on X is an ]R-linear map
Example 4. 2.2. An automorphism of CX is an operator. A vector-
f i e l d on X is an o p e r a t o r . M o r e g e n e r a l l y , a d i f f e r e n t i a l o p e r a t o r on X
is an operator.
L e t OX d e n o t e t he ] R - a l g e b r a of o p e r a t o r s on X . If X' is a n o t h e r
manifold and ~ : X > X' a diffeomorphism, then ~ induces an isomor-
p h i s m ~ : OX > OX' by the d e f i n i t i o n ~ A = @~-1 o A o ~ . T h i s
d e f i n i t i o n m e a n s t h a t t he f o l l o w i n g d i a g r a m c o m m u t e s
CX ~= ~ CX'
A I I r , I
CX < ~0 j CX'
It i s c l e a r t h a t t h e c o r r e s p o n d e n c e X--N--> O X , ~--,----> ~Oa d e f i n e s a c o v a r i a n t
f u n c t o r 0 : ~ i s o ~ > ~ i s o f r o m the c a t e g o r y of m a n i f o l d s a n d d i f f e o m o r -
p h i s m s to t h e c a t e g o r y of ] R - a l g e b r a s and a l g e b r a i s o r n o r p h i s m s .
N o w l e t X be a G - m a n i f o l d w i t h r e s p e c t to a h o m o m o r p h i s m
v : G > Aut X. Then according to proposition I. I. 9, OX is a G-object
in the category of ]R -algebras. Moreover, the invariant elements under
- 4 3 -
t h i s o p e r a t i o n f o r m a n ] R - s u b a l g e b r a of OX , a s f o l l o w s i m m e d i a t e l y .
L e t u s c o n s i d e r a n a r b i t r a r y a s s o c i a t i v e A - a l g e b r a O o v e r a r i n g 1%
w i t h i d e n t i t y . T h e n o n e c a n d e f i n e a n e w m u l t i p l i c a t i o n [ , ] : O x O > O
i n t h e f o l l o w i n g w a y :
[A I, A2] = AIA 2 - AzA I for A I, A z g O
This multiplication is bilinear and satisfies
I) [A , A ] : O for A g O
Z) [A I, [A Z, A3] ] + [Az,[A 3, AI] ] + [A 3. [A l, Az]] = o
for AI, A z, A 3 g O (Jacobian identity)
t u r n i n g t h e r e f o r e O i n t o a L i e - a l g e b r a a c c o r d i n g t o
D E F I N I T I O N 4. 2. 3. A / % - m o d u l e O o v e r a r i n g 1% w i t h a b i l i n e a r
map [ , ] : O x O----> O satisfying [A, A] = O for A e O and the Jacobian
i d e n t i t y i s a L i e z a ! g e b r a o v e r A .
D E F I N I T I O N 4. 2. 31. A h o m 0 m o r p h i s m h : O > O ' of L i e _ a l j e b r a s
O a n d O t o v e r a f i e l d A i s a / % - l i n e a r m a p s a t i s f y i n g
h[A I, AZ] = [hA1, hAg] f o r A1, A g e O .
S t a r t i n g f r o m a n a s s o c i a t i v e / % - a l g e b r a O w e h a v e a s s o c i a t e d t o O
a t% - L i e a l g e b r a . T h i s c o n s t r u c t i o n i s f u n c t D r i a l , i . e . i f h : O > O I
i s a h o m o m o r p h i s m of /% - a l g e b r a , t h e n t% i s a l s o a h o m o m o r p h i s m of t h e
a s s o c i a t e d A - L i e a l g e b r a . A p p l y i n g t h i s t o t h e ] 1 % - a l g e b r a of o p e r a t o r s
on X , w e o b t a i n
-44-
PROPOSITION 4. Z. 4. Let X be a G-manifold and OX the set of
operators on X. T h e definition (Tg) ,~(A) =Tg ~-I
oA o T for A C OX makes g
OK a G-set. This operation conserves the ]R-algebra structure on OX a___s
w e l l a s t h e a s s o c i a t e d s t r u c t u r e of a n ] R - L i e a l g e b r a . In p a r t i c u l a r , t h e
invariant elements under this operation form a ]R-algebra and _a JR-Lie
algebra respectively.
PROPOSITION 4. Z. 5. Let X be a G-manifold, X' G' a -manifold
and ~:X > X' a p-equivariant diffeomorphism with respect to a homo- . . . .
morphism p : G > G'. Then @~ : OX > OX' is a p -equivariance
w i t h r e s p e c t to t h e o p e r a t i o n s d e f i n e d i n p r o p o s i t i o n 4. 2 . 4 . M o r e o v e r ,
X'.
s e n d s G - i n v a r i a n t o p e r a t o r s on X i n t o p ( G ) - i n v a r i a n t o p e r a t o r s on
This follows from remark I. i. I0 and propositions i. Z. 9 and i. g. IZ.
We now apply this to vectorfields. Let X be a manifold and A a
vectorfield on X. Then A is a map A : CX > CX which satisfies
(i) A(f I + fz) = Af I +~ for fl' fz C CX
(ii) A(flf2) = Afl-f z + fI.AIz for fl' f2 ~ CX
(iii) A(?~) = O for X C ]R
T h e r e f o r e A C O X . In f a c t , t h e s e p r o p e r t i e s a r e c h a r a c t e r i s t i c f o r v e c t o r -
f i e l d s . T h e c o m p o s i t i o n of v e c t o r f i e l d s i n OX is n o t a v e c t o r f i e l d , b u t t h e
c o m p o s i t i o n of v e c t o r f i e l d s w i t h r e s p e c t t o t h e a s s o c i a t e d ] R - L i e a l g e b r a
s t r u c t u r e [ , ] : OX x OX ~ OX g i v e s a v e c t o r f i e l d . H e r e ( i i ) i s e s s e n t i a l .
-45-
T h u s t h e v e c t o r f i e l d s f o r m a s u b a l g e b r a of t h i s ] R - L i e a l g e b r a . L e t DX
d e n o t e t h e J R - L i e a l g e b r a of a l l v e c t o r f i e l d s on X .
If X and X' are manifolds and ~: X > X a diffeomorphism, then
the isomorphism @# : OX > OX' defined at the beginning of this section
certainly sends DX into DX' . Applying proposition 4. 2.4 we therefore
obtain
COROLLARY 4. Z. 6. Let X be a G-manifold and DX the ]R-Lie
,-i 7 g ) , ( f o r algebra of vectorfields on X. The definition ( A) = 7g oA �9 7g
A ~ DE makes DE a G-Lie algebra with respect to 7: G > Aut DE.
In p a r t i c u l a r , t h e i n v a r i a n t e l e m e n t s of DX u n d e r t h i s o p e r a t i o n f o r m a
] R - L i e a l g e b r a .
A n d p r o p o s i t i o n 4. Z. 5 g i v e s
COROLLARY 4. Z. 7. L e t X b e a G - m a n i f o l d , X I ! a G -manifold
and ~0 : X !
> X a p - e q u i v a r i a n t d i f f e o m o r p h i s m w i t h r e s p e c t to a h o m o -
morphism p: G > G' Then ~, : DX > DX' �9 is a P-equivariance with
r e s p e c t t o t h e o p e r a t i o n s d e f i n e d i n c o : r o l l a r y 4. Z. 6. M o r e o v e r , g)$ s e n d s
X' G-invariant vectorfields on X into p (G)-invariant vectorfields on .
F o r l a t e r u s e , we m a k e e x p l i c i t t h e e f f e c t of q g , .
L E M M A 4. 2 . 8 . L e t ~0: X !
> X be a diffeomorphism and ~$ : DX
, -1 t h e i n d u c e d i s o m o r p h i s m on v e c t o r f i e l d s , d e f i n e d b y cpsA = r o A o q~
> DX'
�9 L e t
f' = Ax(cp#f' ). I__f cP. : T (X) x6 X and f' C CX'. Then ( ~A)~ (x) x x
! > T (x)(X) denotes, the linear map of tangent spaces induced by r
( ~ , A ) ~ ( x ) = ~ , A . X X
then
-46 -
Proof: ((~,A) f')($(x)) = ~*((r f'))(x) = ((A~*)f')(x) by
definition of ~0,. This means ( ~0,A)~(x)f' = Ax({p*f'). The right side is
exactly the definition of (~x Ax)f' and therefore also (~A)~(x) = ~,xAX.
4. 3. The Lie algebra of a Lie group.
L e t G be a L i e g r o u p � 9
D E F I N I T I O N 4. 3.1. T h e L i e a l s e b r a L G of G i s t h e IR - L i e a l g e b r a
of i n v a r i a n t v e c t o r f i e l d s u n d e r t h e o p e r a t i o n of G on G b y l e f t - t r a n s l a t i o n s .
E x p l i c i t l y s t a t e d , t h i s m e a n s t h a t t% 6 L G i f a n d o n l y i f ( L g ) . A = A
f o r a l l g E G . L G i s a L i e a l g e b r a b y c o r o l l a r y 4. Z . 6 . T h e l e t t e r L s h a l l
r e m i n d u s of l e f t i n v a r i a n t a s w e l l a s t h e f o u n d e r of t h e t h e o r y , S o p h u s L i e .
T h e f o l l o w i n g l e m m a i n s u r e s t h e e x i s t e n c e of m a n y l e f t i n v a r i a n t
vectorfields on a Lie group.
_ t h e L E M M A 4. 3. Z. L e t G be a L i e g r o u p , L G i t s L i e a l g e b r a , G e
t a n g e n t s p a c e of G a t t h e i d e n t i t y e a n d A ~ G e e
a n d on l F one A C L G s u c h t h a t A = A e e
P r o o f : I f A e x i s t s , t h e n ~ = ( L g ) , A f o r g C (3
Ag = ( ( L g ) . A)g . In v i e w of l e m m a 4. Z. 8 t h i s m e a n s
T h e n t h e r e e x i s t s one
a n d i n p a r t i c u l a r
(1) Ag = ( L g ) . e A e "
T h i s n e c e s s a r y c o n d i t i o n f o r X s h o w s t h e u n i q u e n e s s . We n o w d e f i n e A
= 1G b y t h i s f o r m u l a . A s L e , w e c e r t a i n l y h a v e A = A . T h e l e f t e e
invariance of A is seen from
- 4 7 -
((Lg),A)g~ = ( L g ) , A = ( L g ) , ( L ) , e A
= ( L g ) , A = A . e e g'~
T h e r e r e m a i n s to s h o w t h a t t he f a m i l y ( A g ) g ~ G is a v e c t o r f i e l d (i. e.
~ w
a d i f f e r e n t i a b l e v e c t o r f i e l d ) , w h i c h m e a n s t h a t A(CG) c C G . Le t f ~ C G .
By lemma 4.2.8
{ ( L g ) , A ) g f = Ae(L$f)g
and t h e r e f o r e
{Af)(g) = A e ( L g f ) .
Let N: I > G, I an interval of IR o-ontaining
d Nt/t = A Then G wi t h ~-~ = o e
O , a d i f f e r e n t i a b l e c u r v e i n
A e ( L g f) = ~ L f) ( ~ ) t = o = ~'[ f( g ~t) t - 0
w h i c h s h o w s ~ f 6 C G . |
T h e c o r r e s p o n d e n c e A "~> A e
of t he l e m m a d e f i n e s a b i j e c t i v e m a p
~ : G > L G e
w h i c h i s s e e n to be an i s o m o r p h i s m of I R - v e c t o r s p a c e s by (1). We h a v e
p r o v e d
- 4 8 -
G e
T H E O R E M 4. 3. 3. L e t G be a L i e g r o u p , L G i t s L i e a l g e b r a a n d
t h e t a n g e n t s p a c e of G a t t h e i d e n t i t y e . T h e f o r m u l a
( ~ ( A e ) ) g = ( L g ) . e A e f o r g C G , A e C G e
defines a map ~: G ----> 113, which is an isomorphism of ]R-vectorspaces. e
T h i s m a p ~ a l l o w s t r a n s p o r t i n g t h e s t r u c t u r e of ] R - L i e a l g e b r a f r o m
L G to G e
In t h i s s e n s e , G i s o f t e n r e f e r r e d to a s t h e L i e a l g e b r a of G. e
C O R O L L A R Y 4. 3 . 4 . L e t G be a L i e g r o u p of d i m e n s i o n n . T h e
L i e a l g e b r a L G is a L i e a l g e b r a of d i m e n s i o n n .
C o n s i d e r t h e m a p ( L g ) . : G > G , w h i c h i s a n i s o m o r p h i s m f o r e e g
all g6; G. More generally, the maps P(gl' gz ) = (Lgz)*e(Lgl):1~e : Gg I >Ggz
h a v e t h e p r o p e r t i e s :
1) P ( g z ' g3 ) p ( g l ' gz ) = P ( g l ' g3 ) f o r g l ' gz ' g3 6; G
Z) P(g, g) = IGg for g 6; G
(x) > T (x) DEFINITION 4. 3.5. Let X be a manifold and P(gz' gl ) : T 1 gz
a n J R - l i n e a r m a p f o r a l l ( g l ' gz ) 6; X x X , s a t i s f y i n g 1) a n d Z). T h e n X
i s c a l l e d a p a r a l l e l i z a b l e m a n i f o l d .
The maps P(gz' gl ) are then necessarily isomorphisms and it makes
s e n s e t o s p e a k of t h e d i m e n s i o n of X.
L e t e be a f i x e d p o i n t of X a n d A i (i = 1, �9 �9 �9 , n , n , - d i m X) a b a s e e
of t h e v e c t o r s p a c e T X . T h e n P ( e , g) A. = A. d e f i n e s v e c t o r f i e l d s e 1 e lg
A. ( i = 1, �9 . . , n) on X such tha t thex~ec to rs A. ( i = 1, . - - , n) f o r m a base 1 lg
-49 -
of T X for all g C G. g
COROLLARY 4. 3. 6. The manifold of a Lie group G is parallelizable.
Example 4. 3. 7. Consider IR with its additive Lie group structure.
Then LIR ~ IR as vectorspace, because the tangent space of IR at O is
]R . There is only one possible Lie algebra structure on 11% , defined by
= Ofor c m.
By the same argument, L 11 ~ = IR for the additive group ~Ir= IR/~.
Now let V be n-dimensional IR-vectorspace and G = GL(V). We
first remark that considering GL(V) c s = algebra of ]R-linear
endomorphiams of V, the tangent space G is identified to s (V) for all g
g G G. The multiplication in GL(V) is the restriction of the bilinear
map s (V) x s (V) > s defining the multiplication in s This
shows that (Lg)~ A,~ = gay for g ~ GL(V), Ay C Gy identified to s
Y We s h o w n o w
P R O P O S I T I O N 4. 3 . 8 . A f t e r t h e c a n o n i c a l i d e n t i f i c a t i o n of L { G L ( V ) )
w i t h t h e t a n g e n t s p a c e a t t h e i d e n t i t y , w e h a v e L ( G L ( V ) ) = s (V) a s L i e
a l g e b r a s , w h e r e on s w e c o n s i d e r t h e L i e a l g e b r a s t r u c t u r e a s s o c i a t e d
in t h e s e n s e of s e c t i o n 4. 2 t o t h e n a t u r a l a l g e b r a s t r u c t u r e .
P r o o f : L e t A 1, A 2 C L ( G L ( V } } . W e u s e t h e f o r m u l a
[AI'Az] g = Z -
which is valid for the global chart given by the embedding GL(V)c s
-50 -
In view of A i = gA i g e
which shows
w e h a v e
I~ Aigl(g) Ajg = AjgAig
[A I, A z] g = AlgAZg - AZgAlg.
B u t t h e r i g h t s i d e i s j u s t t h e c o m m u t a t o r [ A l g , A z ] in s F o r g
g = e t h i s g i v e s t h e d e s i r e d r e s u l t .
We have defined the Lie algebra LG of a Lie group G by considera-
tion of the operation of G on G by left translation. Doing the same for
the right translations, we obtain another Lie algebra RG. Explicitly:
RG i s the Lie algebra of the right invariant vectorfields. As in theorem
4. 3. 3, we can define an isomorphism O > RG of ]R-vectorspaces, e
obtaining therefore an isomorphism LG ~ RG of ]R-vectorspaces. We
shall see in section 4. 6 that there is also a natural isomorphism LG ~ RG
of the ]R-Lie algebra structure.
Exercise 4. 3.9. Let G be a Lie group, CG the ]R-vectorspace
of real-valued functions on G , DG the ]R-Lie algebra of all vectorfields
on G and LG the Lie algebra of G. Show that D G = CG | LG.
4. 4. E f f e c t of m a p s on o p e r a t o r s a n d v e c t o r f i e l d s .
I n s e c t i o n 4. 2. w e h a v e s e e n t h e e f f e c t of d i f f e o m o r p h i s m s on o p e r a t o r s .
We w a n t to s t u d y n o w t h e e f f e c t of a r b i t r a r y ( i . e . d i f f e r e n t i a b l e ) m a p s .
-51 -
map
v A ! ! Let X, X be manifolds and A, operators on X, X respectively.
!
DEFINITION 4. 4.1. A and A are ~0-related with respect to a
&0: X > X' , if the following diagram commutes.
C X
t AI
I CX (0"
CX'
CX'
If ~ is a diffeomorphism, A and ~0~A are @-related operators.
A ! But in the general case, A does neither determine an such that A and
A' A' are ~0-related, nor is unique, if it exists.
I
LEMMA 4. 4. g. Let ~0: X > X be a map.
(i) mIf A i and A v.1 (i = i, 2) are ~-related operators on
X and X' respectively, then the following operators
are ~-related:
! I
A I + A 2 and A I + A 2 ,
I !
AIA g and AIA 2 ,
[A I, A2] and [ AVl , A'Z]
Proof:
(ii) If A and A' are ~0-related operators on X and X'
res]~ectively, then for k C IR ~IA and XA' are
~0 -related.
(i) Let f' ~ CX'. Then
, �9 . , _ , ~0~(AV2f ') (D*((A I + A'z)f') = @$(A' I f' + A'zf') = ~ (All) +
= AI(~f') + AZ(~*f') = (A I + AZ)(~f'),
-SZ-
! !
showing that A I + A z and A 1 + A z are W-related. The ~-relatedness
' ' is seen by comparing the diagrams serving to define of AIA g and AIA 2
@-relatedness, and the third assertion is a consequence of this and (ii).
(ii) ~*((kA')f') = (D *(k(A f )) = @ k. ~*(A'f')
= k. A(~0*f') = ( XA)(@*f') , q. e. d.
T h e l e m m a a p p l i e s in p a r t i c u l a r to v e c t o r f i e l d s . F o r t h a t we m a k e
explicit the notion of (~-relatedness in
PROPOSITION 4.4. 3. Let X, X' be manifolds, @ : X > X' a
map and A, A' vectorfields on X, X' respectively. Then A and A'
~ x A x A~( f o r e v e r y x g X . a r e ~ - r e l a t e d if and o n l y if = x)
Proof: Let f' g CX' Then
(4. Ax)f' = Ax(~f') = (A(~f'))(x) x
by d e f i n i t i o n of ~)~ . On the o t h e r h a n d x
A ! @(x)f' = (A'f')(@(x)) = (@*(A'f'))(x)
C o m p a r i s o n p r o v e s the l e m m a .
4. 5. T h e f u n c t o r L.
We h a v e d e f i n e d the L i e a l g e b r a LG f o r a n y L i e g r o u p G. We w a n t
to e x t e n d t h i s c o r r e s p o n d e n c e to a f u n c t o r f r o m L i e g r o u p s to L ie a l g e b r a s .
L E M M A 4 . 5 . i. L e t G G' be L i e g r o u p s P : G > G ' , , a h o m o -
m o r p h i s m of L i e g r o u p s and A ~ L G . T h e n t h e r e e x i s t s one and o n l y
A' a r e O - r e l a t e d . one C LG' s u c h t h a t A and A'
-53-
Proof: Suppose A'
sition 4. 4. 3 we obtain
(i)
e x i s t s w i t h t he d e s i r e d p r o p e r t i e s . By p r o p o -
w h e r e e , e is t he i d e n t i t y of G, G'
is on ly one A' 6 IX]' s u c h t h a t ~', e
N o w w e d e f i n e c o n v e r s e l y A' a s t he u n i q u e e l e m e n t of LG '
A' ,eAe e ) = p
respectively. By lemma 4. 3. Z there
A' = , . This proves uniqueness. e
s ati s lying
!
p , g A g = 9 , g ( L g ) , e A e = ( L g ( g ) ) * e ' 9 * e Ae = A p ( g )
w h i c h i s the d e s i r e d r e s u l t in v i e w of p r o p o s i t i o n 4. 4. 3.
In t h e p r o o f of t he l e m m a 4. 5 .1 , w e u s e d on ly t he r e s t r i c t i o n of
to a neighborhood of e C G.
notion.
DEFINITION 4. 5. Z.
neighborhood of e ~ G.
T U is a differerLtiable m a p
It i s u s e f u l to i n t r o d u c e a c o r r e s p o n d i n g
L e t G , G' be L i e g r o u p s and U
A l o c a l h o m o m o r p h i s m G > G'
p : U > G' which satisfies
an open
defined on
p(g lgz ) = p ( g l ) p ( g z ) f o r a l l g l ' gz C U s u c h t h a t g lg z e U .
T h e r e s t r i c t i o n of a h o m o m o r p h i s m p : G > G' to an o p e n
n e i g h b o r h o o d of e 6 G i s a l o c a l h o m o m o r p h i s m G ~ > G ' . If we i d e n t i f y
a m a p w i t h i t s r e s t r i c t i o n s to o p e n s u b s e t s of t h e d o m a i n , we c a n c o m p o s e
l o c a l h o m o m o r p h i s m s , o b t a i n i n g t h u s the c a t e g o r y of L i e g r o u p s a n d
p o Lg - Lp(g) o p i m p l i e s
(1) T h e r e r e m a i n s to s h o w t h a t A a n d A' �9 a r e ~ - r e l a t e d . N o w
- 5 4 -
l o c a l h o m o m o r p h i s m s .
i s o m o r p h i s m . E x p l i c i t l y s t a t e d we h a v e
D E F I N I T I O N 4. 5. 3. T w o L i e g r o u p s , G and G' ,
i s o m o r p h i c if and on ly i f t h e r e e x i s t s o p e n n e i g h b o r h o o d s
l l U I and a d i f f e o m o r p h i s m p: U > U w i t h i n v e r s e p :
b o t h P and P' a r e l o c a l h o m o m o r p h i s m s .
T H E O R E M 4. 5 . 4 . L e t G, G' be L i e g r o u p s ,
h o o d of e in G and P: U ~ G'
A n e q u i v a l e n c e in t h i s c a t e g o r y is c a l l e d a l o c a l
a r e l o c a l l y
U, U t Of e, e'
> U s u c h t h a t
U a n open nei~hbor-
a local homomorphism. The formula
(L(p)A)e, = p~ A e e
defines a homomorphism of Lie algebras
following diagram is commutative,
f o r A 6 LG
L ( p ) : LG > LG ' . T h e
P e t
G > G , e e
I I
LG L(P ) > LG'
w h e r e ~ d e n o t e s the i s o m o r p h i s m of t h e o r e m 4. 3. 3. M o r e o v e r f o r
!
A C LG the v e c t o r f i e l d s A / U and L( p)A ~ LG a r e p - r e l a t e d .
P r o o f : In t he p r o o f of l e m m a 4. 5.1 a l l w a s s h o w n e x c e p t t h e f a c t
t h a t L ( p ) is a h o m o m o r p h i s m . T h i s is a c o n s e q u e n c e of l e m m a 4. 4. Z.
We o b s e r v e t h a t a h o m o m o r p h i s m p : G > G ' d e f i n e s in the
I s a m e w a y a h o m o m o r p h i s m R(p ) : RG > RG o f t h e L ie a l g e b r a s of
right invariant vectorfields.
-55-
Complement to 4. 5.4. If P : G > G' is an isomorphism, then
L(p ) = P~/LG , where p ~ : DG �9 > DG' is the map defined in section 4. Z.
P r o o f : We h a v e to s h o w t h e c o m m u t a t i v i t y of t h e d i a g r a m
LG L(P) ; LG'
N A p~'
DG > DG
L e t A 6 L G . T h e n on one h a n d
(L(p)A)p (g)
and on the other hand
p , A e = p , ( L g ) , e A e = (Lp(g))~e e
(P~A)p(g) : p. Ag g
by lemrna 4. Z. 8. This shows the desired property.
One cannot define U(p) in general by P~, because this map only
makes sense for a diffeomorphism p .
THEOREM 4. 5.5. Let ~ be the cate~or}r of Lie groups and local
homgmorphisms of Lie ~roups, s
homomorphisms of Lie algebras.
t h e c a t e g o r y of ] R - L i e a l g e b r a s a n d
T h e c o r r e s p o n d e n c e G ~ - ~ > L G ,
9 ~ " " > L ( p ) d e f i n e s a c o v a r i a n t f u n c t o r L : s > s
T h i s i s c l e a r by t h e o r e m 4 . 5 . 4 .
We c a n a l s o c o n s i d e r t h e f u n c t o r g i v e n by G " ' " > G , P ~ P e
The commutativity of the diagram in theorem 4. 5.4 expresses that
is a natural transformation of this functor into L, in fact a natural
e
M
e q u i v a l e n c e .
-56-
COROLLARY 4.5.6. The Lie algebras of locally isomorphic
groups are isomorphic.
Proof: L sends equivalences in ~ into equivalences in =s
We apply this to the natural injection G o ~ G of the connected
component of the identity G o into G , which is a local isomorphism.
This shows LG o ~ LG. The Lie algebra is therefore a property of
of an arbitrary neighborhood of the identity.
>It= IR/Z
Therefore LIR ~ L~r , what we already know.
p: "][" > ]R is a homomorphism, then p = 0.
being compact, p (,j[n) is contained in a closed interval
"IF with p(t) # 0. Then there exists a positive integer
G o , in fact,
Example 4. 5. 7. The canonical homomorphism ]R
is a local isomorphism.
LEMIVIA 4.5.8. If
Proof: "Jr
I. Suppose t C
n such that nP(t) ~ l;which is a contradiction.
This proves that there is no homomorphism
the identity isomorphism LT = LIR.
isomorphism has this property.
Example 4.5.9.
and T : G > GL(V)
"ff > IR i n d u c i n g
B u t of c o u r s e t h e n a t u r a l l o c a l
L e t V be a n ] R - v e c t o r s p a c e of f i n i t e d i m e n s i o n ,
a r e p r e s e n t a t i o n of t h e L i e g r o u p G i n V. We
p r o v e d in P r o p o s i t i o n 4. 3. 8 t h a t s (V) i s t h e L i e a l g e b r a of G L ( V ) .
T h e m a p T c a n be s e e n to be d i f f e r e n t i a b l e , a n d i n d u c e s t h e r e f o r e a
homomorphism L(T) : LG > s (V).
- 5 7 -
D E F I N I T I O N 4. 5 . 1 0 .
t h e A - L i e a l g e b r a of A - e n d o m o r p h i s m s of V.
a A - L i e a l g e b r a O i n V i s a h o m o m o r p h i s m
t h e n c a l l e d a n O - m o d u l e w i t h r e s p e c t t o v .
L e t A b e a r i n g , V a A - m o d u l e a n d s
A r e p r e s e n t a t i o n of
: 0 ~> s V is
F o l l o w i n g e x a m p l e 4. 5 . 9 , a r e p r e s e n t a t i o n of a L i e g r o u p G i n
a f i n i t e - d i m e n s i o n a l ] R - v e c t o r s p a c e V d e f i n e s a r e p r e s e n t a t i o n of t h e
L i e a l g e b r a L G in V.
W e n o w c o n s i d e r t h e h o m o m o r p h i s m d e t : G L ( V ) > ]I% $ i n t o t h e
r n u l t i p l i c a t i v e g r o u p ]R $ of t h e r e a l s . T h e L i e a l g e b r a of ]R $ i s JR.
We p r o v e
P R O P O S I T I O N 4. 5 .11. T h e h o m o m o r p h i s m s > IR of L i e
a l g e b r a s i n d u c e d b y t h e h o m o m o r p h i s m G L ( V ) > ]R $ i s t h e t r a c e
m a p .
P r o o f : L e t A ~ s (V) a n d a t a c u r v e in G L ( V ) w i t h a ~ = e ,
= A. Then 0
det, A = d {det at}/t e dt = 0 "
Now for any non-degenerated n-form co on V (n = dim V) and any
n - t u p l e of v e c t o r s Vl , ' ' - , Vn of V w e h a v e
~0(Vl, " ' ' ' V n ) " d e t a t = r t v l , ' ' ' , a t v )n
and therefore
-58-
~(Vl, - . . , Vn) �9 d e t , e d
A = d-t-{U~(0~tVl' " ' " 'atVn)}/t=O
= Z ~ ( a t V l ' " ' ' a t V i - l ' ~ tv i ' a tV i+ l ' ' ' ~ t V n ) f / = O i
= ~, ~(Vl,''',Av i, --., v n)
i
= ~ ( V l , ' ' ' , Vn) - %r A
s h o w i n g e
4 . 5 . 4 .
d e t ~ A = t r A . T h i s i s t h e d e s i r e d r e s u l t i n v i e w o f t h e o r e m
C O R O L L A R Y
P r o o f :
w e h a v e f o r
4. 5.1Z. t r ( A B ) = t r ( B A ) f o r A, B C s .
t r : s > IR b e i n g a h o m o m o r p h i s m of L i e a l g e b r a s ,
A, B C ~(V)
t r ( A B - BA) = t r [ A , B ] = [ t r A, t r B] = O ,
t h e l a t t e r b r a c k e t b e i n g the t r i v i a l one i n 1R .
Example 4.5. 13. Let G be a Lie group and
0 - - - -> G > TG -- - -> G > e e
the sequence of example 3. I. 7. It induces a sequence of Lie algebra
h o m o m o r p h i s m s
O ~ L(Ge) > L ( T G ) ~ LG > O
H e r e LG e - ' G e ( s e e e x a m p l e 4. 6. 4 b e l o w ) . S e e 7 . 5 . 6 f o r t h e e x a c t n e s s
of t h i s s e q u e n c e . T h e i n c l u s i o n G ~ TG i n d u c e s a h o m o m o r p h i s m
- 5 9 -
4. 6. A p p l i c a t i o n s of t he f u n c t o r a l i t y of L.
4. 6 .1 . T h e L i e a l g e b r a of a p r o d u c t g r o u p . L e t G 1, G 2 be Lie
g r o u p s a n d G 1 x G Z t he p r o d u c t g r o u p . T h e c a n o n i c a l p r o j e c t i o n s
Pi : G1 x G 2 > G ( i = 1, 2 ) a r e h o m o m o r p h i s m s of L i e g r o u p s a n d i n d u c e l
L i e a l g e b r a h o m o m o r p h i s m s L(Pi ) : L (G 1 x GZ) > LG.I " L e t e l , e 2 be
t h e i d e n t i t i e s of G 1, G 2 . By t h e o r e m 4 . 5 . 4 , w e h a v e t h e c o m m u t a t i v e
diagram (expressing the naturalit 7 of ~ )
1t
X
G 2
t
LG 2
t h e v e r t i c a l a r r o w s b e i n g i s o m o r p h i s m s . T h e I R - l i n e a r i s o m o r p h i s m
~-- G 1 x (G 1 x G 2 ) e l ' ez e l G 2 e 2
i m p l i e s t h e r e f o r e t h e i s o m o r p h i s m s of I R - v e c t o r s p a c e s
L(G 1 x G2) = LG 1 x LG 2 .
If q i : LGI x L G 2 > LG. (i = I, 2) denotes the canonical projection, 1
this isomorphism is given b y the commutative diagram
-60 -
L(G 1 (%
x G z) > LX31 x LG z
LG 1 LG 2
We w a n t to t r a n s p o r t t he L i e a l g e b r a s t r u c t u r e of
LG 1 x LG z. For A~ L(G 1
with A.I = L(Pi)A (i = I, 2).
a(A') = (A_[, A'Z) with A~: =
a [ A , A ' ] =
x GZ) we h a v e
S i m i l a r l y f o r A' C L(G 1
L (P i )A ' (i = 1, Z). T h e n
(L(PI)[A, A'] , L(pz)[A, A'] )
L(G 1 x G z) to
a(A) = (L(PI)A, L(pz)A) = (A I, A Z)
x G 2) we have
= ( [ A 1, A:] , [ A z , A~] )
as L(p i) a r e L i e a l g e b r a h o m o m o r p h i s m s . We d e f i n e
[ a(A), a (A ' ) ] = a [ A , A ' ]
w h i c h m e a n s
(1) [(A 1, AZ), (A' 1, AZ) ] = ([A 1, A'I], [A Z, A'Z])
f o r A. , A'. ~ LG. 1 1 1
(i = 1, Z) .
Wi th t h i s d e f i n i t i o n , a is an i s o m o r p h i s m .
L e t m o r e g e n e r a l l y A be a r i n g and O 1, O z A - L i e a l g e b r a s .
C o n s i d e r the p r o d u c t m o d u l e O 1 x O Z w i t h the m a p [ , ]~O 1 x Oz) x ( O 1 x O7)
> O 1 x O 2 d e f i n e d by (1). T h e n O 1 x O z i s a A - L i e a l g e b r a .
D E F I N I T I O N 4. 6. Z. T h e d i r e c t p r o d u c t of two A - L i e a l g e b r a s
O1, 0 2 is t he L i e a l g e b r a O 1 x 0 2 w i t h the m u l t i p l i c a t i o n d e f i n e d by (1).
We can now state
- 6 1 -
P R O P O S I T I O N 4. 6. 3.
d i r e c t p r o d u c t . T h e L i e a l g e b r a
t o t h e d i r e c t p r o d u c t L G 1 x L G 2
L e t G 1, G z b e L i e g r o u p s a n d G 1 x G 2 t h e
L ( G 1 x GZ) i s c a n o n i c a l l y i s o m o r p h i c
of t h e L i e a l g e b r a s L G 1 a n d L G 2 .
W e r e m a r k t h a t c o m m u t a t i v i t y f o r t h e m u l t i p l i c a t i o n i n a L i e a l g e b r a
m e a n s t h a t [ A 1, AZ] = 0 f o r a n y p a i r A 1, A z . I t i s t h e n c l e a r t h a t t h e
p r o d u c t of c o m m u t a t i v e L i e a l g e b r a s i s c o m m u t a t i v e .
E x a m p l e 4 . 6 . 4 . L( IR n) = LIR x . . . x L]R. B u t w e h a v e a l r e a d y s e e n
t h a t LIR = IR w i t h t h e t r i v i a l L i e a l g e b r a s t r u c t u r e ( e x a m p l e 4. 3. 7).
T h e r e f o r e L(IR n) = ]R n w i t h t h e t r i v i a l L i e a l g e b r a s t r u c t u r e .
Similarly L(T n) = IR n for the additive group ~.n = ~n/~Tn "
4. 6. 5. The relation between LG and RG. Let G be a Lie group,
G ~ the opposite group and I : G > G ~ the isomorphism defined by
l(g) = g-i for g C G.
L E M M A 4. 6 . 6 . I , e - 1Ge
Proof: Consider the map q~: G >G defined by ~(g) = gg-l.
= O : G > G . B u t r b e i n g c o n s t a n t . r g e
r = ( R g _ l ) , + ( L g ) , g = l I , g g g
a n d t h e r e f o r e -I
= - ( i ~ ) , g = _ ( L g _ 1 ) , e o ( R g _ 1 ) , I , g -1 ~ ( R g = l ) * g g
For g = e w e o b t a i n
l , e = - IGe , q . e . d .
- 6 2 -
R e m a r k . We h a v e s h o w n t h e f o r m u l a
I , = - ( L _ l ) , e ~ ( R g _ l ) , g g g
f o r g g G
T h i s m e a n s t h a t t h e t a n g e n t m a p to t h e m a p I : G > G in e a c h p o i n t i s
a l r e a d y g i v e n by t h e t a n g e n t m a p s of t h e t r a n s l a t i o n s . T h i s c a n be u s e d
to p r o v e t h a t t h e d i f f e r e n t i a b i l i t y of t h e m u l t i p l i c a t i o n G x G
g r o u p m a n i f o l d a l r e a d y i m p l i e s t h e d i f f e r e n t i a b i l i t y of I : G
L e t A
a l g e b r a
b y t h e b r a c k e t [ A 1, A2] =
P R O P O S I T I O N 4 . 6 . 7 .
g r o u p . I d e n t i f y L G w i t h G
i s o m o r p h i s m
> G o n a
> G .
b e a r i n g a n d ~ a L i e a l g e b r a o v e r A �9 T h e o p p o s i t e L i e
i s t h e A - m o d u l e ~ w i t h t h e L i e a l g e b r a s t r u c t u r e d e f i n e d
O [A I,Az] for A I, A ze
L e t G be a L i e g r o u p a n d G ~ t h e o p p o s i t e
a n d L { G ~ w i t h G ~ by t he c a n o n i c a l e
of t h e o r e m 4. 3. 3. T h e n
L(G ~ : (LG) ~
(LG) ~ being the opposite Lie al~ebra of LG.
Proof: After the indicated identification we have L(1) = l•e for
t h e i s o m o r p h i s m I : G > G ~ . T o A. ~ L G (i = 1, 2) c o r r e s p o n d s 1
I, A i = - A i 6 L(G~ To [A I, A2] C LG there corresponds on one e
hand - [A I, AZ]LG and on the other hand also [-A I, -Ag]L(GO) , as
I i s a n i s o m o r p h i s m . T h e r e f o r e
[At' AZ]L{GO ) - [AI, AT]LG
= [A I, AZ](LG)O q . e . d .
- 6 3 -
C O R O L L A R Y 4. 6 . 8 . L e t G be a L i e g r o u p , LG the L i e a l g e b r a
of l e f t i n v a r i a n t v e c t o r f i e l d s and RG the L i e a l g e b r a of r i g h t i n v a r i a n t
v e c t o r f i e l d s . I d e n t i f y LG a n d RG w i t h G e by t h e c a n o n i c a l i s o m o r p h i s m
of t h e o r e m 4. 3. 3. T h e n
aO = (LG) ~
Proof: We observe that left translations of G
of G ~ and vice versa, so that IX] = R(G ~ and RG =
mentioned identifications, by proposition 4. 6. 7 L(G ~
shows RG = (LG) ~
RG
t h e n
P r o o f :
and t h e r e f o r e
a r e r i g h t t r a n s l a t i o n s
L(G~ After the
= (LG) ~ which
This shows, of course, the existence of a natural isomorphism
e~
= LG. Moreover
COROLLARY 4.6.9. Let G be a Lie group. If G is commutative,
LG is commutative.
The commutativity of G implies RG = LG. But RG = (LG) ~
LG = (LG) ~ , q.e.d.
We shall see in chapter 6 that, for connected G, the converse is
als o true.
Example 4. 6.10. Let V be a finite-dimensional ~t-vectorspace
and consider the natural representation of GL(V) in V. We have seen that
identifying L(GL(V)) canonically with the tangent space at the identity,
we obtain s (V). By corollary 4. 6. 8, RG identified with the tangent
i
space at the identity is ( s (V)) ~
-64-
4. 7. T h e a d j o i n t r e p r e s e n t a t i o n of a L i e g r o u p .
C o n s i d e r the o p e r a t i o n of G on G by i n n e r a u t o m o r p h i s m s
~Y : G > Au t G ( s e e e x a m p l e 1 .1 .5 ) . T h e f u n c t o r L t r a n s f o r m s t h e
G - g r o u p G in to a G - L i e a l g e b r a L G a c c o r d i n g to p r o p o s i t i o n 1 .1 .9 .
We r e p e a t the d e f i n i t i o n of the i n d u c e d G - o p e r a t i o n on L G : i t i s t he
c o m p o s e d h o m o m o r p h i s m
G ~ > Aut G L > Aut LG
D E F I N I T I O N 4. 7.1. T h e a d j o i n t r e p r e s e n t a t i o n of a L i e Group G
is the r e p r e s e n t a t i o n of G in LG i n d u c e d by the o p e r a t i o n of G on G by
i n n e r a u t o m o r p h i s m s : A d g = L ( [ [ g ) .
PROPOSITION 4. 7. Z. !
L e t G, G ' be L i e ~ r o u p s and p: G > G a
h o m o m o r p h i s m . T h e n L ( p ) : LG !
> LG is a p - e q u i v a r i a n e e w i t h
r e s p e c t to t he a d j o i n t r e p r e s e n t a t i o n s of G, G' in L G , LG
P r o o f : T h e c o m m u t a t i v i t y of the d i a g r a m
!
G P >G
iI ! I
p I G - > G
( s e e a l s o e x a m p l e 1. Z. 3) a n d the f u n c t o r a l i t y of L p r o v e t h e s t a t e m e n t .
C o n s i d e r the c a n o n i c a l i s o m o r p h i s m ~l : G e
> LG of theorem 4.3. 3
p e r m i t t i n g to i n t e r p r e t G e as t he L i e a l g e b r a of G . F r o m t h e o r e m 4 . 5 . 4
it f o l l o w s t h a t the e f f e c t of Ad g : LG > LG on G e i s g i v e n by the m a p
- 6 5 -
( g g ) * e e
in e x a m p l e 3. Z. 8 is j u s t
: G - - - > G e . T h i s p r o v e s t h a t the o p e r a t i o n of G on Ge d e f i n e d
G and L G . e
t he a d j o i n t r e p r e s e n t a t i o n a f t e r identification of
Another description of the adjoint representation is given in
P R O P O S I T I O N 4. 7. 3. L e t G be a L i e g r o u p ,
t he a d j o i n t r e p r e s e n t a t i o n and A G L G . T h e n
Ad g A = (Rg_l ) CA
P r o o f : A d g A = L ( g g ) A = ( g g ) , A
( [ [ g ) , A = (R _ I ) , ( L g ) , A = ( R g _ I ) , A f o r A G L G . g
Ad : G > Aut LG
by c o m p l e m e n t 4. 5 . 4 . But
T h i s s h o w s t h a t t he o p e r a t i o n of G in G by r i g h t t r a n s l a t i o n s d e f i n e s
a r i g h t o p e r a t i o n of G in LG and the a d j o i n t r e p r e s e n t a t i o n d e s c r i b e s t h e
e f f e c t of t h i s o p e r a t i o n .
- 6 6 -
C h a p t e r 5. VECTORFIELDS AND 1 - P A R A M E T E R GROUPS OF TRANSFORMATIONS.
T h e L i e a l g e b r a of a L i e g r o u p g i v e s a d e e p i n f o r m a t i o n on t h e g r o u p .
T h e k e y f o r t h e u n d e r s t a n d i n g of t h i s is t he r e l a t i o n b e t w e e n v e c t o r f i e l d s
a n d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , w h i c h i s s t u d i e d in t h i s c h a p t e r .
5 ,1 . 1 - p a r a m e t e r g r o u l ~ o f t r a n s f o r m a t i o n s .
D E F I N I T I O N 5 . 1 . 1 . A n I R - m a n i f o l d X i s c a l l e d a 1 - p a r a m e t e r
group of transformations.
Let X be a manifold and y:I >X , t --,--~
H e r e I d e n o t e s a n o p e n i n t e r v a l of ]R c o n t a i n i n g 0 .
_ d ~t d t ~ t t h e t a n g e n t v e c t o r of ~ in t h e p o i n t ~/t"
~t' a curve on X.
We denote by
We then have
~t f _ d f ( • t ) f o r e v e r y f ~ C X dt
w h i c h c h a r a c t e r i z e s ~ t "
N o w l e t q~ : IR x X > X , (t , x) " -"~> (P t(x) , be a 1 - p a r a m e t e r g r o u p
of t r a n s f o r m a t i o n s . We s h a l l a l s o s a y f o r t h i s s i t u a t i o n : '~g~ i s a
1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s of X . " D e f i n e f o r e v e r y x ~ X
/t /t (1) Ax - d-i~Ot(x) = 0 - ~ t Ix) =0
A x i s a v e c t o r f i e l d on X . T h e n A ; ( ) x C X
If t ~- '~> $ t
t ~ ~ s t = @t"
r e s p e c t i v e l y , t h e n
i s a 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s , t h e n a l s o
If A, B a r e t h e v e c t o r f i e l d s i n d u c e d by (Pt ' ~ t
- 6 7 -
d d B x = ~-: { ~ ( x ) } : :[ Gst(X)/t =- s A ,
t = 0 = 0 x
I n t h i s d e f i n i t i o n of t h e v e c t o r f i e l d c o r r e s p o n d i n g t o a 1 - p a r a m e t e r
g r o u p of t r a n s f o r m a t i o n s , w e h a v e n o t m a d e u s e of t h e f a c t t h a t S t i s
g l o b a l l y d e f i n e d .
D E F I N I T I O N 5 . 1 . Z. L e t ( > 0 , I b e a n o p e n i n t e r v a l ( - ( , E ) of E
IR a n d U a n o p e n s e t o f X . A l o c a l l - p a r a m e t e r g r o u p of l o c a l t r a n s -
f o r m a t i o n s of X d e f i n e d on U i s a m a p qg: I ( x U > X , ( t , x ) ~ Ot (x )
s u c h t h a t
I) f o r a l l t 6; I ( , ~ t
2) i f t , s , t + s 6; I (
: U > s t ( U ) i s a d i f f e o m o r p h i s m ,
a n d x , S s (X ) E; U , t h e n ~ t + s ( X ) = opt ( ~ s ( X ) ) .
E q u a t i o n (1) s t i l l m a k e s s e n s e f o r x 6; U a n d d e f i n e s a v e c t o r f i e l d
A on U , w h i c h i s c a l l e d t h e v e c t o r f i e l d i n d u c e d on U b y q9 t . T h e
p r o p e r t i e s of a l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a t i o n s a r e n o t
u s e d f o r t h i s f a c t , b u t t h e y a l l o w t o p r o v e
P R O P O S I T I O N 5 .1 . 3. L e t
~_roup of t r a n s f o r m a t i o n s of X ,
s u b s e t of X , a n d A t h e i n d u c e d v e c t o r f i e l d on U . T h e n f o r
c u r v e t ~ " ~ > ~ t (x) s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n
~: I E x U > X be a local..l-parameter
= (-E,E)C ]R U an open where E > O, I E
x 6; U , the
w i t h t h e i n i t i a l c o n d i t i o n
Ot(x) -- Aot(x )
O o ( X ) -- •
- 6 8 -
P r o o f : L e t f C C X . T h e n f o r f i x e d x C U
d %t (x)f -- = Lira
s-->O _/_Is { f(~t+s (x)) - f(~t (x))}
L i m s - - > 0
__Is {f( Ss(~t (x))) - f( ~~
= d /
d s / s = 0
= Acpt(x) f , q . e . d .
C O R O L L A R Y 5 . 1 . 4 . L e t ~p, qJ : I ( x U > X b e two l o c a l
. . 1 - p a r a m e t e r g r o u p s of l o c a l t r a n s f o r m a t i o n s d e f i n e d on U . I f t h e y
i n d u c e t h e s a m e v e c t o r f i e l d s on U , t h e y c o i n c i d e .
P r o o f : T h i s i s j u s t t h e u n i q u e n e s s t h e o r e m f o r o r d i n a r y d i f f e r e n t i a l
e q u a t i o n s .
A p p l y i n g t h e e x i s t e n c e t h e o r e m f o r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s ,
w e prove n o w
P R O P O S I T I O N 5 . 1 . 5 . L e t A b e a v e c t o r f i e l d on X . F o r a n y x ~ X
t h e r e e x i s t s a n E > 0 , a n o p e n s u b s e t U o f X a n d a l o c a l 1 - p a r a m e t e r
g r o u p of l o c a l t r a n s f o r m a t i o n s cp : I x U > X , w h i c h i n d u c e s on U E
t h e g i v e n v e c t o r f i e l d .
P r o o f : F o r a f i x e d x C X w e d e f i n e t " ' > ~Pt(x) a s t h e s o l u t i o n o f
the differential equation
~o t (x) = A ~ t ( x )
-69 -
w i t h i n i t i a l v a l u e r = x .
~ t (~ps(x)) =
such that both sides are defined.
T h e n
a n d
W e now prove
r (x)
W r i t e
f o r s , t , t + s C I E
cat+s(X) = Ctl(t) , ~at(~as(X)) = a z ( t ) .
a l ( t ) = cbt+s(X) = A t+s(X) = A a l ( t )
~ ( t ) = ~ o t ( ~ s ( x ) ) = A t ( O s ( x ) ) -- A a 2 ( t )
(i = 1, Z) a r e s o l u t i o n s of t h e s a m e d i f f e r e n t i a l e q u a t i o n . A s 1
al (O ) = a Z ( O ) : q )s (X) , w e o b t a i n c 1 = a 2 ,
t h a t oPt(x) d e p e n d s d i f f e r e n t i a b l y o n x .
x C U t h e r e f o r e opt(r = ~ _ t ({Pt(x))
opt i s a d i f f e o m o r p h i s m .
p r o v i n g t h e d e s i r e d r e s u l t .
I t r e m a i n s t o s h o w t h a t t h e m a p x - - - - '~ ~ t (x ) i s a d i f f e o m o r p h i s m
: U > cPt(U ) . C e r t a i n l y ~Po i s t h e i d e n t i t y t r a n s f o r m a t i o n . W e k n o w
F o r s u f f i c i e n t l y s m a l l t a n d
= (Po(X) = X , p r o v i n g t h a t
D E F I N I T I O N 5 . 1 . 6 . A v e c t o r f i ~ l d A on X i s c o m p l e t e , i f i t i s
induced by a 1-parameter group of transformations.
Example 5. I. 7. Consider the vectorfield A on ]R which has in
every point a positively oriented vector of length one. Consider the
submanifold (0, l)c JR. Then A r.~strict'ed to (0, I) is not complete.
A criteria for completeness is given in
-70-
LEMMA 5.1. 7. Let A be a vectorfield on X. Suppose there exists
r > 0 and a local 1-parameter group of local transformations ~0: I xX > X
inducing A. Then ~ has an extension to a 1-parameter group of transforma-
tions and A is therefore complete.
Proof: ~ t is a diffeomorphism f o r It I ~ E. There is only to define
( ~t for It l > f. Write t = k. ~ + r with an integer k and Irl < ~2
-k If k > 0, define __(~ = (r ~ . If k < 0 , define ~t = ((D__ E)
Z z Now ~ ~t satisfies all conditions.
~ ~r
Example 5. I. 8. Let X be a compact manifold. Any vectorfield A
on X is complete.
We remark that the relation between vectorfields and local l-parameter
groups of local transformations described in this section is at the origin
of the denomination of a vectorfield as an infinitesimal transformation.
5.2. 1-parameter groups of transformations and equivariant maps.
Convention on notations. Given a local 1-parameter group of local
transformations on X we denote it ~t and just speak of the induced
vectorfield A on X, without specifying the domains of definition. Also,
for a given vectorfield A on X, we just write %D t for a local 1-parameter
group of local transformations inducing A on some subset of X. The
formulas are valid as soon as they make sense. They make sense in
particular if only l-parameter group of transformations occur.
-71-
P R O P O S I T I O N 5 . 2 . 1 . L e t
t r a n s f o r m a t i o n s on X, X' , A, A '
!
~t' ~t be local 1-parameter groups of
the induced vectorfields and ~: X >X'
! !
a m a p . _If @ o ~ t = ~ t o ~ f o r a l l t , t h e n A and A a r e ~ - r e l a t e d .
P r o o f : We have
!
Ix) A~; (x) = Ax~t(q~(x))
by d i f f e r e n t i a t i n g w i t h r e s p e c t to t . Or
!
,~tlx) AtiYtlx) = A (~t(x))
w h i c h s h o w s the p r o p o s i t i o n in v i e w of l e m m a 4 . 4 . 3.
It is convenient to call a map ~ : X ----> X' satisfying q~ ~ ~t = ~!t " ~
a n e q u i v a r i a n c e w i t h r e s p e c t to t h e g i v e n l o c a l 1 - p a r a m e t e r g r o u p s of
!
l o c a l t r a n s f o r m a t i o n s ~ t and kD t . T h e p r o p o s i t i o n s a y s t h a t the
i n d u c e d v e c t o r f i e l d s A and A' a r e t h e n ~ 0 - r e l a t e d . T h i s is c h a r a c t e r i s t i c
f o r e q u i v a r i a n t m a p s . P r e c i s e l y we h a v e
P R O P O S I T I O N 5 . 2 . 2 . L e t X, X' be m a n i f o l d s , A, A' v e c t o r f i e l d s
on X, X' r ~ e s p e c t i v e l y , S t $ ' c o r r e s p o n d i n g l o c a l 1 - p a r a m e t e r g r o u p s - - ' t - -
!
of l o c a l t r a n s f o r m a t i o n s and ~0: X > X a m a p . If A and A! a r e
~ 0 - r e l a t e d , t h e n
O 0 ~ ,
t t
I
P r o o f : F o r x C X w r i t e a l ( t ) = @ (q~t(x)) , a z ( t ) = qit(~O(x)) .
T h e n al(O ) = a2(O) = ~0(x). We p r o v e e l = aZ b y s h o w i n g t h a t a l ' ag
s a t i s f y t he s a m e d i f f e r e n t i a l e q u a t i o n . But
-7Z-
~l(t) = (~q,t(x) AqSt(x) = ~l(t)
!
b y l e m m a 4 . 4 . 3 , a s A a n d A a r e ~ - r e l a t e d a n d
E2.(t ) ' z(t) = A a , q . e . d .
COROLLARY 5. Z. 3. Let X, X' be manifolds and ~: X > X' a
d i f f e o m o r p h i s m . I f A i s a v e c t o r f i e l d on X , s e n e r a t i n $ a l o c a l 1 - p a r a m e t e r
!
~ r o u p of l o c a l t r a n s f o r m a t i o n s q't ' t h e n t h e v e c t o r f i e l d ~ . A o n X
-1 ~ e n e r a t e s t h e l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a t i o n ~ o ~ t ~ q~ "
P r o o f :
A p p l y i n g t h i s t o a n a u t o m o r p h i s m w e o b t a i n
C O R O L L A R Y 5 . 2 . 4 . L e t X b e a m a n i f o l d ,
g e n e r a t i n g a l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a t i o n s
~0 : X > X a d i f f e o m o r p h i s m . T h e n ~ # A = A i f a n d o n l y i f
{~ ~ q/t = q/t ~ (~ f o r a l l t .
I t i s s u f f i c i e n t t o o b s e r v e t h a t A a n d ~ 0 , A a r e r
A a v e c t o r f i e l d on X
~ t and
W e o b s e r v e t h a t t h e p r e c e d i n g i s s t i l l t r u e f o r a l o c a l a u t o m o r p h i s m
of X , i . e . a m a p ~0 : U > X d e f i n e d on a n o p e n s u b s e t of X a n d h a v i n g
a r e s t r i c t i o n b e i n g a d i f f e o m o r p h i s m . ~o#A i s t h e n t o b e i n t e r p r e t e d a s
a v e c t o r f i e l d on a c o n v e n i e n t s u b s e t of X a n d c a n b e d e f i n e d b y t h e
f o r m u l a of l e m m a 4. 2. 8. C o n s i d e r i n p a r t i c u l a r
a u t o m o r p h i s m of X i n t h i s s e n s e . A s qJ ~ S
w e s e e t h a t ( ~ s ) , A = A b y c o r o l l a r y 5 . 2 . 4 .
, which is a local S
~ t = qJt ~ ~ s f o r a l l t ,
This just means that the
velocity field A of the flow St is invariant by the flow, the characteristic
property of a stationary flow.
-73-
An application of corollary 5. Z. 4 is the following:
LEMMA 5. Z. 5. Let G be a Lie group, A 6 LG and t
l-parameter group of local transformations generated by A.
L o �9 = r o L f o r a l l t , g C G . g t t g
We can now prove
P R O P O S I T I O N 5. Z. 6. L e t G be a L i e ~ r o u p .
a local
Then
E v e r y l e f t i n v a r i a n t
by lemma 5. ~. 5. Defining conversely @ by this formula, we obtain the
desired ~: I E x G > G.
Another consequence of lemma 5.2.5 is
PROPOSITION 5. Z. 7. Let G be a Lie sroup, A ~ LG and q~t the
1-parameter group of transformations generated by A. Then i t = R~(e).
Proof: By lemma 5. ~. 5 a particular case of
I~MMA 5.2.8. Let G be a group (in the sense of algebra). A map
,~: G > G is a right translation (and then necessarily R~) = y) i_~f
and only if
~(g) = (~t ~ Lg)(e) : (Lg o ~t)(e) : Lg(~(e))
v e c t o r f i e l d on G i s c o m p l e t e .
P r o o f : We c o n s i d e r a l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a -
t i o n s ~ t g e n e r a t e d b y A , a n d s h o w t h a t St h a s a n e x t e n s i o n t o a l o c a l
1-parameter group of local transformations ~ : I x G > G for an (
> 0. Then the proposition follows by lernma 5. I. 7. Suppose
: I E x U > G for an E > 0 and a neighborhood U of e ~ G. A
necessary condition for an extension ~: I E x G > G is
- 7 4 -
L ~ ~ = % o L f o r a l l g 6 G . g g
P r o o f : T h e a s s o c i a t i v i t y s h o w s t h a t t h e c o n d i t i o n i s n e c e s s a r y .
S u p p o s e c o n v e r s e l y L o qJ = ~ o L f o r a l l g 6 G . F o r V C G g g
g ~ ( V ) = q J ( g v ) a n d i n p a r t i c u l a r g ~ ( e ) = ~ (g) , i . e .
~e(g) = R ~ ; g , q . e . d . (e
5. 3. T h e b r a c k e t of t w o v e c t o r f i e l d s .
We g i v e n o w a n i n t e r p r e t a t i o n of t h e b r a c k e t of t w o v e c t o r f i e l d s
( t a k e n f r o m K. N o r n i z u a n d S. K o b a y a s h i [11], p. 15).
W e u s e t h e n o t a t i o n ~ . A f o r a v e c t o r f i e l d A on X a n d a l o c a l
a u t o m o r p h i s m r o f X a s e x p l a i n e d i n 5 . 2 .
P R O P O S I T I O N 5 . 3 . 1 . L e t A a n d B b e v e c t o r f i e l d s on t h e m a n i f o l d
X , and ~ t a l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a t i o n s g e n e r a t e d
by A. Then
1 [A, B] = Lirn ~-[B x - ((~t).B)x] for x S X
x t - - ~ O
LEMMA 5. 3. Z. Le___~t E >0, I E = (-E, ()c]R and f : I E xX > ]R
with f(o, x) = 0 for x C X. Then there exists g : I x X > ]R with
8f f ( t , x) = t g ( t , x ) . M o r e o v e r g ( o , x) = ~ (o , x) .
1
P r o o f : D e f i n e g ( t , x) = -~- ( t s , x ) d s
o
-75-
LEMMA 5. 3. 3. Let A generate opt. For any f G CX there exists
gt G CX with f o ~t = f + t gt and go = A/.
The function g(t, x) = gt(x) is defined, for each fixed x G X, in
I t [ < r f o r s o m e ~ > O.
Proof: Consider h(t, x) = f(~0t(x) ) - f(x) and apply lemma 5. 3. Z.
Then f o~)t = f + tgt" We have
1 1 (A.f)(x) = L i r a -~ [f((Pt(x)) - f(x)] = L i r a T f(t, x)
t--->0 t-->0
= Lim g(t, x) = go(X) t--->0
P r o o f of p r o p o s i t i o n 5. 3 .1 .
l e m m a 5. 3. 3. S e t x t = ( p : l ( x ) .
L e t f G C X .
Then
T a k e gt 6 C X as in
((~t)$B)x f = (B(f �9 qgt))(xt) = (Bf)(xt)+ t(Bgt)(xt)
a n d
I Lim -[ [B x - ((~t), B)x ] f = t--->0
i Lira -~ [(Bf)(x) - (Bf)(xt) ] t-->0
L i r a (Bgt)(xt) t-->O
= = A (Bf) - Bx(Af) Ax(Bf) - B xgo x
= [A, B] f , q.e.d. X
C O R O L L A R Y 5. 3 . 4 . L e t A a n d B b e v e c t o r f i e l d s on t h e m a n i f o l d
X and ~ t a local 1-parameter group of local transformations generated
b y A . T h e n f o r a n y v a l u e of s ~ I R , x ~ X
-76 -
= Lira 1 [((~0s),B) ((Os). [A, B] ) 7 x t-->0
- (( ~t+ s), B)x ]
a S
Proof: L e t s em. Then ( ~ s ) , [ A , B] = [ ( ~ s ) . A , ( C ~ s ) . B ] = [ A , ( ~ s ) , B ] ,
( ~ s ) , A = A by t h e r e m a r k a t t h e e n d of 5. Z.
Applying proposition 5. 3. I. , we obtain
1 [A, (~)s),B]x = Lim -[[((~s),B)x - ((~t),(~s),B)x]
t--->0
i [((~s) * _ (( ),B)x] Lim -[ B)x ~t+s " t-->0
P R O P O S I T I O N 5 . 3 . 5 . L e t X be a m a n i f o l d , A a n d B v e c t o r f i e l d s
o__n X g e n e r a t i n g l o c a l 1 - p a r a m e t e r ~ r 0 u p s of l o c a l t r a n s f o r m a t i o n s S t a n d
~ t r e s p e c t i v e l y . T h e n ~ t ~ ~ = ~ ~ ~ s s t
f o r e v e r y s a n d t i f a n d
o n l y if [ A , B] : O .
P r o o f : If ~0 t o @ = q~ ~ opt f o r e v e r y s a n d t , ( ~ t ) . B = B S S
by c o r o l l a r y 5 . 2 . 4 . B y p r o p o s i t i o n 5. 3 .1, [ A , B] = O . S u p p o s e c o n v e r s e l y
d - . [ A , B] = O . B y c o r o l l a r y 5. 3 . 4 , - ~ ( (cp t ) .B)x O f o r a n y t . T h e r e -
f o r e (~p t ) .B = B f o r e v e r y t and by c o r o l l a r y 5 . 2 . 4 ~0 t c o m m u t e s w i t h
e v e r y ~ . S
P R O P O S I T I O N 5. 3 . 6 . L e t t he v e c t o r f i e l d s A and D of X g e n e r a t e
l o c a l 1 - p a r a m e t e r g r o u p s of l o c a l t r a n s f o r m a t i o n s ~ t and ~ t r e s p e c t i v e l y .
S u p p o s e [ A , B] = O . T h e n X t = {~t o ~t = ~ t ~ @t is a l o c a l 1 - p a r a m e t e r
group of local transformations and is generated by A + B.
- 7 7 -
Proof: Proposition 5. 3.5 shows that
group of local transformations. Now
~t(x) = ~t ( ~ t (x))
X is i n d e e d a l oca l 1 - p a r a m e t e r t
+ (r ~'t (x)
= A x t ( x ) + (r ) Bxit(x)
But by p r o p o s i t i o n 5. 3 .5 and c o r o l l a r y 5. 2. 4 (Ot) * B = B .
and
(r Bq't(x) = B(pt(~t(x) ) = BXt(x )
kt(x) = (A + B) Xt(x) , q.e.d.
T h e r e f o r e
5.4. 1-parameter subgroups of a Lie group.
DEFINITION 5.4. I. A l-parameter subgroup a of a Lie group G is
a homomorphism a: ]R > G of Lie groups.
Remark. Let X be a manifold and ~t a 1-parameter group of trans-
formations of X. One would like to consider t ""> (~t as a l-parameter
s u b g r o u p IR > Aut X of Aut X .
of the d i f f e r e n t i a b i l i t y of th i s m a p .
3.2.5.
The trivial homomorphism O : IR
But it does not make sense to speak
See also the remark after example
,>G is a 1 - p a r a m e t e r s u b g r o u p
of G,
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A non-trivial l-parameter subgroup
an injection, as shown by
Example 5.4.2. The canonical homomorphism ]R > ]R/Z = ~r is
a l-parameter subgroup of "Jr .
Lemma 5.4. 5 below shows that a non-trivial l-parameter subgroup
is an immersion.
Let A be a complete vectorfield on G and ~t
of transformations generated by A. Define
a t = ~0t{e ) for e C G
T h e n a : IR > G and ao = ~0o(e) = e , but a
a: IN >G is not necessarily
t h e 1 - p a r a m e t e r g r o u p
is not necessarily a 1-parameter
s u b g r o u p of G . If A g L G , th i s i s t he c a s e , as s t a t e d in
P R O P O S I T I O N 5 . 4 . 3. L e t G be a L i e g r o u p , A C L G , ~0 t th___e_e
l-parameter subgroup of G generated by A and a: IR > G the map
defined by a t = ~t(e). Then a is a l-parameter subgroup of G.
~t = Rat an.d (Dt is completely described by a
Proof: Applying lemma 5.2.5, we obtain
M o r e o v e r
a t l + t 2 = ~ 0 t l + t z ( e ) = ~ t l ( ~ t z { e ) ) = ( ~ t l o L~0t2(e))(e)
= (L~0t2(e) ~ ~0 t l ) (e) = ~t2(e)~0tl(e) = at 2atl
In view of proposition 5. g. 7 we have @t = Rat " |
T h e s t a t e m e n t ~0 t = Ra t i s o f t en p a r a p h r a s e d in t he l i t e r a t u r e by:
- 79 -
" t h e i n f i n i t e s i m a l t r a n s f o r m a t i o n g e n e r a t e d b y a l e f t i n v a r i a n t v e C t o r f i e l d
i s a r i g h t t r a n s l a t i o n " .
W e c a l l a t h e 1 - p a r a m e t e r s u b g r o u p of G
s G d e n o t e t h e s e t of 1 - p a r a m e t e r s u b g r o u p s of G .
m a p ~ : LG > s
LEMMA 5.4.4. Let A C LG, a= ~(A)
solution of the differential equation
a t = An t
w i t h i n i t i a l c o n d i t i o n a = e . o
P r o o f : h t = ~ t ( e ) = A t ( e ) = A a t
a = ~ (e) = e , q . e . d . O o
d e f i n e d b y A E L G . L e t
W e h a v e d e f i n e d a
s . T h e n a i s t h~
a n d s h o w s i t s i n j e c t i v i t y .
by proposition 5. I. 3 and
T h i s l e m m a g i v e s a d i r e c t d e s c r i p t i o n of t h e m a p ~ : L G > s
W e s h a l l s e e t h a t ~ i s b i j e c t i v e . F i r s t w e
prove
LEMM A 5.4.5. L e t a ~ s T h e n
a t = ( L a t ) * e h ~ = ( R a t ) * e a ~
t o
P r o o f :
s , w e o b t a i n
d t + s
By d i f f e r e n t i a t i n g = a t a = a a a t + s s s t
= ( ) * a s t s S @ s
w i t h r e s p e c t
a n d f o r s = 0 t h e d e s i r e d r e s u l t .
- 8 0 -
T H E O R E M 5 . 4 . 6 . L e t G be a L i e g r o u p , L G i t s L i e a l g e b r a a n d
s the set of l-parameter subgroups of G. For any A C LG we define
~'(A) = ct ~ s a s t h e s o l u t i o n of ~ t = A w i t h i n i t i a l c o n d i t i o n ct = e . t o
T h e n ~. : L G > s G i s b i j e c t i v e .
P r o o f : L e t a g s If a= ~'{A) f o r s o m e A ~ L G , t h e n
n e c e s s a r i l y A = ~ w h i c h s h o w s i n j e c t i v i t y of 4 - D e f i n i n g c o n v e r s e l y e o "
A ~ L G a s t h e v e c t o r f i e l d w i t h A = ~ , w e h a v e A a t = (Lct t) tt = tt e o ~ o t
b y l e m m a 5 . 4 . 5, w h i c h s h o w s s u r j e c t i v i t y of ~ .
L e m m a 5 . 4 . 5 s h o w s t h a t w e o b t a i n b y t h e s a m e d e f i n i t i o n a b i j e c t i o n
RG > s G . I n f a c t , t h e t a n g e n t v e c t o r s of t h e c u r v e t , - , ~ a t b e l o n g
a s w e l l t o t h e l e f t a s t o t h e r i g h t i n v a r i a n t v e c t o r f i e l d d e f i n i n g a . T h e
s i t u a t i o n i s d e s c r i b e d p r e c i s e l y b y
P R O P O S I T I O N 5 . 4 . 7. L e t A g L G , a = ~{A) 6 s G a n d g ~ G .
T h e n A - ( g ~ t ) ' t g = 0
P r o o f : L e t ~ t b e t h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s g e n e r a t e d
b y A . T h e n ~ t ( g ) = A ~ t ( g ) f o r a n y g ~ G . N o w b y p r o p o s i t i o n 5 . 4 . 3,
~ t ( g ) = R a t ( g ) = g a t
a n d
~0t(g ) - ( g a t ) ~ .
F o r t = 0 t h i s s h o w s /~p ( g ) = A = ( g a t ) " , q . e . d . o g t = 0
A c o n s e q u e n c e of t h e o r e m 5 . 4 . 6 i s t h e
-81-
PROPOSITION 5.4. 8. Let I be an open interval of IR containing O
a n d a : I > G a l o c a l h o m o m o r p h i s m of L i e g r o u p s . T h e n t h e r e e x i s t s
a u n i q u e 1 - p a r a m e t e r s u b g r o u p a : n~ > G of G w i t h a / I = a .
P r o o f : L e t A E L G b e d e f i n e d b y a = A a n d ~ = ~ ( A ) . L e m m a o e
5 . 4 . 5 s t i l l a p p l i e s t o a , s h o w i n g ~t t = ( L a t ) . e ct o a n d t h e r e f o r e
ao ~tt A a t = ( L a t ) . e = . B u t a i s a l s o a s o l u t i o n of t h i s d i f f e r e n t i a l
e q u a t i o n . N o w a = ct = e s h o w s I = a , a n d a i s a n e x t e n s i o n of a O O
to a 1-parameter subgroup of G. The uniqueness follows from the fact
that there exists only one 1-parameter subgroup a of G with given d O
As shown by the example of the local isomorphism "IF > IR, there
exists not necessarily an extension of a local homomorphism G !
> G of
Lie groups to a homomorphism (see remark following lemma 4. 5.8). It
f o l l o w s f r o m t h e t h e o r y of t o p o l o g i c a l g r o u p s , t h a t a n e x t e n s i o n e x i s t s , i f
G i s s i m p l y c o n n e c t e d ( s e e a l s o l e m m a 7. Z. 5). P r o p o s i t i o n 5 . 4 . 8 i s a
particularly simple case of this situation.
L E M M A 5 . 4 . 9 . L e t G b e a L i e g r o u p , A a n d B 6 L G , ~ a n d ~ t h e
c o r r e s p o n d i n g e l e m e n t s C s ~ t a n d ~ t h e 1 - p a r a m e t e r g r o u p s of
t r a n s f o r m a t i o n s g e n e r a t e d b y A a n d B r e s p e c t i v e l y . T h e n
~ q = @ ~ ~ t f o r e v e r y t a n d s i f a n d o n l y i f a t ~ s = ~ a f o r ~ t s s s t
e v e r y t and a .
P r o o f : For g C G ,
( ~ t ~ @s )(g) = ( ~ t ~ @s ~ L ) (e ) = ( L g o ~ t ~ @ )(e) b y l e m m a 5. g 5. g s
-82 -
Now @(e)s = -~s = L ~ s ( e ) and t h e r e f o r e f o r the s a m e r e a s o n
o o @ s ) ( e ) = ( L g o ) ( e ) = ( L o L o r ) ( e ) ( L g opt ~ t ~ L ~ s g 1~ s
t h u s p r o v i n g
( ~ t a ~ @ s ) { g ) : g[5 s t
Similarly
( ~ o ~ t ) ( g ) = g a t [Bs S
T h i s p r o v e s the l e m m a .
C o n s i d e r the e x p r e s s i o n ( ~ t oK s)(g) = g ~ s a t ' w h i c h o c c u r r e d in
the p r o o f . In p a r t i c u l a r (@ t o qJt)(e ) = ~ t a t " If [A , B] = O , by
p r o p o s i t i o n 5. 3. 6 Xt = ~ t o St = qJt o ~ t i s a 1 - p a r a m e t e r g r o u p of
t r a n s f o r m a t i o n s , X t (e) = ~t a t = a t ~t a 1 - p a r a m e t e r s u b g r o u p , and
A + B the c o r r e s p o n d i n g v e c t o r f i e l d . T h e r e f o r e
P R O P O S I T I O N 5 . 4 . 1 0 . L e t A, B C LG and
p o n d i n g 1 - p a r a m e t e r s u b g r o u p s . S u p p o s e [A, B] = O . T h e n
at ~ t = ~ t a t = Nt d e f i n e s a 1 - p a r a m e t e r s u b g r o u p and A + B
c o r r e s p o n d i n g l e f t i n v a r i a n t v e c t o r f i e l d .
a,~ C s the corres-
i s the
Together with proposition 5. 3.5 we obtain from lemma 5.4.9
PROPOSITION 5.4. II. Le__~t A, B ~ LG, @t and ~t the 1-parameter
groups of transformations generated by A and B respectively and a,
the corresponding 1-parameter subgroups of G. Then the following
s t a t e m e n t s a r e e q u i v a l e n t :
-83-
I) [ A , B ] = O
2) ~ t o ~ = S S
3) at~ s ~s = r t
~ ~ t f o r e v e r y s a n d t
f o r e v e r y s a n d t
We s h a l l s e e i n c h a p t e r 6 t h a t t h e s e c o n d i t i o n s a r e e v e n e q u i v a l e n t
t o a t ~ t = ~ t a t f o r e v e r y t ( s e e p r o p o s i t i o n 6 . 5 . 3 . ) .
Now we consider a homomorphism p : G > G of Lie groups. A
1 - p a r a m e t e r s u b g r o u p a ~ s G g i v e s b y c o m p o s i t i o n w i t h p a n e l e m e n t
!
p o a E s . I f p i s a l o c a l h o m o m o r p h i s m , p o a i s a l o c a l h o m o -
m o r p h i s m ]R > G ' , b u t c a n b y p r o p o s i t i o n 5 . 4 . 8 b e u n i q u e l y e x t e n d e d
t o a 1 - p a r a m e t e r s u b g r o u p of G ' , w h i c h w e a l s o d e n o t e b y p o a . T h e
m a p s : s > s s o d e f i n e d i s c o m p a t i b l e w i t h t h e m a p ~ of
t h e o r e m 5 . 4 . 6. M o r e p r e c i s e l y w e h a v e
PROPOSITION 5.4. iZ. Let G, G' b e Lie ~roups and 9 : G > G'
a local homomorphism. Then the followin$ diasram
L G -- L(p) --:> L G '
s (0) >,s
commutes, where s is the composition with p and ~G' ~G' the
m a p s of t h e o r e m 5 . 4 . 6 .
Proof: For a ~ s G we have ( 9 ~ a)t/t=0 = P *e ~t~ This
m e a n s t h a t t h e d i a g r a m (without t h e d o t t e d l i n e )
- 8 4 -
P . e i
G > G e
LG L(P)
LG
s s ~ s
c o m m u t e s . L ( 9 ) i s d e f i n e d by f i l l i n g in t he d o t t e d l i n e in t h e u p p e r ha l f .
G b e i n g s u r j e c t i v e , ( in f a c t b i j e c t i v e ) , t h i s p r o v e s t he p r o p o s i t i o n .
D e f i n e t he f o r g e t - f u n c t o r V : s > E n s , a s s i g n i n g to a n y L i e a l g e b r a
i t s u n d e r l y i n g s e t and to a n y L i e a l g e b r a h o m o m o r p h i s m the c o r r e s p o n d i n g
m a p of the u n d e r l y i n g s e t s . P r o p o s i t i o n 5 . 4 . 12 s t a t e s t h e n t h a t
: V o L > s is a n a t u r a l t r a n s f o r m a t i o n , in f a c t a n a t u r a l e q u i v a l e n c e .
5 . 5 . K i l l i n g v e c t o r f i e l d s .
In t h i s s e c t i o n , t h e r e l a t i o n b e t w e e n 1 - p a r a m e t e r s u b g r o u p s of a L i e
g r o u p G and 1 - p a r a m e t e r g r o u p s of t r a n s f o r m a t i o n s of a G - m a n i f o l d X
is s t u d i e d .
L e t X be a G - m a n i f o l d w i t h r e s p e c t to a h o m o m o r p h i s m T : G
and ~: ]R
> Aut X
> G a 1 - p a r a m e t e r s u b g r o u p of G . T h e c o m p o s e d h o m o m o r p h i s m
]R ~ ~ G T ~ Au t X
d e f i n e s a 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s @ t of X . I n d e e d , the
m a p
- 8 5 -
IR x X > G x X > X
( t , x ) ~ (a t , x ) - - - ~ > T a ( x ) = ~t(x) t
i s d i f f e r e n t i a b l e .
D E F I N I T I O N 5 . 5 . 1 . T h e K i l l i n ~ v e c t o r f i e l d A on X d e f i n e d bY
a G s G is t h e v e c t o r f i e l d i n d u c e d by the 1 - p a r a m e t e r g r o u p of t r a n s -
f o r m a t i o n s ~ t "
R e m a r k . As a l r e a d y o b s e r v e d in s e c t i o n 5 . 4 , one w o u l d l i k e to
c o n s i d e r t he 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s ~ t as a 1 - p a r a m e t e r
s u b g r o u p
on Aut X
as i n d i c a t e d a b o v e .
]R > Au t X of A u t X . T h e n ~ t w o u l d d e f i n e a v e c t o r f i e l d
as e x p l a i n e d in 5. 4. As t h i s p r e s e n t s d i f f i c u l t i e s , one p r o c e e d s
T h e d i f f e r e n t i a l e q u a t i o n
~t (x)
d e s c r i b i n g the r e l a t i o n b e t w e e n ~0 t
= A * t (x)
and A* c a n h e u r i s t i c a l l y b e w r i t t e n as
~ t = A~ t
i n t e r p r e t i n g n o w A* as a v e c t o r f i e l d on A u t X . T h i s d e s c r i b e s ,
c o u r s e , A* on ly a l o n g the c u r v e t ~ ~ t on Au t X .
of
E x a m p l e 5 . 5 . 2 . If T: IR > Au t X d e f i n e s a 1 - p a r a m e t e r g r o u p of
t r a n s f o r m a t i o n s T t of X , t h e n t h e K i l l i n g v e c t o r f i e l d A* d e f i n e d by
111% : IR > IR i s j u s t the v e c t o r f i e l d i n d u c e d by T t .
- 8 6 -
P R O P O S I T I O N 5 . 5 . 3. L e t G o p e r a t e on G
T h e K i l l i n $ v e c t o r f i e l d on G d e f i n e d by a ~ s
v e c t o r f i e l d B 6 RG c h a r a c t e r i z e d by ao = Be "
by l e f t t r a n s l a t i o n s .
i s t h e r i g h t i n v a r . i a n t
P r o o f : T h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s ~ t = L a t on
G i n d u c e s by p r o p o s i t i o n 5 . 4 . 3 a n d t h e o r e m 5 . 4 . 6 ( r e s p e c t i v e l y t h e
a n a l o g u e f o r RG) a r i g h t i n v a r i a n t v e c t o r f i e l d B on G . I t i s c h a r a c t e r i z e d
= h . b y B e o
E x a m p l e 5 . 5 . 4 . L e t p : G > G ' b e a h o m o m o r p h i s m a n d
7 : G > B i j G ' t h e o p e r a t i o n of G o n G ' d e f i n e d by 7 = L �9 g P (g)
S u p p o s e a C s G a n d c o n s i d e r t h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s
of G ' d e f i n e d by a : ~ = = t Lp ( a t ) L(p oa) t .
G' i n v a r i a n t v e c t o r f ~ e l d B ' on c h a r a c t e r i z e d by
It i n d u c e s t h e r i g h t
B', = (p o a)(t)~ = p, ao e t = O e
C o m p o s i n g t h e c o r r e s p o n d e n c e a ~v.~> B ' w i t h t h e c a n o n i c a l m a p R G
we o b t a i n o b v i o u s l y j u s t t h e h o m o m o r p h i s m
!
p:G >G .
Example 5.5.5. Consider a finite dimensional IR -vectorspace V
and the natural representation of GL(V) in V. Let a be a l-parameter
subgroup of GL(V) and v G V.
d e f i n e d by a s a t i s f i e s A* = v O
> s
R(p) : RG >RG' defined by
by l e m r n a 5 . 4 . 5. T h e r e f o r e t h e K i l l i n g v e c t o r f i e l d A* d e f i n e d by t h e
1 - p a r a m e t e r s u b g r o u p a s a t i s f i e s A * -- h v , i . e . i s t h e v e c t o r f i e l d V O
Then the Killing vectorfield A* on V
, where v t = atv. But ~'t = ~ttv = &oat v
-87-
canonically defined by the endomorphism ~ C s O
W e n o w a p p l y t h e r e s u l t s of s e c t i o n 5. 3 t o K i l l i n g v e c t o r f i e l d s a n d
prove
PROPOSITION 5.5.6. Let X be a G-manifold with respect to a
homomor}~hism 7: G > Aut X, a a 1-parameter subgroup of G and
A # t h e K i l l i n g v e c t o r f i e l d on X d e f i n e d b y a . I f C i s a n a r b i t r a r y
v e c t o r f i e l d on X , t h e n
, : Lim 1 t)~C)x ] [A* c ] x t - > 0 c x - f o r x C X .
P r o o f :
g e n e r a t e d b y A ~ .
5 . 3 . 1 .
C O R O L L A R Y 5 . 5 . 7 .
t h e c o r r e s p o n d i n g r i g h t i n v a r i a n t v e c t o r f i e l d .
f i e l d on G , t h e n
( T o a ) t i s t h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s of X
T h e f o r m u l a i s t h e r e f o r e a p a r t i c u l a r c a s e of p r o p o s i t i o n
Let G be a Lie group, a ~ s and B ~ RG
If C is an arbitrary vector-
1 [B, C] = Lira ?[Cg g t-->0 - ((L%),C)g] f o r g C G .
Proof: Let G operate on G by left translations. By proposition
5.5. 3 B is then the Killing vectorfield defined by a ~ s G with respect
to this operation. Now we are in the situation of proposition 5.5.6. i
Note that in particular for C C RG this formula expresses the bracket
in RG with the aid of L~ . Of course, we have a similar formula for left t
invariant vectorfields. We deduce the following interesting formula:
-88-
a G
P R O P O S I T I O N 5 . 5 . 8 . L e t G b e a L i e ~ r o u p a n d A, C 6 L G .
s G i s the 1-parameter subgroup defined by A, then
I f
w h e r e Ad : G
P r o o f :
N o w C 6; L G
a_ /
[A, C] = {Ad( at)}-LL C dt = 0
> A u t L G d e n o t e s t h e a d j o i n t r e p r e s e n t a t i o n of G .
A s i n c o r o l l a r y 5 . 5 . 7 , w e f i r s t o b t a i n f o r g C G
1 [A, C]g = t--->0Lim ~-[Cg - ((Rat) ,C)g]
a n d t h e r e f o r e ( L a t l ) , c = C , i . e .
This shows
(R at) ,C = Ad(atl)c
[A, C]g
which can be written
- - dtd {Ad(atl) Cg}/t =0
[ -A, C] = d (Ad(~l) Cg}/ . g dt t=0
B u t t h e s u b g r o u p -I
t ------> a = a t -t c o r r e s p o n d s to t h e v e c t o r f i e l d - A 6 L G ,
s h o w i n g t h u s t h e d e s i r e d r e s u l t . |
We h a v e s u p p o s e d A d : G > A u t L G to be a h o m o m o r p h i s m of L i e
g r o u p s . T h i s f o l l o w s f r o m t h e c o n t i n u i t y of A d ( s e e s e c t i o n 6. 3).
-89 -
5. 6. The homomorphism ~: RG > DE for a G-manifold X.
T h e k n o w l e d g e of t h e f o l l o w i n g t w o s e c t i o n s i s n o t n e c e s s a r y f o r t h e
understanding of the subsequent developments.
W e s h a l l s h o w t h a t an o p e r a t i o n T: G > A u t X d e f i n e s a h o m o -
morphism v: RG > DE of Lie algebras. First we prove
L E M M A 5.6. I. Let X be a G-manifold, X' !
a G - m a n i f o l d ,
' X' p : G > G a homomorphism and ~0: X > a p-equivariance.
! C o n s i d e r a C s a = p o a E s a n d t h e K i l l i n g v e c t o r f i e l d s
' A # A * , A ' * d e f i n e d b y a , a . T h e n a n d A ' * a r e g ~ - r e l a t e d .
Proof: Let %D q] be the l-parameter groups of transformations t' t
l I I !
= "rat ' ~ t = Td t " of X , X d e f i n e d b y a , a : q~t It is sufficient to
pr OF e
!
~o ~ = ~ o~ t t
in view of proposition 5. Z. i.
Now the p -equivariance of r signifies the commutativity of the
d i a g r a m
T G x X > X
[ I
G' ' w' X ' x X >
As al = 9 o a, the following diagram is also commutative:
-90 -
axl x
IRxX > G x X
i I
a Xlx, Xf I I IRx - >G xX
C o m p o s i n g t h e s e d i a g r a m s , we o b t a i n t h e c o m m u t a t i v e d i a g r a m
IRxX ~ >X
I I
X' ~I" X' ]Rx )
w h i c h p r o v e s ~ . x~ = x~' t t ~ ~0 " |
We a r e n o w in t he p o s i t i o n to p r o v e t h e f u n d a m e n t a l
T H E O R E M 5 . 6 . Z. L e t G be a L i e g r o u p , X a m a n i f o l d , RG t h e
L i e a l g e b r a of r i g h t i n v a r i a n t v e c t o r f i e l d s on G, and DX the L i e a l g e b r a
of v e c t o r f i e l d s on X . A n o p e r a t i o n v : G > A u t X d e f i n i n g X a_s_s
G - m a n i f o l d i n d u c e s a h o m o m o r p h i s m
v : R G > D X �9
I f B ~ RG a n d a ~ s G the 1 - p a r a m e t e r s u b g r o u p d e f i n e d by
t h e n o-(B) is t he K i l l i n g v e c t o r f i e l d on X d e f i n e d by a .
at = Ba t '
P r o o f : L e t B ~ R G , er(B) 6 D X . We s h o w t h a t B and ~ B ) a r e
p - r e l a t e d u n d e r t he e f f e c t of a m a p p : G > X . T h e t h e o r e m w i l l t h e n
f o l l o w by l e m m a 4. 4. Z.
-91-
C h o o s e
t h e d i a g r a m
x 6 X a n d define 0
p : G > X by p(g) = Vg(Xo) . T h e n
h g G = > G
I I pl
X g > X
i s c o m m u t a t i v e . C o n s i d e r i n g t h e o p e r a t i o n of G on G by l e f t t r a n s l a t i o n s ,
t h i s m e a n s t h a t p i s a n e q u i v a r i a n c e . L e t Q ~ s T h e c o r r e s p o n d i n g
K i l l i n g v e c t o r f i e l d s on G a n d X a r e p - r e l a t e d b y l e m m a 5 . 6 . 1 . T h e
K i l l i n g v e c t o r f i e l d on G d e f i n e d b y a i s by p r o p o s i t i o n 5 . 5 . 3 t h e e l e m e n t
B ~ R G . B u t t h e K i l l i n g v e c t o r f i e l d on X d e f i n e d b y a i s j u s t ~(B) a n d
t h e r e f o r e B a n d �9 (B) a r e p - r e l a t e d . I
D e n o t e by KX t h e s e t of K i l l i n g v e c t o r f i e l d s on t h e G - m a n i f o l d X .
T h e n KX = 3 r n ~ a n d by t h e o r e m 5 . 6 . 2 KX i s a L i e a l g e b r a .
E x a m p l e 5 . 6 . 3. L e t G o p e r a t e on G by l e f t t r a n s l a t i o n s . By
p r o p o s i t i o n 5 . 5 . 3 t h e L i e a l g e b r a of K i l l i n g v e c t o r f i e l d s i s RG and t h e
h o m o m o r p h i s m o-: R G > R G t h e i d e n t i t y 1 R G .
E x a m p l e 5 . 6 . 4 . L e t p : G > G ' b e a h o m o m o r p h i s m a n d
e I T: G > B i j G ' t h e o p e r a t i o n of G o n d e f i n e d by Tg - L p ( g } .
B y e x a m p l e 5 . 5 . 4 t h e L i e a l g e b r a of K i l l i n g v e c t o r f i e l d s i s a s u b a l g e b r a
of R G ' a n d t h e h o m o m o r p h i s m R ( p ) : R G > R G ' t h e h o m o m o r p h i s m
of t h e o r e m 5 . 6 . 2 .
- 9 2 -
E x a m p l e 5 . 6 . 5 . L e t V b e a f i n i t e - d i m e n s i o n a l ] R - v e c t o r s p a c e a n d
c o n s i d e r the n a t u r a l r e p r e s e n t a t i o n of G = G L ( V ) in V. We i d e n t i f y R G
and IX] w i t h G e b y t h e c a n o n i c a l i s o r n o r p h i s m s . W e k n o w f r o m
p r o p o s i t i o n 4. 3. 8 t h a t t h e n L G = s B u t b y c o r o l l a r y 4 . 6 . 8 , w e a l s o
h a v e RG = (LG) ~ T h e r e f o r e RG = ( s (V)) ~ a f t e r i d e n t i f i c a t i o n w i t h G e
N o w w e s e e f r o m e x a m p l e 5 . 5 . 5 , t h a t t he h o m o m o r p h i s m ~: R G > KV
O in t h i s c a s e i s j u s t t h e c a n o n i c a l m a p ( s (V)) > D r , a s s i g n i n g t o a n y
e n d o m o r p h i s m A ~ s (V) the v e c t o r f i e l d A* ---- . ~ V �9
c o n s e r v e s t h e b r a c k e t , i s s e e n d i r e c t l y as f o l l o w s .
Then their bracket in (s (V)) ~ is
T h a t t h i s map
L e t A I, A 2 6 s
AzA 1 - ARA I. On the other hand
* * d A* )(v)A* _ (d * )(v) * [A 1, A2 ] = (iv 2 v Iv dvAl A2
v v v
b y t he s a m e f o r m u l a a s in e x a m p l e 4. 3. 8.
Bu t
d , , * * = A A = A2AIV ( d-vA2v)lV)Al v 2 v i v
and t h e r e f o r e
[A~, A 2 ] = (A2A 1 AIA2)v , q.e.d. v
R ( v ) : R G
�9 : R G
one (s 0
C o n s i d e r m o r e g e n e r a l l y a r e p r e s e n t a t i o n of t he L i e g r o u p G in V.
T h e h o m o m o r p h i s m 7 : G > G L ( V ) i n d u c e s a h o m o m o r p h i s m
0 > R(GL(V)) = (s (V)) and the induced homornorphism
> DV is just the composition of this homomorphism with the
> DV d e s c r i b e d b e f o r e .
- 9 3 -
W e o b s e r v e t h a t t h e g e n e r a l s i t u a t i o n f o r a G - m a n i f o l d X i s i n s o m e
s e n s e s i m i l a r t o t h a t o f e x a m p l e 5 . 6 . 5 . T h e o p e r a t i o n T: G > A u t X
i n d u c e s b y c o m p o s i t i o n w i t h t h e n a t u r a l h o m o m o r p h i s m ( i t ' s e v e n a n
i s o m o r p h i s m ) a r e p r e s e n t a t i o n G > A u t C X of G in C X . T h e t r o u b l e
i s t h a t C X i s v e r y b i g a n d one c a n n o t j u s t s p e a k of t h e d i f f e r e n t i a b i l i t y
of t h i s m a p . B u t i f w e d o n ' t c a r e a b o u t t h i s , w e c a n v i e w t h e i n d u c e d
h o m o m o r p h i s m R G > D X ~- > OX i n t o t h e L i e a l g e b r a of o p e r a t o r s
on X a s t h e r e p r e s e n t a t i o n of R G i n C X i n d u c e d b y t h e r e p r e s e n t a t i o n
of G i n C X , e x a c t l y a s i n e x a m p l e 5 . 6 . 5 .
T h e L i e a l g e b r a of K i l l i n g v e c t o r f i e l d s on a G - m a n i f o l d i s b y d e f i n i t i o n
a L i e a l g e b r a of c o m p l e t e v e c t o r f i e l d s . I t i s n a t u r a l t o a s k i f a n y f i n i t e
d i m e n s i o n a l L i e a l g e b r a of c o m p l e t e v e c t o r f i e l d s on a m a n i f o l d X c a n b e
i n t e r p r e t e d a s t h e L i e a l g e b r a of K i l l i n g v e c t o r f i e l d s c o r r e s p o n d i n g t o t h e
o p e r a t i o n of s o m e L i e g r o u p G on X . W e d i s c u s s t h e p a r t i c u l a r c a s e of
a c o m m u t a t i v e L i e a l g e b r a . S e e R . P a l a i s [ 13 ] f o r a p o s i t i v e a n s w e r
t o o u r q u e s t i o n i n g e n e r a l .
L e t a c o m m u t a t i v e L i e g r o u p G o p e r a t e on X . A s T : R G > KX
i s a s u r j e c t i ve h o m o m o r p h i s m , t h e L i e a l g e b r a K X of K i l l i n g v e c t o r f i e l d s
is commutative.
We prove a converse
PROPOSITION 5.6.6.
Lie alge_bra of complete vectorfields on X.
Let K be a finite dimensional, commutative
T h e n t h e r e i s a n o p e r a t i o n
- 9 4 -
of t h e a d d i t i v e g r o u p of K on X , s u c h t h a t t h e L i e a l g e b r a K i s t h e L i e
a l g e b r a of K i l l i n g v e c t o r f i e l d s of t h i s o p e r a t i o n .
P r o o f : C o n s i d e r t h e a d d i t i v e s t r u c t u r e on K , m a k i n g i t a L i e
g r o u p w i t h L i e a l g e b r a K . We d e f i n e an o p e r a t i o n of K on X as f o l l o w s .
L e t A ~ K and ~0 t t h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s of X g e n e r a t e d
be p r o v e d .
v: K - - - > A u t X . We by A . T h e n we s e t 7 A = ~0 ~ , d e f i n i n g t h u s a m a p
s h o w 7 to be a h o m o m o r p h i s m .
L e t A, B ~ K g e n e r a t e ~ t ' ~ t r e s p e c t i v e l y . T h e n by p r o p o s i t i o n
5. 3. 6, X t = ~ t " ~t i s t h e 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s g e n e r a t e d
by A + B . T h e r e f o r e TA+ B = X1 = ~1 �9 ~1 = v A o 7 B , as w a s to
We o b s e r v e t h a t 7 t A = ~O t . N o w t h e 1 - p a r a m e t e r s u b g r o u p of K
c o r r e s p o n d i n g to A is t ~ - - - > tA and t h e r e f o r e t h e c o r r e s p o n d i n g K i l l i n g
vectorfield A $ on X satisfies
A* d { }/ x = d-i T tA (x) "t
d = - (x) = A �9
=0 dt ~Pt =0 x
T h i s s h o w s t h a t t he h o m o m o r p h i s m of t h e o r e m 5 . 6 . Z i s j u s t t h e i d e n t i t y
in t h i s c a s e . T h i s f i n i s h e s t h e p r o o f .
T h e h o m o m o r p h i s m ~: RG > DX d e f i n e d f o r a G - m a n i f o l d X
t h e o r e m 5 . 6 . Z r e f l e c t s p a r t i c u l a r p r o p e r t i e s of t he o p e r a t i o n 7 : G
N a m e l y
P R O P O S I T I O N 5 . 6 . 7 . L e t 7: G > Aut X d e f i n e X as a
G-manifold and ~: RG > DX be t h e i n d u c e d homomorphism.
i n
> Au t X .
- 9 5 -
(i) I f ~" i s i n j e c t i v e ( i . e . a n e f f e c t i v e o p e r a t i o n , t h e n ~ i s i n j e c t i v e .
( i i ) If ~ i s a f r e e o p e r a t i o n , t h e n a K i l l i n g v e c t o r f i e l d i s e i t h e r
e v e r y w h e r e z e r o o r n o w h e r e z e r o .
P r o o f : L e t B ~ RG a n d r 6 ~ G t h e c o r r e s p o n d i n g 1 - p a r a m e t e r
s u b g r o u p . T h e n ( ~ ' B ) ~ t ( x ) = ~%(x) , w h e r e t ~ t = T'0{t "
(i) S u p p o s e G'B = O .
q" b e i n g i n j e c t i v e , T '~ t
B = O .
D
T h e n ~ t ( x ) = O a n d ~ t ( x ) = x f o r e v e r y x a n d t .
= e a n d B = ~ = O , i . e . = 1 i m p l i e s r t e X O
( i i ) X
f o r e v e r y t ( x f i x e d ) b e c a u s e of t h e u n i q u e n e s s of t h e s o l u t i o n of
q t ( x ) = ( s w i t h ~o(X) = x . 'l" b e i n g f r e e , ~ t ( x ) = " ~ t
f o r s o m e x i m p l i e s ~ = e a n d a s b e f o r e B = O a n d ~" B = O . t
t h a t a f r e e o p e r a t i o n i s i n j e c t i v e , s o b y (i) B = O a n d 0" B = O a r e
S u p p o s e ( ~ B ) = 0 f o r s o m e x ~ X . T h e n ~o(X) = 0 a n d ~ t ( x ) = x
( X ) ----" X
( R e m e m b e r
e q u i v a l e n t s t a t e m e n t s . )
N o t e t h a t t h e i n j e c t i v i t y of %" : RG > DX d o e s n o t i m p l y t h e
i n j e c t i v i t y of T : G > A u t X .
E x a m p l e 5 . 6 . 8 . A h o m o m o r p h i s m ~ : G !
> G i n d u c e s a n o p e r a t i o n
G I "r = L o ~ of G on ( s e e e x a m p l e 5 . 6 . 4 ) a n d t h e i n d u c e d h o m o m o r p h i s m
�9 " i s j u s t R { ~ ) �9 R G > R G ' . R ( ~ ) c a n be i n j e c t i v e w i t h o u t T = L o
b e i n g i n j e c t i v e . I t i s s u f f i c i e n t t o e x h i b i t a n o n - i n j e c t i v e ~ : G > G
w i t h i n j e c t i v e ~ , : Ge > G'e " e
T h e c a n o n i c a l h o m o m o r p h i s m
IR > I R / Z = l r i s s u c h a c a s e .
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R e m a r k . L e t us d i s c u s s the r e s u l t s of t h i s s e c t i o n f r o m t h e h e u r i s t i c
po in t of v i e w a l r e a d y m e n t i o n e d s e v e r a l t i m e s , c o n s i d e r i n g Aut X a s a
L i e g r o u p . T h i s i s an e f f e c t i v e o p e r a t i o n on X and d e f i n e s t h e r e f o r e by
~ w
5 . 6 . 7 an i n j e c t i o n R ( A u t X) ~> D X . I t i s n a t u r a l to t h i n k t h a t t he i m a g e
is t h e s e t of a l l c o m p l e t e v e c t o r f i e l d s on X .
a L i e a l g e b r a , w h i c h d e s t r o y s m a n y h o p e s .
Bu t t h i s s e t is n o t n e c e s s a r i l y
O t h e r w i s e one w o u l d d e c r e t e
t h i s a l g e b r a to be R ( A u t X ) . In c a s e X is c o m p a c t , t h e r e is h o w e v e r no
p r o b l e m , e v e r y v e c t o r f i e l d b e i n g c o m p l e t e . Now a n y o p e r a t i o n T: G > Au t X
c a n be t h o u g h t to d e f i n e a h o m o m o r p h i s m R ( v ) : RG > R ( A u t X ) . T h e n
the h o m o m o r p h i s m ~: RG > DX of t h e o r e m 5 . 6 . Z i s j u s t the c o m p o s i t i o n
= r176 R(T).
Exercise 5.6.9. Let G be a commutative group operating on the
manifold X and KX be the Lie algebra of Killing vectorfields on X. Show
that every element of KX is invariant under the action of G on KX.
5. 7. Killin~ vectorfields and equivariant maps.
We shall study the compatibility with equivariant maps of the
homomorphism ~: RG > DX defined in section 5.6 for a left operation.
It is clear that considering a right operation of G on X, one obtains
similarly a homomorphism 0-: LG
We prove first the
LEMIV~A 5. 7. I. Let X, X'
> DX.
f
be manifolds, {P: X > X a map and
A, A I' , A, A z' pairs of <~-related vectorfields on X, X' . _If ~0 is
-97 -
r I
surjective, then A I = A 2 .
Proof: ~p is injective, because ~P~fl = O fz or fl * ~ = fz * ~
i m p l i e s f l = f2 i f q~ i s s u r j e c t i v e . B u t by d e f i n i t i o n of ~ 0 - r e l a t e d n e s s ,
w e h a v e c o m m u t a t i v e d i a g r a m s
C X ~ C X
A I i C X < C X
(i = i , z)
!
Injectivity of (0 $ clearly implies /~ = A Z
P R O P O S I T I O N 5. 7 . 2 . L e t X b e a G ~ w i t h r e s p e c t t o a
r i g h t o p e r a t i o n G ~ ' G ,~ I": > Aut X, X a - m a n i f o l d w i t h r e s p e c t t o a
right operation 7': G'~ > A u t X ' , p : G > G' a homomorphism and
cp : X > X' a__ p-equivariant map. Consider the induced homomorphisms
I I ! I
: L G > K X e . . . - > D X , �9 : L G > K X ~ D X of t h e o r e m 5 . 6 . Z.
I f e i t h e r G o p e r a t e s e f f e c t i v e l y on X o1" q~ i s s u r j e c t i v e , t h e r e i s a
u n i q u e m a p ~/: K X > K X ' m a k i n g c o m m u t a t i v e t h e d i a g r a m
(5" LG > KX
I I L ( P ) "y[
~ t I I
LG > KX
and this map is a homomorphism of Lie algebras.
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P r o o f : L e t A G L G . By p r o p o s i t i o n 5 . 6 . 1 , the v e c t o r f i e l d s
A* = ~(A) and oJ(L(p)A) are ~0-related.
Suppose first that G operates effettively on X, i.e. that
7: G ~ > Aut X is injective. By proposition 5.6. 7, o" is then also
injective. For A* 6 KX there is therefore a unique A C LG with
�9 (A) = A*. We define y(A*) = ~'(L(p)A).
Suppose now that (~ is surjective and let A 6 IX3.
and ~'(L(p)A)
defined by A ~ .
As A* = ~(A)
are (P-related, by lemma 5.7. I, o"(L(p)A) is uniquely
We can therefore define N as before.
and
l_f ~: X > X' is a
diagram commute s.
T h e u n i q u e n e s s of y f o l l o w s f r o m the s u r j e c t i • i t 7 of ~ : LG > KX
Y is a h o m o m o r p h i s m by l e m m a 4 . 4 . Z, q. e. d.
C O R O L L A R Y 5. 7. 3. L e t t he s i t u a t i o n be as in p r o p o s i t i o n 5 . 7 . Z.
p - e q u i v a r i a n t d i f f e o m o r p h i s m , t h e n t h e f o l l o w i n ~
LG > KX g > DX
LG' > KX'~ > DX'
Proof: If C ~ DX, then C and r are
~,/KX = Y , q.e.d.
We w a n t to a p p l y t h i s to t h e d i f f e o m o r p h i s m
by a r i g h t - o p e r a t i o n T: G ~
O - r e l a t e d .
7g_ I : X
> Aut X. First we remark
T h e r e f o r e
> X d e f i n e d
- 9 9 -
L E M M A 5. 7 . 4 . T g _ l : X---> X i_~s
Proof: The diagram
T -I X g L)
I
T -1 X g >
g - equivariant. g
X
I i T
X
c o m m u t e s f o r Y 6 G : T g ~ T - I = v OT g(Y) g g yg-1
B y c o r o l l a r y 5. 7. 3 t h e r e f o r e w e h a v e
P R O P O S I T I O N 5 . 7 . 5 . L e t X b e a G ~ w i t h r e s p e c t t o a
r i g h t o p e r a t i o n v : G ~ > A u t X a n d if: IX] > K X > D X t h e i n d u c e d
h o m o m o r p h i s m of t h e o r e m 5 . 6 . Z. T h e n t h e f o l l o w i n g d i a g r a m c o m m u t e s .
g-1 = "r.yg-1 = .rg_l o "ry
Adg
LG ~ >KX e" > DX
f r (
LG ~ > KX < > D X
W e j u s t r e m e m b e r t h a t A d g = L ( g g ) . F u r t h e r i t i s c l e a r t h a t
(T - 1 ) , = a g g d e f i n e s D X a s G - L i e a l g e b r a , v b e i n g a r i g h t o p e r a t i o n .
We o b t a i n t h e r e f o r e
T H E O R E M 5. 7 . 6 . L e t X b e a G ~ w i t h r e s p e c t t o a
G ~ homomorphi sm T : > Aut X. Consider the adjoint representation_
--of G __in LG and the operation of G __~ DX defined by ag = ( Tg_l), �9
Then the induced homomorphism ~: LG > DX is an equivariance.
- I 0 0 -
F o r A 6
t he f o r m u l a
LG and the c o r r e s p o n d i n g A $ = ~(A) w e h a v e t h e r e f o r e
( V g _ l ) , A *
T g
T h i s s h o w s in p a r t i c u l a r t h a t
K i l l i n g v e c t o r f i e l d s .
E x a m p l e 5. 7. 7.
c o m m u t a t i v i t y of t he d i a g r a m
= ~ ( A d g A ) .
t r a n s f o r m s K i l l i n g v e c t o r f i e l d s in
L e t G o p e r a t e on G by r i g h t t r a n s l a t i o n s . The
LG
Adg I
LG
> DG
I ~ (Rg -1) ,
> DG
e x p r e s s e d by t h e t h e o r e m is j u s t p r o p o s i t i o n 4. 7. 3,
F o r e f f e c t i v e o p e r a t i o n s , �9 i s a n i n j e c t i o n by p r o p o s i t i o n 5 . 6 . 7 .
T h e c o m m u t a t i v e d i a g r a m
LG
A d g I
L G
0" > DX
P
0" > DX
s h o w s t h a t
a d j o i n t r e p r e s e n t a t i o n of G .
L E M M A 5. 7. 8. T h e h o m o m o r p h i s m
if a n d on ly if t h e h o m o m o r p h i s m a: G
a: G > Au t DX c a n be i n t e r p r e t e d a s an e x t e n s i o n of t he
G ~ 7: > A u t X is i n j e c t i v e
> Aut DX d e f i n e d by
- I 0 1 -
(1 g
(Vg_l) * is injective.
Proof: a is the composition
G I > G T #
> Au t X > Aut DX
g ~ . . . . . . . . ~ > g-1 ~ r -1 ~ ' ~ ~ ( V - l ) , g g
a n d
a n d on ly if "r i s i n j e c t i v e , q. e . d.
R e m a r k . F o r a l e f t - o p e r a t i o n
a c o m m u t a t i v e d i a g r a m
I is b i j e c t i v e .
* is i n j e c t i v e , b e c a u s e r C Aut X w i t h ~ , = 1DX i m p l i e s ~0,x = 1Tx(X }
= 1 x . T h e r e f o r e a is i n j e c t i v e i f a n d o n l y i f * o v i s i n j e c t i v e i f
T: G > Au t X w e h a v e s i m i l a r l y
RG v > DX
f f
cr
RG > DX
w h e r e g = ( T ) , a n d ~ is t h e r e f o r e an e q u i v a r i a n c e w i t h r e s p e c t to t h e g g
( l e f t ) o p e r a t i o n s of G on RG and D X . L e t us t a k e up a g a i n o u r h e u r i s t i c
v i e w p o i n t of l o o k i n g at A u t X as a L i e g r o u p .
f r o m t h e l e f t on X , d e f i n i n g a h o m o m o r p h i s m
( s e e r e m a r k a t t he e n d of s e c t i o n 5 . 6 ) .
Aut X o p e r a t e s n a t u r a l l y
: R ( A u t X) > DX
As j u s t o b s e r v e d , f o r ~0 C Aut X
we h a v e a c o m m u t a t i v e d i a g r a m
-102-
R(Aut X)
t R%tt
R ( A u t X)
O"
. ~
O"
> DX
r
> DX
b e i n g an i n j e c t i o n , w e s e e t h a t (p~ : DX - - ~ DX i s a n e x t e n s i o n of
R ( % ) , w h i c h i s t h e a d j o i n t r e p r e s e n t a t i o n of A u t X in R ( A u t X ) . B y t h e
a r g u m e n t i n t h e p r o o f of l e m m a 5 . 7 . 8 w e s e e t h a t A u t X > R u t D X ,
s e n d i n g ~ t o ~ , , i s i n j e c t i v e . T h i n i s i n a c c o r d a n c e w i t h l e m m a 5. 7. 8,
r b e i n g a n i n j e c t i o n .
C h a p t e r
-103-
6. T H E E X P O N E N T I A L M A P O F A L I E G R O U P
T h e r e l a t i o n b e t w e e n 1 - p a r a m e t e r s u b g r o u p s of a L i e g r o u p G
a n d i n v a r i a n t v e c t o r f i e l d s on G s t u d i e d in s e c t i o n 5 . 4 i s u s e d to d e f i n e
a m a p exp : G e -- G , w h i c h t u r n s out to h a v e w o n d e r f u l p r o p e r t i e s .
6. l,
we i d e n t i f y LG wi th G e
e .
D E F I N I T I O N 6.1 . 1,
m a p d e f i n e d by
exp A
and w r i t e A E G e
D e f i n i t i o n a n d na tura l i t~r of exp . In the f o l l o w i n g , f o r c o n v e n i e n c e ,
f o r a t a n g e n t v e c t o r at
The exponential map e x p : G e - * G is t he
-- a 1 f o r A ~ G e
w h e r e a i s the 1 - p a r a m e t e r s u b g r o u p of
to t h e o r e m 5 . 4 . 6 . Wi th o u r c o n v e n t i o n
L e t a E s a nd d e f i n e fo r a t
c l e a r l y ~ ~ s a n d m o r e o v e r we h a v e
G d e f i n e d by A a c c o r d i n g
A = h 0 ,
~s = a s t " T h e n
L E M M A 6 . 1 . 2 . ~0 = th0 "
P r o o f : F o r f ~ CG we h a v e
d f(l~s) [ = ~sf(Cts t ) [ = th0 f ' ~0f = " ~ s = 0 s = 0
q, e. d.
This shows that exp(tA) = a t .
PROPOSITION 6. I. 3. exp((tl+tz)A )
Proof: a is a homomorphism. |
- e x p ( t l A ) � 9 e x p ( t z A )
- 1 0 4 -
T h e p r o p o s i t i o n 5 . 4 . 1 0 s h o w s t h a t f o r
[ A , B ] = 0 we h a v e t h e f o r m u l a
A , B ~ G w i t h e
e x p ( t ( A + B ) ) = exp( tA)" e x p ( t B )
a n d in p a r t i c u l a r f o r t = 1
e x p ( A + B ) = e x p A . e x p B , s h o w i n g
P R O P O S I T I O N 6 . 1 . 4. I f t h e L i e a l g e b r a L G o f G i.~s
c o m m u t a t i v e , t h e n e x p : G e -~ G i s a h o m o m o r p h i s m o f t h e a d d i t i v e
v e c t o r g r o u p G e i n t o G .
T o j u s t i f y t h e n o t a t i o n e x p , l e t u s c o n s i d e r t h e
a n d G
LG w i t h
a ~ s
t h i s c a s e
E x a m p l e 6 .1 . 5. L e t V be a f i n i t e d i m e n s i o n a l ~ - v e c t o r s p a c e
- G L ( V ) . In p r o p o s i t i o n 4 . 3 . 8 we h a v e s e e n t h a t i d e n t i f y i n g
G e , we h a v e LG = s N o w l e t A ~ s a n d
t h e c o r r e s p o n d i n g 1 - p a r a m e t e r s u b g r o u p , We s h o w t h a t in
(2O
e x p ( t A ) = a t = e t a = Z n"~', ( t A ) l n
nffi0
T o p r o v e t h i s , c o n s i d e r ~t = e t a = Zoo 1 n--0 ~ ( tA )n T h e n
~t = XOO n ( t A ) n - l A a n d ~0 1V B u t a l s o h t n=0 n-~. = ~t A = " = a t a 0 = a t A
a n d a 0 = i V , T h e r e f o r e a = ~ , a s b o t h s a t i s f y t h e s a m e d i f f e r -
e n t i a l e q u a t i o n w i t h t h e s a m e i n i t i a l c o n d i t i o n , q. e . d.
C o n s i d e r in p a r t i c u l a r V -" ~ . T h e n G L ( V ) " R * , t h e
m u l t i p l i c a t i v e g r o u p of r e a l n u m b e r s d i f f e r e n t f r o m z e r o , T h e L i e
a l g e b r a o f ~ * i s ~t w i t h t h e ( o n l y p o s s i b l e ) t r i v i a l L i e a l g e b r a
- 1 0 5 -
s t r u c t u r e .
m a p .
T h e m a p e x p : ~ -~ l~* i s j u s t t he o r d i n a r y e x p o n e n t i a l
We now s h o w t h e n a t u r a l i t y of e x p , i . e .
P R O P O S I T I O N 6.1 . 6. L e t p: G -~ G ' be a l o c a l h o m o m o r p h i s m .
T h e n t h e f o l l o w i n g d i a g r a m ( t a k e n in t h e s e n s e of l o c a l m a p s ) i s c o m m u t a t i v e .
P ~ G e .> C~
e e
G P > G'
P r o o f : L e t A ~ G e . I f a t = e x p ( t A ) i s t h e c o r r e s p o n d i n g !
s u b g r o u p of G , t h e n (p �9 a)(t)" It = 0 = p* ~0 = p~' 1 - p a r a m e t e r A e e "
T h e r e f o r e
a n d f o r t = 1
e x p ( t p , e A )
e x p ( p , A) e
A s an a p p l i c a t i o n ,
"- p ( e x p ( tA))
= p ( e x p A ) , q . e . d .
we o b t a i n
C O R O L L A R Y 6 . 1 . 7 . F o r g E G , A E LG
-1 e xp ( A dg A) = g exp A g .
we h a v e
of LG
P r o o f : Adg
wi th G e
= L(~g) by d e f i n i t i o n . N o t e t h a t a f t e r i d e n t i f i c a t i o n
the a d j o i n t r e p r e s e n t a t i o n o p e r a t e s in G e .
-106-
A n o t h e r a p p l i c a t i o n of p r o p o s i t i o n 6.1. 6 is the fo l lowing .
C o n s i d e r a f i n i t e - d i m e n s i o n a l F t - v e c t o r s p a c e V . The L i e a l g e b r a
of GL(V) is by p r o p o s i t i o n 4 . 3 . 8 e q u a l to s . Now the h o m o -
m o r p h i s m det : GL(V) -~ l~* i n d u c e s by p r o p o s i t i o n 4 .5 .11 the Lie
a l g e b r a h o m o m o r p h i s m t r :~(V) -~ ~t . The n a t u r a l i t y of the e x p o n e n t i a l
m a p p i n g p r o v e s
C O R O L L A R Y 6.1. 8. F o r any A ~ s
det e x p A -- e x p t r A
The i m a g e of the map e x p : G e -~ G is c o n t a i n e d in G o , the
c o n n e c t e d c o m p o n e n t of the i d e n t i t y of G . T h e f o l l o w i n g e x a m p l e
shows tha t exp n e e d not be s u r j e c t i v e e v e n f o r c o n n e c t e d G .
E x a m p l e 6.1. 9. Le t SL(2, •) be the g r o u p of 2 - r o w e d
q u a d r a t i c m a t r i c e s wi th d e t e r m i n a n t 1 . It i s a c o n n e c t e d L ie g roup .
We show that t h e r e is an e l e m e n t in SL(2, ~t) wh ich is not a s q u a r e � 9
T h i s wi l l i m p l y tha t exp is not s u r j e c t i v e .
Le t g E SL(2, R ) and c o n s i d e r i t s c h a r a c t e r i s t i c p o l y n o m i a l
de t (X3"d- g) = X 2 X t r g + det g w h e r e t r g d e n o t e s the t r a c e
of g . Now det g = 1 and t h e r e f o r e , by a t h e o r e m of l i n e a r
2 a l g e b r a , g - t r g . g + ; I d -- 0
ob ta in t r g2 2 = ( t r g ) - 2 > - 2
Z
C o n s i d e r the e l e m e n t
L =
�9 A p p l y i n g the t r a c e func t ion , we
D
of SL(2, i t) .
A s t r s < -2 , t h e e q u a t i o n
- 1 0 7 -
2 ~ - g h a s no s o l u t i o n .
R e m a r k .
t r a n s f o r m a t i o n s
C o n s i d e r i n g (Pt
to w r i t e a s f o r L i e g r o u p
r = e x p tA*
V i e w n o w , a s b e f o r e , a n o p e r a t i o n o f
C o n s i d e r a m a n i f o l d X a n d a 1 - p a r a m e t e r g r o u p o f
r o f X g e n e r a t e d b y a v e c t o r f i e l d A* o n X .
a s a 1 - p a r a m e t e r s u b g r o u p of A u t X , i t i s s u g g e s t i v e
G o n X a s a h o m o m o r p h i s m
G T > A u t X
a n d l e t a ~ s a n d
T h e n by d e f i n i t i o n
A ' b e t h e K i l l i n g v e c t o r f i e l d d e f i n e d b y a .
~- = e x p tA* a t
o r i f A ~ G e w i t h a t - e x p t A
T e x p tA = e x p tA*
T h i s e x p r e s s e s j u s t t h e c o m m u t a t i v i t y o f t h e d i a g r a m
G o- :> KX e
e x ~ I e x p
G r > A u t X
w h e r e 6: G e -*KXa- .DX i s t h e h o m o m o r p h i s m i n d u c e d by t h e o p e r a t i o n
G -~ A u t X . In t h e u p p e r r i g h t , we c a n n o t w r i t e DX , b e c a u s e o n l y
t h e c o m p l e t e v e c t o r f i e l d s a r e s e n t i n t o A u t X b y t h e e x p o n e n t i a l m a p .
e x p i s t h e r e f o r e , e v e n in t h i s e a s e , a n a t u r a l t r a n s f o r m a t i o n (o f s u i t a b l y
-I08-
d e f i n e d functors) .
6, Z. exp is a l o c a l d i f f e o m o r p h i s m at t he i d e n t i t y . We show now
P R O P O S I T I O N 6 . 2 . 1 . T h e t a n g e n t l i n e a r m a p
induced by exp: G e -. G .is the identity map.
exp, , 0: G e -~ G e
Proof: For A ~ G we have e
exp~ 0 A = exP, tAA It=O =
d exp ( tA) i =-gt-
t
d (tA)} {exP*tn t-0
;0
-" A , q . e , d ,
By the inverse function theorem we therefore have
THEOREM 6.2.2. There is an openneighborhood N O of
in G e and an open ..neighborhood N e of e in G such that
exp: N O -. N e .is an analytic diffeomorphism.
O
We d e n o t e by l o g : N e - . N O
d e f i n e s a c h a r t o f G at e .
t h e i n v e r s e m a p , T h e m a p log
D E F I N I T I O N 6. Z. 3. A c a n o n i c a l c h a r t of G i s a p a i r
(N e, log) of a n o p e n n e i g h b o r h o o d N e of e in G a n d a d i f f e o -
m o r p h i s m
e x p / N 0 .
log: N e - . l o g ( N e ) -- N O ~ G e w h i c h i s a n i n v e r s e of
-109-
An i m m e d i a t e a p p l i c a t i o n of t h e o r e m 6 . 2 . 2 is the fo l lowing
PROPOSITION 6. Z. 4, Let
c o n n e c t e d c o m p o n e n t of the identity,.
G O is c o m m u t a t i v e .
G be a Lie ~roup and G O the
If LG is c o m m u t a t i v e , t h e n
Proof : The i m a g e of exp c o n t a i n s an open n e i g h b o r h o o d U
of e in G . Any two e l e m e n t s in U c o m m u t e , as e x p : G e - . G
is a h o m o m o r p h i s m by p r o p o s i t i o n 6 .1 .4 . But U g e n e r a t e s G O ,
so that any two e l e m e n t s of G O c o m m u t e ,
T o g e t h e r with c o r o l l a r y 4 .6 , 9 we have t h e r e f o r e
G
T H E O R E M 6 . 2 . 5 . Let
is c o m m u t a t i v e if and only if
G be a c o n n e c t e d Lie ~roup.
LG ks c o m m u t a t i v e .
T h e n
LEMMA 6 . 2 . 6 , Let p : G -~ G' be a h o m o m o r p h i s m .
L(p): LG -~ LG' is i n j e c t i v e ( s u r j e c t i v e ) if and only if p•g is i n j e c t i v e
( s u r j e c t i v e ) for every, g ~ G ,
P roo f : p(g~/) = p(g)p(~/) i m p l i e s s ~/--e p , o (Lg ) , lg e
(Lp(g) )*e ' P~'e and psg = (Lp(g ) ) . e , ~ p - e ~ ( L g ~ e , q . e . d .
PROPOSITION 6 . 2 . 7 , Let G be a c o m m u t a t i v e c o n n e c t e d
Lie group . T h e n exp: L G - . G is s u r j e e t i v e .
P roof : We have s e e n in 6.1. 4 that for c o m m u t a t i v e LG
exp: LG -. G is a h o m o m o r p h i s m . Let G' be the i m a g e , Now
- I I 0 -
e x p , 0 i s t h e i d e n t i t y m a p a n d t h e r e f o r e e x p , A
i s o m o r p h i s m f o r e v e r y A E LG . T h e r e f o r e
( and c l o s e d ) s u b g r o u p of G , i . e . G' = G .
�9 by6, Z.6, an
G' is an open
(N e,
exp
C o n s i d e r a l o c a l h o m o m o r p h i s m
log} be a c a n o n i c a l c h a r t o f G .
, we h a v e f o r g @ N e
p : G -. G' . L e t
B y t h e n a t u r a l i t y 6 , 1 . 6 of
( * ) p(g) = e x p ( L ( p ) i o g g)
T h i s n e c e s s a r y c o n d i t i o n d e t e r m i n e s p / N e by L(p) a n d h a s t he
f o l l o w i n g a p p l i c a t i o n s ,
P R O P O S I T I O N 6 . 2 . 8 . L e t Pi: G -~ G' (i : 1, 3) be l o c a l
h o m o m o r p h i s m s , If t h e i n d u c e d a l g e b r a h o m o m o r p h i s m s
L(Pi ) : LG -. LG ' (i : 1, 2) c o i n c i d e , t h e n t h e r e e x i s t s a n o p e n
n e i g h b o r h o o d U o f e in G , on w h i c h P l a n d P2 c o i n c i d e .
P r o o f : T a k e U = N e a s t he d o m a i n of a c a n o n i c a l c h a r t
of G . T h e n ( * ) s h o w s p l (g ) = p2(g) f o r g E U .
C O R O L L A R Y 6, 2, 9. Let G
Pi: G - G' (i = I, 2) h o m o m o r p h i s m s ,
be c o n n e c t e d a n d
If L(Pl) : L(p2) {
o.{
t h e n Pl : PZ
P r o o f : A n y n e i g h b o r h o o d of e g e n e r a t e s
T h i s c o r o l l a r y of 6 . 2 . 8 i s e x p r e s s e d by s a y i n g t h a t t he
f u n c t o r L: s s i s f a i t h f u l on t he s u b c a t e g o r y of c o n n e c t e d L i e
g r o u p s a n d g l o b a l h o m o m o r p h i s m s .
- l l l -
A n a p p l i c a t i o n o f t h i s l a s t r e s u l t i s t h e
k e r A d
P R O P O S I T I O N 6 . 2 . 1 0 . L e t G b e c o n n e c t e d . T h e n
- Z G , w h e r e Z G i s t h e c e n t e r o f G .
L ( ~ g )
q . e . d .
P r o o f : B y d e f i n i t i o n A d = L o ~ , B u t a s s e e n b e f o r e ,
--- 1LG i m p l i e s ~g = 1G . T h e r e f o r e k e r A d -- k e r ~"
N o w c o n s i d e r t h e p r o b l e m of c o n s t r u c t i n g a l o c a l h o m o -
m o r p h i s m G --G' i n d u c i n g a g i v e n L i e a l g e b r a h o m o m o r p h i s m
LG - - L G ' . R e m e m b e r t h a t t h e i s o m o r p h i s m L qr -~ L ~ is
n o t i n d u c e d b y a n y h o m o m o r p h i s m ~ -- R , s o we o n l y c a n e x p e c t
t h e e x i s t e n c e o f a l o c a l h o m o m o r p h i s m w i t h t h e d e s i r e d p r o p e r t y .
-- Z G ,
P R O P O S I T I O N 6 . 2 . 1 1 , L e t G, G'
h: LG -* LG ' a L i e a l g e b r a h o m o m o r p h i s m .
a c a n o n i c a l c h a r t a t e
l o c a l h o m o m o r p h i s m
f o r m
be L i e g r o u p s . . a n d
L e t (N e , l o g ) be
in G . T h e r e s t r i c t i o n to N o f a e
G - - G' i n d u c i n g h i s n e c e s s a r i l ~ r o f t h e
p = e x p ~ h o l o g : N e - > G'
I f L G a n d LG ' a r e c o m m u t a t i v e , t h e n 9: N e -~G' d e f i n e d b y
t h i s f o r m u l a i s a l o c a l h o m o m o r p h i s m i n d u c i n g h.
P r o o f : W e h a v e a l r e a d y s e e n t h a t t h e r e s t r i c t i o n o f a l o c a l
h o m o m o r p h i s m G --G' t o N e i s n e c e s s a r i l y o f t h i s f o r m , By
6.1~ 4 f o r c o m m u t a t i v e L G e x p i s a h o m o m o r p h i s m G e -- G , s o
t h a t l o g i s a l s o a l o c a l h o m o m o r p h i s m G --G e . T h e r e f o r e p
i s a h o m o m o r p h i s m , L e t L(p ): L G -- LG ' b e t h e i n d u c e d h o m o m o r p h i s m .
It i s c l e a r t h a t P * e " G e -~ Ge
T h e r n a p p:N e -~G'
h: L G -~ L G ' i s a l o c a l h o m o m o r p h i s m G -~ G '
w i t h o u t s u p p o s i n g L G a n d LG ' c o m m u t a t i v e ,
o f t h i s r e q u i r e s a d e e p e r a n a l y s i s o f t h e s i t u a t i o n ,
-112-
i s j u s t h . T h i s s h o w s L(p ) - h , I
d e f i n e d b y a L i e a l g e b r a h o m o m o r p h i s m
i n d u c i n g h e v e n
B u t a d i r e c t p r o o f
S e e a l s o t h e c o m m e n t s
in s e c t i o n 6 . 4 , a f t e r p r o p o s i t i o n 6, 4 . 2 , We s h a l l c o n s t r u c t in C h a p t e r
7, by a d i f f e r e n t m e t h o d , a l o c a l h o m o m o r p h i s m G -~ G' i n d u c i n g a
g i v e n L i e a l g e b r a h o m o m o r p h i s m
t h i s w i l l p r o v e t h a t t h e p : N e -~ G'
l o c a l h o m o m o r p h i s m ,
h: LG -~ LG ' ( s e e 7 . 2 . 3 ) . B y u n i c i t y ,
d e f i n e d in p r o p o s i t i o n 6 . 2 , 11 i s a
S u p p o s e t h a t a L i e g r o u p G E x e r c i s e 6 . 2 . 1 2 . o p e r a t e s
e f f e c t i v e l y on t h e m a n i f o l d X w i t h r e s p e c t t o 1": G -* A u t X , a n d
l e t KX be t h e L i e a l g e b r a o f K i l l i n g v e c t o r f i e l d s o n X , S h o w
t h a t g ~ G s a t i s f i e s ( V g ) . A = A f o r e v e r y A ~ KX i f a n d
o n l y i f g i s in t h e c e n t r a l i z e r o f t h e i d e n t i t y c o m p o n e n t G O in G .
6 . 3 , U n i c i t ~ o f L i e g r o u p s t r u c t u r e , We b e g i n by p r o v i n g t h e f o l l o w i n g
i m p o r t a n t
P R O P O S I T I O N 6 . 3 , l, L e t G be a L i e g r o u p a n d a: R -~ G
a h o m o m o r p h i s m in t h e a l g e b r a i c s e n s e , w h i c h i s c o n t i n u o u s . T h e n
t h e r e e x i s t s a n A E L G , s u c h t h a t a t = e x p tA a n d h e n c e a i s
a n a l y t i c , i . e . a 1 - p a r a m e t e r s u b g r o u p of G .
P r o o f : L e t (U, l og ) b e a c a n o n i c a l c h a r t o f
n e i g h b o r h o o d of e i n G w i t h VV c U ,
G a n d V a
-113-
2 2 L e t g ~ V . T h e n g ~ VV c U a n d l o g g , l o g g
a r e d e f i n e d . C o n s i d e r t h e 1 - p a r a m e t e r s u b g r o u p o f G t - - - - ~ > e x p ( t l o g g) .
F o r t = 1 in p a r t i c u l a r e x p log g = g . g2 i s o n t h i s 1 - p a r a m e t e r
s u b g r o u p a n d g2 = e x p ( 2 log g ) . O n t h e o t h e r h a n d , g2 = e x p l o g g2 ,
2 2 1 g2 a s g ~ U . T h e r e f o r e l o g g = 2 l o g g o r g = e x p ( - ~ l o g ) ,
2 w h i c h m e a n s t h a t g i s u n i q u e l y d e t e r m i n e d b y g .
N o w c o n s i d e r t h e c o n t i n u o u s h o m o m o r p h i s m a: R -~ G . T h e r e
e x i s t s a n E > 0 s u c h t h a t a t E V f o r e v e r y t w i t h It l ..< E �9
W e c a n s u p p o s e ~ = 1 ( o t h e r w i s e c h a n g e t h e p a r a m e t e r t b y Xt
s u c h t h a t t h e n e w p a r a m e t e r i s d e f i n e d f o r a b s o l u t e v a l u e s < 1 ) .
D e f i n e a 1 = g ~ V . N o w e x p ( ~ l o g g) i s a s q u a r e r o o t o f g
in V , b y t h e p r e c e d i n g t h e r e f o r e t h e u n i q u e o n e . T h i s s h o w s
a l / z = e x p ( l og g ) , o r w i t h l o g a l = A a l s o l og a l / 2 = -~ A .
1 B y i t e r a t i o n o n e o b t a i n s l o g a ( l / Z n ) = ~'~ A a n d b y a d d i t i o n
tog a/n \- --P--- A f o r 0 < p _-- < 2 n 2 n = ,
p G I N *
T h i s s h o w s
0 < r < 1
l o g a r = r A f o r e v e r y d y a d i c r a t i o n a l r w i t h
a n d b y c o n t i n u i t y l o g a t = tA . T h i s p r o v e s a t = e x p tA . I
T o g e n e r a l i z e 6 . 3 . 1 to a r b i t r a r y h o m o m o r p h i s m s , we s h a l l m a k e
u s e o f
L E M M A 6, 3 . 2 . L e t G b e a L i e g r o u p . S u p p o s e G e i s a
d i r e c t p r o d u c t M • N o f v e c t o r s u b s p a c e s M, N. T h e n t h e m a p
r N ~ G d e f i n e d b y ~ ( A , B ) = e x p A e x p B f o r A E M ,
B C N i s a l o c a l d i f f e o m o r p h i s m a t 0 .
-114-
P r o o f : In v i e w of the i n v e r s e f u n c t i o n t h e o r e m , we o n l y h a v e
to show tha t r M~( N -~ G e is an i s o m o r p h i s m . Now
~) = m ~ ( e x p / M • e x p / N ) , w h e r e m d e n o t e s t he m u l t i p l i c a t i o n
m : G • G -~ G . T h e r e f o r e f o r ( X , Y ) E M ~ , N we h a v e
r ( X , Y ) = m . 0 ( e , e ) ( exPa0X 'exP$0 Y) = exP~0X + exPa0Y
= X + Y ,
a s e x p . 0 = i d e n t i t y by 0 . 2 . 1 .
l e m m a is p r o v e d .
H e n c e ~$0 i s t he i d e n t i t y and the
R e m a r k . T h e l e m m a g e n e r a l i z e s of c o u r s e to the c a s e of a
d e c o m p o s i t i o n G e = M 1 ~ . . . X M n
s p a c e s M i c G e .
f o r a f i n i t e n u m b e r of v e c t o r -
We a l s o s t a t e t he f o l l o w i n g
L E M M A 0, 3 . 3 . Le t G, G' be L i e g r o u p s and p : G -~ G'
a h o m o m o r p h i s m in t he a l g e b r a i c s e n s e , I f p i s d i f f e r e n t i a b l e
( a n a l y t i c ) a t e , t h e n p i s e v e r y w h e r e d i f f e_ ren t i ab le ( a n a l y t i c ) .
P r o o f : C l e a r f r o m p~ Lg = Lp(g)0 p o |
We a r e now a b l e to p r o v e
T H E O R E M 6 . 3 . 4 . L e t G , G ' be L i e g r o u p s and p : G -~ G'
a h o m o m o r p h i s m of g r o u p s in t h e a l g e b r a i c sen_se, w h i c h i s c o n t i n u o u s .
T h e n p i s a n a l y t i c , i . e . a h _ o m o m o r p h i s m of L i e g r o u p s .
-115 -
Proof: L e t A ~ G �9 e
T h e c o r r e s p o n d e n c e t ~--~>p (exp tA)
i s a c o n t i n u o u s h o m o m o r p h i s m lit -~ G , H e n c e t h e r e e x i s t s a n
A' ~ G' such that p(exp tA) = exp tA' e �9
define
Then
Now le t A i ( i = l , . . , , n = d i m G) be a b a s e of Ge , a n d
A'. a s t he v e c t o r in G' w i th p ( e x p tAi) = exp tA'. t e l "
rl n P( i= l exp t iA i ) = II n e x p t.A'. i=l t t '
Now the m a p ~ : IR n - , G d e f i n e d by ~(t 1, t n ) - I I n e x p t . A , �9 " " ' i=l I t
is a local diffeomorphism at 0 by the remark following 6, 3. Z.
T h e r e f o r e t h e r e e x i s t s a n e i g h b o r h o o d V of e in G s u c h t h a t
n e v e r y g C V m a y be w r i t t e n in t h e f o r m g - I I i _ l e x p t iA i , w i th
�9 H n t i d e p e n d i n g a n a l y t i c a l l y of g T h e f o r m u l a p( i=l exp t iA i )
= l'l n exp t.A'. now s h o w s t h a t p i s a n a l y t i c a t t h e n e u t r a l e l e m e n t . i:l t
T h e r e f o r e p i s a n a l y t i c by 6 . 3 . B.
: G -~ G' R e m a r k . T h e m a p P* e e e
P~'Ai~'e = A: f o r i = l , n ~. s �9 �9 , �9
i s j u s t g i v e n by
C O R O L L A R Y 6 . 3 . 5 � 9 L e t G, G' b e L i e g r o u p s � 9 If
a s t o p o l o g i c a l g r o u p s , t h e n G = G' a s L i e g r o u p s .
G = G I
P r o o f : If G - G' a s t o p o l o g i c a l g r o u p s , t he i d e n t i t y m a p
i s a h o m e o m o r p h i s m , a n d t h e r e f o r e a d i f f e o m o r p h i s m b y 6, 3 . 4 � 9
T h i s s h o w s t h a t t he L i e a l g e b r a of a L i e g r o u p i s in f a c t a
p r o p e r t y of t h e u n d e r l y i n g t o p o l o g i c a l g r o u p � 9
T h i s r a i s e s t h e q u e s t i o n : w h i c h t o p o l o g i c a l g r o u p s c a n be
t u r n e d i n to L i e g r o u p s , i, e . h a v e a n a n a l y t i c s t r u c t u r e c o m p a t i b l e
w i th the g r o u p s t r u c t u r e a n d s u c h t ha t the c o r r e s p o n d i n g t o p o l o g y
-116-
coincides with the given one ?
It has been proved by A. M. Gleason, Ann. of Math. 56
(1952), 193-212, that a topological group G which is locally com-
pact, locally connected, metrisable and of finite dimension, is a
Lie group.
6.4. Application to fixed points on G-manifolds. As an application
we give, in this section, a characterization of fixed points on a
G-manifold by the Lie algebra of Killing vectorfields. We begin with
the following
L E M M A 6 . 4 . 1 . L e t X be a m a n i f o l d . A a v e c t o r f i e l d
a n d r a l o c a l 1 - p a r a m e t e r g r o u p of l o c a l t r a n s f o r m a t i o n s ~ e n e r a t e d
b_~ A . A p o i n t x E X is a f i x p o i n t of ever~r t r a n s f o r m a t i o n @t
= 0 if a n d onl~r if A x
Proof: ~ 0 t ( x ) = x f o r e v e r y t i m p l i e s ~t(x)[ t___0-0 a n d
t h e r e f o r e A x -- 0 . C o n v e r s e l y , l e t Ax = 0 . T h e n t h e d i f f e r e n t i a l
e q u a t i o n ~ t (x ) -- A t(x ) h a s t h e s o l u t i o n (;.t(x) = x f o r e v e r y t ,
a n d the s o l u t i o n i s u n i q u e .
E x a m p l e 6 . 4 . 2 . On the t w o - s p h e r e S 2 e v e r y v e c t o r f i e l d h a s
a z e r o . T h e r e f o r e e v e r y 1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s h a s a
f i x p o i n t .
M o r e g e n e r a l l y , le t X be a c o m p a c t m a n i f o l d . T h e v a n i s h i n g
of t h e E u l e r - P o i n c a r ~ c h a r a c t e r i s t i c % i X ) is n e c e s s a r y a n d s u f f i c i e n t
f o r t he e x i s t e n c e of a v e c t o r f i e l d w i t h o u t z e r o s . ( R e m e m b e r t h a t
-117-
v e c t o r f i e l d m e a n s d i f f e r e n t i a b l e v e c t o r f i e l d ) . T h e r e f o r e e v e r y
1 - p a r a m e t e r g r o u p of t r a n s f o r m a t i o n s of a c o m p a c t m a n i f o l d X
wi th ~,,(X) ~ 0 h a s a f i x p o i n t .
P R O P O S I T I O N 6 . 4 . 3 . L e t t h e c o n n e c t e d L i e g r o u p G
o p e r a t e on X ~ v: G -~Aut X a n d le t KX be the L i e a l g e b r a
of K i l l i n g v e c t o r f i e l d s on X . A po in t x r X i s G - i n v a r i a n t
if a n d o n l y if A* = 0 f o r e v e r y A* ~ KX. X
P r o o f : S u p p o s e x G - i n v a r i a n t . F o r a n y a ~ s we
h a v e v iX) - X a n d t h e r e f o r e A* = 0 f o r t he c o r r e s p o n d i n g a t x
K i l l i n g v e c t o r f i e l d on X . S u p p o s e c o n v e r s e l y A* = 0 f o r x
e v e r y A* E K X , B y l e m m a 6 . 4 . 1 , f o r e v e r y a ~ s we h a v e
t h e r e f o r e TatiX ) = x . exp b e i n g a l o c a l d i f f e o m o r p h i s m , t h e r e
i s a n o p e n n e i g h b o r h o o d U of e in G s u c h t h a t VgiX) = x
f o r g ~ U . A s G is c o n n e c t e d , U g e n e r a t e s G a n d
VglX) = x f o r e v e r y g ~ G . |
C o n s i d e r in p a r t i c u l a r a f i n i t e - d i m e n s i o n a l E - v e c t o r s p a c e
V a n d a r e p r e s e n t a t i o n T : G -~ G L i V ) of t h e c o n n e c t e d L i e g r o u p
G in V . A p o i n t v ~ V is G - i n v a r i a n t i f a n d o n l y i f
ASv = 0 f o r e v e r y A* ~ KV o Btlt , by e x a m p l e 5 . 5 . 5 t he K i l l i n g
v e c t o r f i e l d A* c o r r e s p o n d i n g to a E s is d e f i n e d by
A* = v.~ h0V.re . T h e r e f o r e v E V is G - i n v a r i a n t i f a n d o n l y i f v
A ) v = 0 f o r e v e r y A E G C o n s i d e r i n g the i n d u c e d r e p r e s e n t - (T , e e �9
a t i o n L i T ) : L G - ~ s of LG in V , we s e e t h a t v ~ V i s
G - i n v a r i a n t i f a n d o n l y if ( L ( v ) A ) v = 0 f o r e v e r y A ~ LG .
-I18-
T h i s m o t i v a t e s t h e f o l l o w i n g
V a
V ,
(A)v
D E F I N I T I O N 6 . 4 . 4 . L e t /k be f i e l d , M a A - L i e a l g e b r a ,
A - v e c t o r s p a c e a n d o-: M -~ s a r e p r e s e n t a t i o n of M in
A n e l e m e n t v ~ V i s c a l l e d i n v a r i a n t o r M - i n v a r i a n t if
= 0 f o r e v e r y A ~ M .
We h a v e t h e r e f o r e
P R O P O S I T I O N 6 . 4 . 5 . L e t V be a f i n i t e d i m e n s i o n a l
~-vectorspace, G a c o n n e c t e d L i e g r o u p , "r:G -. G L ( V ) a
r e p r e s e n t a t i o n of G i n V a n d L(T): LG -" s t h e i n d u c e d
r e p r e s e n t a t i o n of L G in V . A n e l e m e n t v ~ V i s G - i n v a r i a n t
i f a n d o n l y i f i t i s L G - i n v a r i a n t .
We a p p l y n o w p r o p o s i t i o n 6 . 4 . 3 f o r t h e c a s e o f a c o m m u t a t i v e
G a n d p r o v e
P R O P O S I T I O N 6 . 4 . 6 . L e t X be a m a n i f o l d . T h e f o l l o w i n g
c o n d i t i o n s a r e e q u i v a l e n t .
(1) F o r a n y n - d i m e n s i o n a l c o m m u t a t i v e , c o n n e c t e d L i e g r o u p
G a n d anY__opera t ion T : G -* A u t X t h e r e i s a G - i n v a r i a n t p o i n t x
(Z) A n y o p e r a t i o n o f t h e a d d i t i v e g r o u p • n o n X h a s a
X .
f i x p o i n t .
(3) For any n-tuple
[Ai, Aj] = 0 ( i , j = 1 , . . . , n )
for i = I .... , n .
A 1 , . . , , A n of c o m p l e t e v e c t o r f i e l d s w i t h
t h e r e e x i s t s a p o i n t x ~ X w i t h A. t
X
= 0
-119-
on X wi th
a d d i t i v e g r o u p
5 . 6 . 6 t h e r e i s a n o p e r a t i o n of
Proof: (i) =>(2) is clear.
(2) ---->(3) L e t A1,
[Ai, A j ] = 0 f o r
K g e n e r a t e d by
K
. . . , A n be c o m p l e t e v e c t o r f i e l d s
i, j = 1 , . . . , n . C o n s i d e r t he
A 1 �9
o n
� 9 A n . B y p r o p o s i t i o n
X , s u c h t h a t the c o m m u t a t i v e
L i e a l g e b r a K is t he a l g e b r a of K i l l i n g v e c t o r ' f i e l d s of t h i s o p e r a t i o n .
K i s i s o m o r p h i c to t he a d d i t i v e g r o u p R k f o r s o m e k < n .
T h e o p e r a t i o n of K on X
a d d i t i v e g r o u p ~ n on X
Now by p r o p o s i t i o n 6 . 4 . 3 ,
p a r t i c u l a r A. = 0 f o r l x
(3) =>(1).
i n d u c e s t h e r e f o r e a n o p e r a t i o n of t he
�9 w h i c h h a s a f i x p 0 i n t
x i s a z e r o f o r a n y A
i = l . . . . , n �9
L e t G be a n - d i m e n s i o n a l c o m m u t a t i v e ,
x by h y p o t h e s i s .
K a n d in
[Ai, A j
i, j = 1, . . . . n. B y h y p o t h e s i s , t h e r e e x i s t s a c o m m o n z e r o
X f o r t h e s e v e c t o r f i e l d s . T h e n x i s a z e r o f o r a n y K i l l i n g
T : G - ~ A u t X a n o p e r a t i o n of G on X
the L i e a l g e b r a of K i l l i n g v e c t o r f i e l d s . We h a v e d i m KX < n .
] = 0
c o n n e c t e d L i e g r o u p G,
a n d KX
L e t A 1 , � 9 n be a s y s t e m of g e n e r a t o r s of i KX. T h e n
f o r
x
a n d by p r o p o s i t i o n 6�9 4 . 3 x is G ~ i n v a r i a n t . | v e c t o r f i e l d
If one (and h e n c e a n y ) of t h e c o n d i t i o n s in p r o p o s i t i o n 6, 4 . 6 i s
s a t i s f i e d f o r a c e r t a i n n , t h e n it i s c l e a r l y a l s o s a t i s f i e d f o r e v e r y
m < n .
T h e u n d e r l y i n g m a n i f o l d of a c o m m u t a t i v e L i e g r o u p G of
d i m e n s i o n n c l e a r l y d o e s n ' t s a t i s f y a n y of t h e c o n d i t i o n s of t h e
p r o p o s i t i o n 6 . 4 . 6 . T h e o p e r a t i o n of G O on G by t r a n s l a t i o n s
h a s no f i x p o i n t s a n d a n y n - t u p l e of i n v a r i a n t v e c t o r f i e l d s A 1, . . . , A n
- 1 2 0 -
s a t i s f i e s [Ai , A j ] = 0 w i t h o u t a n y one of t h e v e c t o r f i e l d s h a v i n g
a z e r o .
E x a m p l e 0 . 4 . 7 . C o n s i d e r t h e t w o - s p h e r e S 7 F o r n= l
t he c o n d i t i o n (3) of p r o p o s i t i o n 6 . 4 . 6 j u s t s a y s t h a t e v e r y v e c t o r f i e l d on
S 2 h a s a z e r o , w h i c h i s a c o n s e q u e n c e of X.(S z) ~ 0 . F o r
n = 2, t he c o n d i t i o n (Z) w a s s h o w n to be s a t i s f i e d by E . L . L i m a ,
P r o c . A M S , Vol . 15 (1964), p. 138-141.
6 . 5 . T a y l o r ' s f o r m u l a . In t h i s s e c t i o n we m a k e e s s e n t i a l u s e of
t h e a n a l y c i t y of G . We r e c a l l t h a t in t h i s c h a p t e r we i d e n t i f y
LG wi th G e .
P R O P O S I T I O N 0 . 5 . 1 . Le t_ f ~ CG be a f u n c t i o n a n a l y t i c
a t g E Gj a n d A E LG . T h e n t h e r e e x i s t s a n ~ > 0 s u c h t h a t
O0 tn
f (g exp tA) = ~ [ A n f ] ( g ) fo_r Itl <
n--0
P r o o f :
T h i s p r o v e s
(,)
F i r s t � 9 l e t f ~ CG . B y p r o p o s i t i o n 5 . 4 . 7
[Af ] (g ) = - - ~ f ( g e x p tA) t - 0
] [ A n f ] ( g ) = f(g exp tA t = 0
f o r n = 1. We p r o v e ( * ) f o r a r b i t r a r y n by i n d u c t i o n .
-121-
[An+If](g) = [An(Af)](g) = (Af)(g exp tA)
= f(g exp tA exp uA t=0 u=0
= [(~v)n ~v f(g exp VAgv=0
t=0
with t + u = v, showing thus ( * ) ,
If f i s now a n a l y t i c at g , t hen t h e r e e x i s t s a n r > 0
such tha t fo r It I <
cO
f ( g e x p t A ) = ~ tn --6!-.
n = 0
[(+)~ t] f(g exp tA t -'0
(3O
- t n . q . e . d .
n=0
We app ly t h i s to p r o v e the
P R O P O S I T I O N 6 . 5 . Z. Le t O(t 3) deno te a v e c t o r in
such tha t fo r an E > 0 t- ~ O(t 3) i s bounded and a n a l y t i c for
T h e n for A , B E LG and s u f f i c m n t l y s m a l l t
t z (i) exp tA exp tB = exp [t(A+B) + -'Z" [A, B] + O(t3)}
(ii) exp tA exp tS exp (-tA) = exp [tB + tZ[A,B]+ O(t3)]
(iii) exp (-tA) exp (-tB) exp tA exp tB
LG
Itl
= exp [tZ[A,B]+ O(t3)]
< E
Pr oo f : Le t f be a n a l y t i c a t e .
[Anf](e) ; f (exp tA t - -O
We have shown
- 1 2 2 -
We ob ta in t h e r e f o r e
d n d m [AnBmf](e) "[(~[~") ('~) f(exp tA exp s B ) ] t-O s--O
The T a y l o r s e r i e s fo r f(exp tA exp sB) is t h e r e f o r e
t n f(exp tA exp sB) = ~ --nT
n , m > 0
m s [ A n B m f ] ( e )
and for t = s
(1) f (exp tA exp tB) = /. n,m >0
tn+m [AnBmf](e)
The c o e f f i c i e n t of t
the c o e f f i c i e n t of t 2
is {[Af](e) + [Bf](e)} ,
[ l-~.-[A2f](e) + [ABf](e) + is ~ [ B Z f ] ( e ) }
On the o the r hand , by t h e o r e m 6 . 2 . 2 for s u f f i c i e n t l y s m a l l t
e x p t A exp tB = exp Z(t)
with Z: I -~G e , I an open i n t e r v a l of
a n a l y t i c a t O , Z(O) = 0 . T h e n
Z(t) = tZ 1+ t 2 Z z + O(t 3)
E c o n t a i n i n g 0 , Z
fo r f ixed Z l , Z 2 @ G e .
Take any func t ion f which i s l i n e a r in a c a n o n i c a l c h a r t
at e o T h e n it i s a n a l y t i c at e and
-123-
(z) f( e x p tA exp tB) = f (exp{ tZ 1 + tZZz + O(t3)})
= f ( e x p [ t Z 1 + tZZz }) + O' (t 3) (30
= ~= i [(tZ I+ tZzz)nf](e) + O'(t 3) 0-~.
O' (t) b e i n g a r e a l n u m b e r s u c h t h a t f o r a n E
i s b o u n d e d a n d a n a l y t i c f o r It[ < E .
i s
T h e c o e f f i c i e n t of t i s [ Z l f ] ( e )
[[Z2f](e) + ~-[Z~f](e)} .
C o m p a r i n g t h i s w i th t he c o e f f i c i e n t of
we o b t a i n
> 0 i O' t- 7 (t)
t h e c o e f f i c i e n t of t z
t a n d t z in (1) ,
[Zlf](e ) = [(A+B)f}(e)
1 [Zzf](e) = [ ~[A, B]f}(e)
T h i s b e i n g t r u e f o r a n y f u n c t i o n f
c a n o n i c a l c h a r t a t e , we h a v e t h e r e f o r e
T h i s s h o w s
exp tA exp tB
Z 1 = A+B
1 Z z = TEA, B]
t 2 = eXp Z(t) = exp [t(A+B) +-,2-
( i i ) i s o b t a i n e d by (i) a s f o l l o w s
w h i c h i s l i n e a r i n a
[A,B] + O(t3)}
i.e. (i)
exp tA exp tB exp( - tA)
-124-
t [ A , B ] + O(tZ)}l ) exp ( - tA) = exp(t[(A + B ) + ~--
t z = exp(t([ }l-A)+ --2-[[ ]I' -A]+ O(t3)) t 2 t 2
= exp ((tB + - T [ A , B]) + -~--[A, B] + O(t3))
exp (tB + t2[A, B] + O(t3))
( i i i ) is shown s i m i l a r l y by
exp( - tA) exp ( - tB ) exp tA exp tB = exp (t{-(A+B) + ,B] +O(t 2 ,
exp(t[(A+B) + ~[A, B] + O(t2)]l )
t 2 - exp(t([ ]2 + [ }1 )+2 --[[ }2"[}1 ]+O( t3 :
= exp (t2[A, B] + O(t3)) , q . e . d .
R e m a r k . Le t N O
such tha t the r e s t r i c t i o n e x p / N 0 : N O
one can def ine a c o m p o s i t i o n
A o B = l o g ( e x p A �9
if e x p A �9 exp B
N O fo r which O
c o m p o s i t i o n , exp
N e
be an open n e i g h b o r h o o d of O in G e
-. N is a d i f f e o m o r p h i s m . T h e n e
exp B) f or A , B ~ N O
�9 T h i s d e f i n e s a ( p a r t i a l ) c o m p o s i t i o n law in
i s an iden t i t y . In fac t , by the v e r y d e f i n i t i o n of t h i s
i s an i s o m o r p h i s m of N O with N e equ ipped
with the c o m p o s i t i o n i n h e r i t e d f r o m G . Now look at the f o r m u l a (i)
of p r o p o s i t i o n 6 . 5 . 2 . It can be r e w r i t t e n ( for a r b i t r a r y A, B g LG
and s u f f i c i e n t l y s m a l l t ) a s
o tB = (tA + tB)+ 1-~-[tA, tB] + O(t 3) . tA I -
-125-
The fundamental fact can be proved that (for sufficiently small t)
the term O(t 3) also is expressable by operations in LG on A, B .
This means that the composition law in the neighborhood N e of e in
G is completely determined by the Lie algebra LG. The formula (i)
of 6 . 5 . 2 j u s t g i v e s t h e f i r s t t w o t e r m s of t h i s d e v e l o p m e n t .
M o r e o v e r o n e c a n s h o w t h a t a L i e a l g e b r a h o m o m o r p h i s m
h: LG -~ LG ' i s a h o m o m o r p h i s m w i t h r e s p e c t t o t h e c o m p o s i t i o n d e f i n e d
in N O . T h i s i n c i d e n t a l l y s h o w s t h a t t h e m a p p : N e -~G' d e t e r m i n e d
by h: L G -~ L G ' a c c o r d i n g to p r o p o s i t i o n 6 . 2 . 9 i s in f a c t a l o c a l
h o m o m o r p h i s m G -~ G' i n d u c i n g h: L G -~ L G ' ,
We a p p l y p r o p o s i t i o n 6, 5 . 2 to p r o v e
PROPOSITION 6.5.3.
conditions are equivalent
( i ) [ A , B ] = 0
(hi) e x p sA e x p tB
( i i i ) e x p tA e x p tB
L e t A, B ~ LG . T h e n t h e f o l l o w i n g
= e x p t B e x p sA f o r e v e r y s a n d t
= e x p tB e x p tA f o r e v e r y t .
P roof: (i) ---> (ii) by proposition 5.4. II. (ii) ---->(iii) is trivial.
To see (iii) --->(i) observe that (iii) implies by proposition 6.5, 2
t 2 t 2 e x p [ t ( A + B ) + - ~ [ A , B ] + O ( t 3 ) ] = e x p { t ( B + A ) + " z - [ B ' A ] + O ( t 3 ) ]
for sufficiently small t . This implies [A, B] = [B,A] and
[ A , B ] = 0 .
The condition (i) and (iii) of proposition 6.5, Z also imply the
following convenient formulas.
-IZ6-
C OROLLARY 6 . 5 . 4 . Unde r the c o n d i t i o n s of p r o p o s i t i o n
6 . 5 . Z , one h a s
t } (i) e x p t ( A + B ) exp tA e x p t B exp - 2 - . [ A , B ] + O(t 3) ; e x p t A e x p t B e x p O ( t z)
(it) exp [ t Z ~ , B ] ] - exp ( - tA) exp ( - tB) exp tA exp tB exp O(t 3) .
P r o o f : (i) fo l lows f r o m
exp ( - tA) exp ( - tB) exp t(A + B) = exp ( t [ - ( A + B) + ~ [ 'A ,B] + O(t2)]) exp t(A + B)
t 2 = exp (t([ } + ( A + B ) ) + -.2-[{ } , A + B ] + O(t3))
t 2- = exp (T[A, B] + O(t3)) .
(ii) fo l lows f r o m
exp (-tB) exp (-tA) exp tB exp tA exp(tZ[A, B])= exp [tZ[B,A] + O(t3)]exp [tZ[A, B]]
= exp O(t 3) .
The f o r m u l a (i) shows tha t the c u r v e t - - , -~>exp tA exp tB h a s
the s a m e t angen t v e c t o r at e than the 1 - p a r a m e t e r subg roup
t ~-~-~> exp t(A + B) . F r o m p r o p o s i t i o n 5 . 4 . 1 0 it fo l lows tha t the t e r m
O(t 3) i s v a n i s h i n g fo r A , B with [ A , B ] = 0 o
The f o r m u l a (ii) d e s c r i b e s [A, BJ as the t angen t v e c t o r at e
of the c u r v e t ,-,~> exp (-~/'tA) exp ( - ~ B ) exp~rtA e ~ p ~ B .
A n o t h e r c o n s e q u e n c e of 6 . 5 . Z u s e f u l for l a t e r a p p l i c a t i o n is
the fo l lowing
COROLLARY 6 . 5 . 5 . Let A, B ~ LG . T h e n for any
t E ~t we have
-127-
l t (i) e x p t ( A + B ) = l im exp H
n ~ ( x )
(ii) exp [ t2[A, B] } - l im n ~ c o
n
A exp "H B
t t t t exp(--~tnA) exp(-~B) exp ~A exp~B} n2
P roof :
6 . 5 . 2 fo r f ixed t
t A exp t exp ~
and t h e r e f o r e
exp ~
Let t ~ ~ and n
t (A B = exp
t}n A exp ~ B - exp
s u f f i c i e n t l y g r e a t . By p r o p o s i t i o n
+ B ) + 2nZ[A, B] + O
t (A+ B ) +'2"~ [A, B] + O
thus showing (i) . To see (ii) it s u f f i c e s to o b s e r v e tha t by 6 . 5 . 2
2 2 l n l t ~ exp - exp - ~ e x p ~ A e x p ~ B = exp n- 2 - [ A , B ] +
- exp [ t 2 [ A , B ] + O(1)} .
C H A P T E R 7.
-iZ8-
S U B G R O U P S AND S U B A L G E B R A S
7.1. L i e s u b g r o u p s . B e f o r e d e f i n i n g the n o t i o n of L i e s u b g r o u p s , we
p r o v e
L E M M A 7.1. 1. L e t
i n ) e c t i v e h o m o m o r p h i s m .
i s i n j e c t i v e .
H , G be L i e ~ r o u p s a n d L:H-~ G an
T h e n the i n d u c e d h o m o m o r p h i s m L(~): LH -~ LG
P r o o f : L e t a i
w i th L a a 1 = L o a 2 .
t h e m a p s163 -~ s
s h o w s t h e i n j e c t i v i t y of L(r , q . e . d .
N o t e t ha t by l e m m a 6 . 2 . 6 e v e r y t a n g e n t l i n e a r m a p
i s i n j e c t i v e .
s = 1, Z) be 1 - p a r a m e t e r s u b g r o u p s of H
b e i n g i n j e c t i v e , a 1 = a 2 . T h e r e f o r e
i n d u c e d by r i s i n j e c t i v e . By 5 . 4 . lZ, t h i s
C * h : H h ~ Gc(h)
A s u b g r o u p H of D E F I N I T I O N 7.1, Z. Le t G be a L ie g r o u p .
G i s a L i e s u b g r o u p of G if
(i) H i s a L i e g r o u p
( i i ) t he i n j e c t i o n I,: He,.-, G i s a n a l y t i c .
L e t H be a L i e s u b g r o u p of G By l e m m a 7, 1. 1 a n d t he r e m a r k
f o l l o w i n g it , t h e p a i r (H, L) i s a s u b m a n i f o l d of G / a c c o r d i n g to t he
D E F I N I T I O N 7.1. 3. Le t G be a m a n i f o l d . A s u b s e t H of
G i s a s u b m a n i f o l d of G if
(i) H i s a m a n i f o l d
( i i ) t he i n j e c t i o n L:H ~--~ G i s a n i m m e r s i o n , i . e . t d i f f e r e n t i a b l e
and L . h : Hh -" G~(h) i n j e c t i v e fo r any h ~ H .
- 1 2 9 -
L e t H b e a s u b g r o u p o f
l o c a l l y t h e a n a l y t i c s t r u c t u r e o f
t h a t t h e g r o u p o p e r a t i o n s in H
s u b g r o u p of G .
N o t e t h a t a 1 - p a r a m e t e r s u b g r o u p
if a n d o n l y i f a i s i n j e c t i v e .
L e t
: He--. G
i d e n t i f y
H b e a L i e s u b g r o u p of
i n d u c e s a n i n j e c t i v e m a p
LH w i t h a s u b a l g e b r a o f
G . I f H i s a s u b m a n i f o l d o f G,
H i s i n d u c e d f r o m t h a t o f G , s o
a r e a n a l y t i c . H i s t h e r e f o r e a L i e
a:lR -~ G i s a L i e s u b g r o u p
G . B y 7 .1 . 1, t h e i n j e c t i o n
Lit,): LH -~ LG. We c a n t h e r e f o r e
L G a n d w r i t e L(~): LH~--. LG .
L E M M A 7 . 1 . 4 . L e t H be a L i e s u b g r o u p of
e x p : H e - * H i s t h e r e s t r i c t i o n of e x p : G e - ~ G .
G . T h e m a p
P r o o f : A f t e r t h e c a n o n i c a l i d e n t i f i c a t i o n s , t h i s i s j u s t t h e n a t u r a l i t y
6 .1 . 6 o f t h e e x p o n e n t i a l m a p ,
P R O P O S I T I O N 7, 1. 5. L e t
o_f G . LH 1 - 2 , t h e n
( i -- 1 , 2 )
H 1 = H 2 �9
be c o n n e c t e d L i e s u b g r o u p s
P r o o f : T h e r e i s a n o p e n n e i g h b o r h o o d o f e in H 1 w h i c h i s a l s o
a n o p e n n e i g h b o r h o o d o f e in H Z ( t a k e a c a n o n i c a l c h a r t a t e a n d
u s e 7 . 1 . 4 . ) .
We u s e w i t h o u t p r o o f t h e f o l l o w i n g
L E M M A 7.1 . 6. L e t X , Y be m a n i f o l d s r
a n d ~a-X -* Y a d i f f e r e n t i a b l e m a p w i t h ~p(X) c
6 : X --, S i s c o n t i n u o u s , i t i s d i f f e r e n t i a b l e .
S a s u b m a n i f o l d o f Y
S . I f t h e i n d u c e d m a p
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L E M M A 7.1. 7. L e t G be a L i e g r o u p a n d H a L i e s u b g r o u p .
T h e n LH = [A ~ L G / t ~ exp tA i s a c o n t i n u o u s m a p ~ - ~ H } .
P r o o f : A
m a p ]R-~ H .
a c o n t i n u o u s m a p
a n d t h e r e f o r e
LH i m p l i e s t h a t t ~ e x p t A i s a d i f f e r e n t i a b l e
S u p p o s e c o n v e r s e l y A ~ LG wi th t,-,--~> exp tA
-~ H , T h i s m a p is d i f f e r e n t i a b l e by l e m m a 7, 1.6
A E LH .
P R O P O S I T I O N 7.1, 8. Le__t H 1 , H 2 b e two L i e s u b g r o u p s of
I f H 1 a n d H 2 c o i n c i d e a s t o p o l o g i c a l g r o u p s , t h e y c o i n c i d e a s L i e
g r o u p s .
G ,
P r o o f : 7. l, 7 c h a r a c t e r i z e s t he L i e a l g e b r a by a i d of t he t o p o l o g i c a l
s t r u c t u r e al one . B y 7 . 1 . 5 , H10= H20 " T h e i d e n t i t y m a p H 1--, H z
i s t h e r e f o r e a n i s o m o r p h i s m ,
T h i s i s of c o u r s e a l s o a c o n s e q u e n c e of 6 . 3 , 5, but we h a v e p r e f e r r e d
a s i m p l e , d i r e c t p r o o f ,
We s t a t e now
T H E O R E M 7.1, 9. L e t G be a L i e g r o u ~ . If H is a L i e s u b -
g r o u p of G , t h e n t h e L i e a l g e b r a of H i s a s u b a l g e b r a of LG .
E a c h s u b a l g e b r a of LG i s t he L i e a l g e b r a of a u n i q u e c o n n e c t e d L i e
s u b g r o u p of G ,
of
P r o o f : T h e r e o n l y r e m a i n s to s h o w , t h a t f o r a g i v e n s u b a t g e b r a ~-~
LG t h e r e e x i s t s a c o n n e c t e d L i e s u b g r o u p H of G wi th LH = ~ ,
-131-
S u p p o s e t h e r e e x i s t s s u c h a L i e s u b g r o u p H . T h e n exp(~) ~ H
M o r e o v e r exp ~ f c o n t a i n s a n o p e n n e i g h b o r h o o d of e in H a n d
t h e r e f o r e g e n e r a t e s H .
T h i s p e r m i t s c o n v e r s e l y to d e f i n e H a s the s u b g r o u p g e n e r a t e d
by exp ~ . T h e p r o b l e m i s to m a k e H a s u b m a n i f o l d of G , We
do not show t h i s h e r e , but s k e t c h i n s t e a d a n o t h e r p r o o f ( s e e C h e v a l l e y
[ 3 ] , p. 109, t h e o r e m 1), m a k i n g u s e of t he e x i s t e n c e t h e o r e m f o r i n t e g r a l
m a n i f o l d s of a n i n v o l u t i v e f i e l d of v e c t o r s p a c e s on a m a n i f o l d .
L e t H be a L i e s u b g r o u p of G . T h e n the le f t c o s e t s of G
m o d u l o H a r e t he m a x i m a l i n t e g r a l m a n i f o l d s of t he f i e l d of v e c t o r s p a c e s
W on G d e f i n e d by the t a n g e n t s p a c e s of t h e c o s e t s . Now g i v e n c o n -
v e r s e l y a s u b a l g e b r a ~ c LG one c a n r e c o n s t r u c t t he f i e l d of v e c t o r s p a c e s
W . Wg is n a m e l y the v e c t o r s p a c e { A g / A ~ } ~ b e i n g a s u b a l g e b r a ,
W is t h e n i n v o l u t i v e , L e t H be t he m a x i m a l i n t e g r a l m a n i f o l d of W
p a s s i n g t h r o u g h e . To s e e t h a t H is a s u b g r o u p of G , f i r s t o b s e r v e
t h a t the f i e l d of v e c t o r s p a c e s W i s i n v a r i a n t by l e f t t r a n s l a t i o n s . T h e r e -
f o r e t h e m a x i m a l i n t e g r a l m a n i f o l d s a r e j u s t p e r m u t e d a m o n g t h e m s e l v e s
by lef t t r a n s l a t i o n s .
If conversely
a = { g / L ~l ~ g m a n i f o l d of
Now if h C H, t h e n L 1 h = e , so h"
L g _ I H = H f o r g E G , t h e n g ~ H ,
= H} a n d H is a s u b g r o u p of G , H
G, we s e e t h a t H is a L i e s u b g r o u p of G ,
L h _ l H = H 0
T h e r e f o r e
b e i n g a s u b -
E x e r c i s e 7 .1 , 10. L e t X be a G - m a n i f o l d a n d H a L i e s u b -
g r o u p of G . T h e n X is a H - m a n i f o l d . S u p p o s e v : G -~ A u t X
-132-
t o be a n e f f e c t i v e o p e r a t i o n o f G on X a n d l e t S be a s u b a l g e b r a
o f t h e L i e a l g e b r a KX o f K i l l i n g v e c t o r f i e l d s . T h e n t h e r e i s a u n i q u e
c o n n e c t e d L i e s u b g r o u p H
H d e f i n e s a n o p e r a t i o n on
v e c t o r f i e l d s .
o f G s u c h t h a t t h e r e s t r i c t i o n o f T to
X w i t h S a s L i e a l g e b r a o f K i l l i n g -
7, 2. E x i s t e n c e of l o c a l h o m o m o r p h i s m s , We b e g i n w i t h
L E M M A 7 . 2 , 1. L e t
g r o u p s . I f L(p) : LG -. L G '
i s o m o r p h i s m .
p: G -~ G' be a l o c a l h o m o m o r p h i s m of L i e
i_s a n i s o m o r p h i s m , t h e n p i s a l o c a l
P r o o f : I f L(p) i s a n i s o m o r p h i s m , t h e r e e x i s t s o n a n o p e n
n e i g h b o r h o o d o f e ' in G a l o c a l i n v e r s e m a p # o f p : G - ~ G ' .
p b e i n g a l o c a l h o m o m o r p h i s m , ~ i s n e c e s s a r i l y a l o c a l h o m o m o r p h i s m
a n d p t h e r e f o r e a l o c a l i s o m o r p h i s m . |
E x a m p l e 7, 2 . 2 , I f G i s c o m m u t a t i v e , e x p : LG -. G i s a
h o m o m o r p h i s m by 6 .1 . 4, N ow L ( e x p ) = 1LG: LG -, LG a n d e x p
m o r e o v e r s u r j e c t i v e by 6 . 2 . 7 , T h i s i s s u f f i c i e n t to d e t e r m i n e t h e
s t r u c t u r e o f c o m m u t a t i v e c o n n e c t e d L i e g r o u p s ( s e e s e c t i o n 7 . 3 ) ,
i s
We h a v e a l r e a d y p r o v e d in 6 . 2 . 1 1 t h e e x i s t e n c e o f a l o c a l h o m o m o r p h i s m
p: G -. G ' i n d u c i n g a g i v e n h o m o m o r p h i s m h: L G -. LG ' f o r c o m m u t a t i v e
g r o u p s . We p r o v e i t n o w in t h e g e n e r a l c a s e .
-133-
T H E O R E M 7 . 2 . 3 . Le t G , G ' be L ie g r o u p s and h: LG -. LG'
a h o m o m o r p h i s m of Lie a l g e b r a s . T h e n t h e r e e x i s t s a l o c a l h o m o m o r p h i s m
p : G -* G' with L(p) = h .
Note tha t by 6 .2 .11 , on the d o m a i n of a c a n o n i c a l c h a r t p m u s t
n e c e s s a r i l y c o i n c i d e with exp o h o log.
of
Le t
P r o o f : Le t k = { ( A , h ( A ) ) / A ~ LG} . T h e n k is a s u b a l g e b r a
LG x LG' e q u i p p e d with the L ie a l g e b r a s t r u c t u r e of de f i n i t i on 4 . 6 . 2 .
K be the c o n n e c t e d L ie s u b g r o u p of G x G' with Lie a l g e b r a k .
If p: G x G' -* G is the n a t u r a l p r o j e c t i o n , c o n s i d e r the h o m o m o r p h i s m
X - p / K : K - . G . L ( k ) : k - * LG is the m a p g i v e n by L(k) (A,h(A) ) = A
and t h e r e f o r e an i s o m o r p h i s m , By l e m m a 7 . 2 . 1 , k i s a l o c a l i s o -
m o r p h i s m with l o c a l i n v e r s e D: G -~ K , M o r e o v e r L(/z)A = (A ,h (A) )
for A E LG . The c o m p o s i t i o n of D :G -0 K wi th the p r o j e c t i o n
G X G' -, G' g i v e s a l o c a l h o m o m o r p h i s m p: G -. G' . By c o n s t r u c t i o n
L(p)(A) = h(A) for A r LG, i.e. L(p) = h , q.e,d.
T o g e t h e r wi th t h e u n i c i t y p r o p e r t y of 6. Z. 8, the t h e o r e m e x p r e s s e s
tha t L is a c o m p l e t e l y fa i th fu l f u n c t o r on L ie g r o u p s and l o c a l h o m o -
m o r p h i s m s to L ie a l g e b r a s a n d L i e a l g e b r a h o m o m o r p h i s m s .
To be ab le to s p e a k s t r i c t l y of u n i c i t y , we s h a l l c o n s i d e r g e r m s of
l o c a l h o m o m o r p h i s m s , i . e . we s h a l l i d e n t i f y h o m o m o r p h i s m s c o i n c i d i n g
on a n e i g h b o r h o o d of the i den t i t y .
We have a l r e a d y s e e n in 4 . 5 . 6 tha t a l o c a l i s o m o r p h i s m of Lie g roups
-134-
i n d u c e s an i s o m o r p h i s m of Lie a l g e b r a s , T h i s is a t r i v i a l c o n s e q u e n c e of
the f u n c t o r i a l i t y of L , We a r e now ab le to show
T H E O R E M 7 . 2 . 4 , Two L i e ~ r o u p s G and G' a r e locall~r
i s o m o r p h i c if and only if the Lie a l g e b r a s LG and LG' a r e i s o m o p r h i c .
P r o o f : If h: LG -. LG' is an i s o m o r p h i s m , t h e r e e x i s t s by 7, 2 .3
a loca l h o m o m o r p h i s m p:G-- , G' induc ing h , and p [s a loca l
i s o m o r p h i s m by 7 .2 .1 , q . e . d,
T h i s t h e o r e m is the m o s t i m p o r t a n t fact we have p r o v e d up to now.
It t e l l s e x a c t l y which type of i n f o r m a t i o n o n e c a n hope to ob ta in by the Lie
a l g e b r a of a Lie group . Note tha t e . g . t h e o r e m 6 .2 , 5 is an e a s y c o n -
s e q u e n c e of 7, 2 .4 , To c o m p l e t e the s tudy o n e w o u l d l ike to know if e v e r y
f i n i t e - d i m e n s i o n a l Lie a l g e b r a o v e r R is o c c u r i n g as the Lie a l g e b r a of
s o m e Lie group. Th i s is in fact so, but we s h a l l not p rove th i s h e r e . A
p r o o f i s ob ta ined by the fo l lowing t h e o r e m due to Ado: E v e r y f in i te d i m e n s i o n a l
JR-Lie a l g e b r a ~ is i s o m o r p h i c to a s u b a l g e b r a of the Lie a l g e b r a
~ ( n , I~) of GL(n, IR) for s o m e n . The c o n n e c t e d s u b g r o u p of
GL(n, ~ ) c o r r e s p o n d i n g to th i s s u b a l g e b r a i s a Lie g roup with Lie a l g e b r a
i s o m o r p h i c to ~ .
T h i s shows by the way, that any Lie g roup is l o c a l l y i s o m o r p h i c to
a Lie s u b g r o u p of a g roup GL(n, ~ ) fo r s o m e n .
A n o t h e r point to p r e c i s e is the r e l a t i o n b e t w e e n loca l h o m o m o r p h i s m s
and (g loba l ) h o m o m o r p h i s m s . Let p: G -~ G' be a loca l h o m o m o r p h i s m .
If G is c o n n e c t e d , we know by 6. Z. 9 that t h e r e is at m o s t one e x t e n s i o n
- 1 3 5 -
to a g l o b a l h o m o m o r p h i s m G -* G' . We s h a l l m a k e u s e of t h e f o l l o w i n g
l e m m a on t o p o l o g i c a l g r o u p s .
LEM1VLk 7 . 2 . 5 . L e t G be a c o n n e c t e d , local l~r c o n n e c t e d a n d
s i m p l y c o n n e c t e d t o p o l o g i c a l g r o u p , G' a n a r b i t r a r y t o p o l o g i c a l g r o u p
a n d p: G ~ G' a l o c a l h o m o m o r p h i s m (of t o p o l o g i c a l g r o u p s ) . T h e n
t h e r e e x i s t s a u n i q u e e x t e n s i o n of p to a h o m o m o r p h i s m 9 : G -. G' .
P r o o f :
a t o p o l o g y on
e on w h i c h
of n e i g h b o r h o o d s i s d e f i n e d by N(g , g ' , W) = [ (x, x ' )Jx = wg, x '
w h e r e W is a n o p e n n e i g h b o r h o o d of e in G wi th W
s h o w t h a t the p r o j e c t i o n p : G N G ' -~ G i s a c o v e r i n g of G .
w -,~-,-~> (wg, p (w)g ' ) i s a h o m e o m o r p h i s m W -~ N(g, g', V~ �9 If
c o n n e c t e d , t h e r e f o r e N(g, g' ,W)
h o m e o m o r p h i s m : N(g , g' ,W) -~ W g
the N ( g , g ' , W ) wi th g' ~ G' .
of t h i s u n i o n . T h e r e f o r e G ~ G'
L e t "~ d e n o t e t h e c o n n e c t e d c o m p o n e n t of (e , e ' ) in G x G ' , T h e n
(G, p / G ) i s a c o v e r i n g s p a c e of G a n d p / G a h o m e o m o r p h i s m , G
b e i n g s i m p l y c o n n e c t e d . L e t /~ be the i n v e r s e a n d d e f i n e p = qo/~ ,
w h e r e q : G ~ G' --. G' i s t he c a n o n i c a l p r o j e c t i o n . If v E V , t h e n
p'(v) = p(v) , a n d p i s a n e x t e n s i o n of p .
ks a h o m c m o r p h i z m . B y d e f i n i t i o n of p , f o r
7 ( v g ) = p(v)'~(g) = "~(v) '~(g) . F o r v i ~ V
U n i q u e n e s s is c l e a r . T o p r o v e the e x i s t e n c e , we d e f i n e
G X G ' . L e t V ~ G be a c o n n e c t e d n e i g h b o r h o o d of
p i s d e f i n e d � 9 If I~, g. ) ~ G ~ c G ' , a f u n d a m e n t a l s y s t e m
= p ( w ) g ' , w w } ,
c V . We
T h e m a p
W is
i s c o n n e c t e d a n d p / N ( g , g' ,W) a
�9 p ' l ( w g ) is t he d i s j o i n t u n i o n of
N(g, g ' , W) a r e o p e n c o n n e c t e d s u b s e t s
i s l o c a l l y c o n n e c t e d a n d p a c o v e r i n g .
It r e m a i n s to s h o w t h a t p
v E V a n d g ~ G we h a v e
by i n d u c t i o n
-136-
i vi)g) = (H i'p(vi))~(g ) and in particular
Therefore "~((l'[ i vi)g ) = y(II i vi)~(g ) . As
"p is therefore a homomorphism.
Y(n i v i) : n i Y(v i)
V generates G
T o g e t h e r wi th l e m m a 6 . 3 . 3 fo l lows
P R O P O S I T I O N 7. Z. 6.
L ie group, G' . .
m o r p h i s m .
--- G I p:G-~
Let G be a c o n n e c t e d and s i m p l y c o n n e c t e d
an a r b i t r a r y Li e g r o u p and p: G -~ G' a l o c a l h o m o -
T h e n t h e r e e x i s t s a un ique e x t e n s i o n of p to a h o m o m o r p h i s m
Note tha t we h a v e p r o v e d in p r o p o s i t i o n 5 . 4 . 8 a p a r t i c u l a r c a s e of
t h i s p r o p o s i t i o n .
C O R O L L A R Y 7 . 2 . 7 . Le t G, G'
a h o m o m o r p h i s m of L ie a l g e b r a s . If G
then t h e r e e x i s t s a u n i q u e h o m o m o r p h i s m
If, m o r e o v e r . G' is c o n n e c t e d and s i m p l y c o n n e c t e d , and
i s o m o r p h i s m , t hen p is an i s o m o r p h i s m .
be Lie groups and h: LG -- LG'
is connected and simply connected,
p:G-~G' with L(p) -- h
h an
P r o o f : To a h o m o m o r p h i s m h: LG -~ LG' t h e r e e x i s t s by t h e o r e m
7 . 2 . 3 a l o c a l h o m o m o r p h i s m p : G - * G' i n d u c i n g h . If G is
c o n n e c t e d and s i m p l y c o n n e c t e d , p can , by 7. Z. 6, be e x t e n d e d u n i q u e l y
to a h o m o m o r p h i s m .
Suppose now a l s o G' c o n n e c t e d and s i m p l y c o n n e c t e d . If h
i s an i s o m o r p h i s m , i t s i n v e r s e k is i n d u c e d by a h o m o m o r p h i s m
k: G' -~ G . Now L(kop) = ILG and by unicity Z.ap = i G . Similarly
-137-
p o X = 1G, a n d p i s a n i s o m o r p h i s m , q . e . d .
A s a n a p p l i c a t i o n , c o n s i d e r a c o m m u t a t i v e L i e g r o u p G . ]By
6 .1 . 4 t h e m a p exp : LG -. G i s a h o m o m o r p h i s m . I t i s i n d u c e d by t h e
i s o m o r p h i s m 1LG: L G -. L G o f L i e a l g e b r a s . C o r o l l a r y 7. Z. 7 s h o w s
P R O P O S I T I O N 7 . 2 . 8 .
s i m p l e r c o n n e c t e d L i e ~ r o u p ,
If G i s a c o m m u t a t i v e , c o n n e c t e d a n d
exp : L G -. G i s a n i s o m o r p h i s m .
R e m a r k . We m e n t i o n h e r e ~ w i t h o u t p r o o f ; t h e e x i s t e n c e of a
u n i v e r s a l c o v e r i n g g r o u p f o r a n y c o n n e c t e d L i e g r o u p . M o r e p r e c i s e l y
l e t G b e a c o n n e c t e d L i e g r o u p . T h e n t h e r e i s a c o n n e c t e d a n d s i m p l y
c o n n e c t e d L i e g r o u p G a n d a h o m o m o r p h i s m a n d l o c a l i s o m o r p h i s m
~ : G -. G s u c h t h a t (G, cO) i s a c o v e r i n g m a n i f o l d o f G . (G, ~) h a s
t h e f o l l o w i n g u n i v e r s a l p r o p e r t y . F o r a n y c o n n e c t e d a n d s i m p l y c o n n e c t e d
L i e g r o u p H a n d h o m o m o r p h i s m
p : H - * ~ w i t h r o p = p .
If' G i s a c o m m u t a t i v e c o n n e c t e d L i e g r o u p , t h e p a i r
i s t h e u n i v e r s a l c o v e r i n g g r o u p of G .
N o w le t G be a c o n n e c t e d L i e g r o u p ,
a n d p: G -. G' a l o c a l h o m o m o r p h i s m . L e t
9- H -* G t h e r e i s a u n i q u e h o m o m o r p h i s m
( L G , e x p )
G' a n a r b i t r a r y L i e g r o u p
( ~ ~) be t h e u n i v e r s a l
c o v e r i n g g r o u p o f G . T h e n t h e l o c a l h o m o m o r p h i s m
h a s b y 7 . 2 . 6 a u n i q u e e x t e n s i o n to a h o m o m o r p h i s m @: G -~ G'
G ' i s c o n n e c t e d a n d (G', r ) a u n i v e r s a l c o v e r i n g g r o u p of
e x i s t s a u n i q u e h o m o m o r p h i s m @:G - .G ' w i t h ~ ' o ~ = e,, .
S u p p o s e in p a r t i c u l a r
c o n n e c t e d L i e g r o u p s
p o ~: G -. G '
. If
G' t h e r e
p: G -~ G' to be a l o c a l i s o m o r p h i s m of
G , G ' . T h e p r e c e d i n g s h o w s t h a t ~ : ( ~ - . (~'
- 1 3 8 -
is a local isomorphism. By corollary 7.2.7, {~ is an isomorphism.
Therefore a local isomorphism of connected Lie groupStnduces an iso-
morphism of the universal covering groups. This means that to every
class of locally isomorphic connected Lie groups there corresponds a
unique Lie group (up to isomorphisms), which is a universal covering
group of any member of the class, Every member of the class is obtained
from this universal covering group by dividing by a discrete normal
subgroup (see section 7,3) . By theorem 7.2.4 there is an injecttve
map of the classes of locally isomorphic Lie groups into the classes of
isomorphic Ft-Lie algebras, By the above mentioned theorem of Ado
this map is bijective. The problem of classifying all possible connected
Lie groups is therefore decomposed in two steps, First find all R-Lie
algebras. Second find all discrete normal subgroups of a simply connected
Lie group.
Consider the restricted problem of classifying all possible commutative
connected Lie groups. A commutative Lie algebra is characterized by its
dimension. The classification problem reduces therefore to find all discrete
subgroups of a simply connected commutative Lie group. By 7. Z. 8 , this
is just the problem of finding the discrete subgroups of a finite-dimensional
Ft-vectorspace. We shall do this in the next section.
7. 3.
Can
of H
D i s c r e t e s u b g r o u p s . Le t G be a L ie g r o u p and H a s u b g r o u p .
H be de f ined a s a L ie s u b g r o u p of G ?
F o r a g iven t opo logy on H (not n e c e s s a r i l y the r e l a t i v e t opo logy
in G), such tha t H is a t o p o l o g i c a l g roup , t h e r e is by 7. 1. 8 at
-139-
m o s t o n e L i e g r o u p s t r u c t u r e i n d u c i n g t h i s t o p o l o g y a n d m a k i n g H a
L i e s u b g r o u p of G .
T h e e x a m p l e of t h e r ~ t i o n a l s Q ~ ~ s h o w s t h a t i f we t a k e t h e
i n d u c e d t o p o l o g y o n H , t h e r e d o e s n o t n e c e s s a r i l y e x i s t a L i e g r o u p
s t r u c t u r e on H i n d u c i n g t h i s t o p o l o g y a n d m a k i n g H a L i e s u b g r o u p
of G .
We c a n a l w a y s c o n s i d e r H a s a 0 - d i m e n s i o n a l m a n i f o l d , m a k i n g
H a L i e s u b g r o u p of G . T h e L i e a l g e b r a o f a 0 - d i m e n s i o n a l L i e g r o u p
i s 0 , a n d t h e s u b a l g e b r a o f L G c o r r e s p o n d i n g to a 0 - d i m e n s i o n a l L i e
s u b g r o u p t h e r e f o r e 0 . T h e e x a m p l e Q~,~ ~t a g a i n s h o w s t h a t a p a r t
f r o m t h i s t r i v i a l m a n n e r t h e r e i s l p o s s i b l y ~ n o w a y o f t u r n i n g a s u b g r o u p
of a L i e g r o u p i n t o a L i e s u b g r o u p .
D E F I N I T I O N 7 . 3 . 1 . L e t G be a t o p o l o g i c a l g r o u p . A d i s c r e t e
s u b g r o u p H of G i s a s u b g r o u p w h i c h i s a d i s c r e t e s u b s p a c e of G.
W h e n G i s a L i e g r o u p , i t i s n a t u r a l to v i e w a d i s c r e t e s u b g r o u p H
a s a 0 - d i m e n s i o n a l L i e s u b g r o u p of G . N o t e t h a t a d i s c r e t e s u b g r o u p
of a L i e g r o u p G i s a c l o s e d s u b g r o u p ( u s e t h e f a c t t h a t G i s a
H a u s d o r f f s p a c e ) ,
E x a m p l e 7o 3 . 2 .
s u b g r o u p of • n .
We s h o w now
L e t 0 _~ p ~ n . T h e n ~ P i s a d i s c r e t e
P R O P O S I T I O N 7 . 3 . 3 , L e t G, G ' be t o p o l o g i c a l g r o u p s a n d , m
p : G -~ G' a h o m o m o r p h i s m a n d l o c a l i s o m o r p h i s m : The_n t h e k e r n e l o f
- 1 4 0 -
p i s a d i s c r e t e n o r m a l s u b g r o u p o f G ,
G, G'
k e r p
l a t i o n s b e i n g h o m e o m o r p h i s m s , e v e r y p o i n t o f
k e r p i s d i s c r e t e , |
P r o o f : T h e r e e x i s t s o p e n n e i g h b o r h o o d s N, N' o f e , e ' in
s u c h t h a t p / N : N -, N' i s a h o m e o m o r p h i s m . T h e r e f o r e
N N -- {e} a n d e i s a n i s o l a t e d p o i n t o f k e r p . T h e t r a n s -
k e r p i s i s o l a t e d a n d
C O R O L L A R Y 7 . 3 . 4 .
k e r n e l o f t h e h o m o m o r p h i s m
additive group of LG .
Let G be a commutative Lie group. Th__._e
exp : L G -. G i s a d i s c r e t e s u b g r o u p o f t h e
Proof: The homomorphism exp: LG -. G is by example 7.2. Z a
local isomorphism.
T h i s r a i s e s t h e p r o b l e m of f i n d i n g a l l d i s c r e t e s u b g r o u p s o f t he
a d d i t i v e v e c t o r g r o u p o f a f i n i t e d i m e n s i o n a l • - v e c t o r s p a c e V , E v e r y
s u c h s u b g r o u p i s i s o m o r p h i c t o a g r o u p 2E p, w h e r e p ~ d i m V . M o r e
p r e c i s e l y we s h o w
D
d i m e n s i o n o f t h e s u b s p a c e g e n e r a t e d b~"
l i n e a r l y i n d e p e n d e n t v e c t o r s ,
L E M M A 7 . 3 . 5 . L e t V be a n - d i m e n s i o n a l
a d i s c r e t e s u b g r o u p o f t h e a d d i t i v e v e c t o r g r o u p . . p _< n
D , T h e n t h e r e e x i s t s
v I . . . . . Vp _in V g e n e r a t i n g
P r o o f :
i n d u c t i o n .
E. - v e c t o r s p a c e a n d
L e t be t h e
P
D .
We a s s u m e k n o w n t h e c a s e p = 1 a n d p r o v e t h e l e m m a by
S u p p o s e t h e l e m m a t r u e f o r a l l k < p a n d l e t D g e n e r a t e
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a p-dimensional subspace U of V . There is a (p-l)-dimensional
subspace A of U generated by e lements of D , Let v I , , . . , Vp_ l
be linearly independent vectors in V generating D N A . Now
D + A[A ~ D/D N A . That this algebraic isomorphism is a topological
i somorphism follows from the fact that these groups are locally compact
and that D + A has a countable base (for a proof we re fe r to corol lary 3, 3
of S. Helgason [ 6 ], p. Ill). Using this, we see that D + A/A is
discrete . Being a subgroup of the l -dimensional vectorspace U/A , the
group D + A/A is generated by an element Vp + A . Then v I �9 . . . . Vp
are linearly independent and generate D .
We a r e now a b l e to d e t e r m i n e the s t r u c t u r e of t he c o m m u t a t i v e
c o n n e c t e d L i e g r o u p s .
T H E O R E M 7 . 3 . 6 . L e t G be a c o m m u t a t i v e c o n n e c t e d L i e g r o u p
of d i m e n s i o n n . T h e n t h e r e i s a n i n t e g e r p , 0 _~ p ~ n , s u c h t h a t
--~ n - p ~ T p O - - E
Proof: The homomorphism exp: LG -~ G is surjective by proposition
6 . 2 . 7 . We h a v e t h e r e f o r e an i s o m o r p h i s m L G / k e r exp ~ G in t he
a l g e b r a i c s e n s e . It i s no t d i f f i c u l t to s e e d i r e c t l y t h a t it i s an i s o m o r p h i s m
of L i e g r o u p s . We o m i t t h i s h e r e ,
g e n e r a l s t a t e m e n t , Now k e r e x p
by l e m m a 7 . 3 . 5 . T h e r e f o r e G
a s we s h a l l p r o v e � 9 in 7, 7 . 6 , a m o r e
Z p f o r s o m e p wi th 0 ~_ p _~ n
E n / z P w h i c h p r o v e s t h e t h e o r e m
- 1 4 2 -
C O R O L L A R Y 7, 3 . 7 , L e t G be a c o m p a c t c o n n e c t e d L i e g r o u p
o f d i m e n s i o n n . T h e n G = T n .
A s m e n t i o n e d a t t h e e n d of s e c t i o n 7 . 2 , o n e s t e p in t he c l a s s i f i c a t i o n
p r o b l e m f o r L i e g r o u p s c o n s i s t s in f i n d i n g a l l d i s c r e t e n o r m a l s u b g r o u p s
o f a s i m p l y c o n n e c t e d L i e g r o u p . T h i s i s g r e a t l y s i m p l i f i e d by
P R O P O S I T I O N 7, 3 . 8 . Le t H
t h e c o n n e c t e d t o p o l o g i c a l g r o u p G
c e n t e r o f G .
be a d i s c r e t e n o r m a l s u b g r o u p of
T h e n H i s c o n t a i n e d in t h e
P r o o f : L e t h ~ H . T h e m a p G - ~ H d e f i n e d b y g ~ g h g "1
i s c o n t i n u o u s . T h e i m a g e b e i n g c o n n e c t e d , i t m u s t be a p o i n t a n d t h e r e f o r e
e q u a l t o h , q . e . d .
7 . 4 . O p e n s u b g r o u p s , c o n n e c t e d n e s s . L e t G b e a L i e g r o u p a n d
H a s u b g r o u p w h i c h i s a n o p e n s u b s e t of G , H i s a s u b m a n i f o l d a n d
t h e r e f o r e a L i e s u b g r o u p o f G . T h e L i e a l g e b r a o f a n o p e n s u b g r o u p
i s LG, t h e i n j e c t i o n b e i n g a l o c a l i s o m o r p h i s m . T h e r e f o r e a n o p e n s u b -
g r o u p H o f G n e c e s s a r i l y c o n t a i n s G O , a s L H = LG i m p l i e s
H 0 = G O . R e m e m b e r t h a t a n o p e n s u b g r o u p i s n e c e s s a r i l y c l o s e d .
E x a m p l e 7 . 4 . 1 . L e t V be a f i n i t e - d i m e n s i o n a l l : t - v e c t o r s p a c e
a n d de t : G L ( V ) -~ ~ * t h e d e t e r m i n a n t h o m o m o r p h i s m . R * b e i n g n o t
c o n n e c t e d , G L ( V ) i s n o t c o n n e c t e d , O n t h e o t h e r h a n d , a n y t w o b a s e s
o f V w i t h t h e s a m e o r i e n t a t i o n ( a f t e r t h e c h o i c e o f a n o r i e n t a t i o n )
can b e c o n t i n u o u s l y t r a n s f o r m e d o n e i n t o t h e o t h e r b y a u t o m o r p h i s m s o f
-143-
V. T h i s s h o w s r e a d i l y t h a t d e t - l ( ~ +) i s t he c o n n e c t e d c o m p o n e n t
of the i d e n t i t y in G L ( V ) , w h e r e ~ + : {x ~ ~ * / x > 0] .
L e t H be an o p e n s u b g r o u p of G a n d G / H the s e t of
l e f t c0sets m o d u l o H , A l l l e f t ocsels b e i n g o p e n in G, the q u o t i e n t
t o p o l o g y i s d i s c r e t e a n d G / H c a n be c o n s i d e r e d a s a 0 - d i m e n s i o n a l
m a n i f o l d . If in p a r t i c u l a r H i s a n o r m a l s u b g r o u p of G , t h e n
G / H c a n be c o n s i d e r e d a s a 0 - d i m e n s i o n a l L i e g r o u p � 9
T h i s a p p l i e s to t he c o n n e c t e d c o m p o n e n t of t he i d e n t i t y and
~/-- G / G 0 is a 0 - d i m e n s i o n a l L i e g r o u p .
E x a m p l e 7�9 4 . 2 . L e t G = G L ( V )
of a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e V
7, 4 .1.
A s a n y c o n n e c t e d c o m p o n e n t of G
m a n i f o l d G is d i f f e o m o r p h i c to G O x
c a n o n i c a l d i f f e o m o r p h i s m . A n y s p l i t t i n g
be t he g r o u p of a u t o m o r p h i s m s
�9 T h e n ~ = Z 2 by e x a m p l e
i s d i f f e o m o r p h i c to G O , t he
. T h e r e is , h o w e v e r , no
s: ~ - . G of t he e x a c t s e q u e n c e
e -~G0 -. G -. ~ - . e
d i r e c t p r o d u c t G0xT~ , w h e r e
d e f i n e d by Tr = ~ S ( r f o r
m a n y p a r t i c u l a r c a s e s .
g i v e s r i s e to an i s o m o r p h i s m of G wi th t h e s e m i -
T: ~ -~ Au t G O is the h o m o m o r p h i s m
~ ~ , S u c h a s p l i t t i n g e x i s t s in
E x a m p l e 7 .4 �9 3.
V . T h e r e f l e c t i o n at t he o r i g i n of V t o g e t h e r wi th t he i d e n t i t y of
f o r m s an i s o m o r p h i c i m a g e of Z 2 -" G / G 0 in G �9
In t h e c a s e of a c o m m u t a t i v e g r o u p G , a s p l i t t i n g s: ~ -~ G
the e x a c t s e q u e n c e e -. G O -. G -~ ~ -. e d e f i n e s an i s o m o r p h i s m of
w i th G0 x~/ .
G = GL(V) f o r an o d d - d i m e n s i o n a l ~ - v e c t o r s p a c e
V
of
G
-144-
E x a m p l e 7, 4 . 4 . Le t V be a f in i te d i m e n s i o n a l
C o n s i d e r a v e c t o r s u b s p a c e U and a v e c t o r a ~ V , a
un ion of U and i t s t r a n s l a t e s by i n t e g e r m u l t i p l e s of
1 R - v e c t o r s p a c e .
U . T h e n the
a is a L ie
g r o u p G in the r e l a t i v e t opo logy of V . U is the c o n n e c t e d c o m -
ponen t of the i d e n t i t y of G and i t s g r o u p of c o n n e c t e d c o m p o n e n t s i s
i s o m o r p h i c to • . The e x a c t sequemce 0 -~ U -*G - ~ -~ 0 h a s an
e v i d e n t s p l i t t i n g h o m o m o r p h i s m s : Z -~ G and G ~ U x Z .
7 . 5 . C l o s e d s u b g r o u p s . We have s e e n tha t d i s c r e t e and open s u b g r o u p s
of a L ie g r o u p G a r e L ie s u b g r o u p s . Both t y p e s of s u b g r o u p s a r e
c l o s e d in G . We s t a t e now m o r e g e n e r a l l y
T H E O R E M 7 .5 , 1. Le t G be a L ie g r o u p and H a s u b g r o u p
o.f G . Suppose H is a c l o s e d s u b s e t of G o T h e n t h e r e e x i s t s
a un ique L i e Group s t r u c t u r e on H such tha t the c o r r e s p o n d i n g t o p o l o g y
is the i n d u c e d topo log F on H and s u c h tha t H is a L i e s u b g r o u p of G
P r o o f :
H
T h e u n i q u e n e s s s t a t e m e n t f o l l o w s f r o m 7.1, 8, Now let
At
G . Le t ~ c LG be d e f i n e d by
A ~ implies
A,B ~ ~ . Then by 6.5.5,
H is closed. Therefore
be a c l o s e d s u b g r o u p of
~ = [A ~ L G / e x p t A ~ H for t e v e r y
We f i r s t p r o v e tha t ~ is a s u b a l g e b r a of ~ . i u
tA ~ by d e f i n i t i o n of ~ , Suppose now
(i), exp t(A+B) g H for e v e r y t ~ l R , a s
A+B ~ ~ , and by 6 . 5 . 5 , ( i i) , e x p t 2 [ A , B ]
s h o w i n g ~ , B ]
H for any t
is t h e r e f o r e a s u b a l g e b r a of LG .
IR,
- 1 4 5 -
C o n s i d e r now the c o n n e c t e d L i e s u b g r o u p H$ of G wi th
LH'~ = ~ . B y c o n s t r u c t i o n of ~ we h a v e e x p ~ H a n d t h e r e f o r e
H* c H , H* b e i n g g e n e r a t e d by e x p ~ z .
L e t H be e q u i p p e d wi th the r e l a t i v e t o p o l o g y of G . We s h a l l
p r o v e t h a t a n e i g h b o r h o o d V of e in H* is a n e i g h b o r h o o d of
e in H This will prove that H$ is a topological subgroup of
H (using that H~'r H is continuous), and taking V = H~' j moreover .
that H$ ks open in H ( e being an inner point of H$ in H) , Then
H$ = H 0 as topological groups. H 0 is therefore a Lie subgroup of G.
H can now be turned into a submanifold of G with the aid of translations,
It is then clear that the multiplication H ~ H -~ H ,~ill be differentiable.
This is in fact sufficient to see that H is a Lie subgroup of G ,
There remains to show, that a neighborhood V of e in H$ is
a neighborhood of e in H . Suppose V is not a neighborhood of e
in H . We show that this leads to a contradictLon. There exists a
s e q u e n c e c 1 , . . . c k , . . . in H - V wi th K-.co'lim c k = e . L e t M
be a c o m p l e m e n t a r y s u b s p a e e of ~ in LG . B y 6 . 3 . 2 , t h e r e e x i s t
b o u n d e d , o p e n , c o n n e c t e d n e i g h b o r h o o d s U 1 , U 2 of O in M a n d t /
r e s p e c t i v e l y , s u c h t h a t 4 : ( A , B ) ~ e x p A exp B f o r A ~ M , B ~ ~,~
i s a d i f f e o m o r p h i s m of U I • U 2 on to a n o p e n n e i g h b o r h o o d of e in
G
A k
�9 We c a n , t h e r e f o r e , a s s u m e tha t c k = exp A k exp B k wi th \
U 1 , B k E U 2 a n d e x p B k E V . T h e n A k~r 0 a n d l i ra A k
S i n c e A k ~ 0 , t h e r e e x i s t s a n i n t e g e r r k > 0 s u c h t h a t
r k A k ~ U 1 a n d ( r k + l ) A k ~ U 1 . Now U 1 is b o u n d e d , s o we c a n
- - 0 .
a s s u m e , p a s s i n g to a s u b s e q u e n c e , t h a t t h e s e q u e n c e ( r k A k ) c o n v e r g e s
-146-
t o a l i m i t A ~ U 1 . S i n c e ( r k + l ) A k ~ U 1 a n d A k-~ 0 , A
t h e b o u n d a r y of U 1 , in p a r t i c u l a r A 4 0 .
L e t p, q be a n y i n t e g e r s (q > 0) . T h e n we c a n w r i t e
p r k = q s k + t k , w h e r e s k , t k a r e i n t e g e r s a n d 0 _~ t k < t k
T h e n l i m - - q A k = 0 , so
P r k s k exp-P--A = itkm exp A k = l~m (expAk) q q
q .
w h i c h b e l o n g s to H . B y c o n t i n u i t y exp tA 6 H f o r e v e r y t G
But t h e n A 6 ~ , in c o n t r a d i c t i o n to A ~ U 1 C M a n d A ~ 0 . |
T h e p r e v i o u s l y d i s c u s s e d p a r t i c u l a r c a s e s of 7 . 5 . 1 , w h e r e H
e i t h e r a d i s c r e t e o r a n o p e n s u b g r o u p of G , c o r r e s p o n d to t h e c a s e
w h e r e ~ " is e i t h e r 0 o r e q u a l to LG .
COROLLARY 7 . 5 . Z. L e t G
s u b g r o u p . L e t LH be t he L i e a l g e b r a of
u n i q u e L i e g r o u p s t r u c t u r e d e f i n e d in 7 . 5 . 1 .
i s on
be a L i e g r o u p a n d H a c l o s e d
H wi th r e s p e c t to the
T h e n
LH = [A ~ L G / e x p t A ~ H f o r e v e r y t ~ IR} .
P r o o f : T h e L i e a l g e b r a of H w a s d e f i n e d i n t he p r o o f of 7 . 5 . 1
by t h i s p r o p e r t y ,
R e m a r k .
s u b g r o u p H of G w h i c h h a s c o u n t a b l y m a n y c o m p o n e n t s .
S. H e l g a s o n [ 6 ] , p, 108.
A n i m p o r t a n t c l a s s of c l o s e d s u b g r o u p s of a L i e g r o u p G
k e r n e l s of h o m o m o r p h i s m s s t a r t i n g f r o m G.
T h e c o r o l l a r y 7, 5, Z i s e v e n t r u e f o r a n a r b i t r a r y L i e
S e e
is
a r e t he
-147-
P R O P O S I T I O N 7 .5 , 3.
L~e g r o u p s . T h e n
= k e r L(p), w h e r e
of L ie a l g e b r a s .
Le_~_t p : G -. G' be a h o m o m o r p h i s m of
k e r p is a L ie subgroup , of G and L ( k e r p)
L(p): LG -~ LG' is the i n d u c e d h o m o m o r p h i s , m
P r o o f :
s u b g r o u p of G. By 7 .5 , Z,
e ' d e n o t i n g the i d e n t i t y of
p(exp tA) = e' fo r e v e r y
k e r p is a c l o s e d s u b g r o u p of G and t h e r e f o r e a Lie
L ( k e r p) = ~A ~ L G / p ( e x p tA) = e' fo r e v e r y t ~ •],
G' . By the n a t u r a l i t y 6, 1. 6 of exp,
t ~ l~ is e q u i v a l e n t to exp(L(p) tA) - e '
fo r e v e r y t ~ Ft . T h i s a g a i n is e q u i v a l e n t to the e x i s t e n c e of an
E > 0 , such tha t L(p)tA = 0 fo r e v e r y I t ] < E . The l a t t e r p r o p e r t y
s i g n i f i e s L(p)A = 0 . T h e r e f o r e L ( k e r p ) = k e r L ( p ) , q. e, d.
T h i s s h o w s in p a r t i c u l a r tha t the k e r n e l of the h o m o m o r p h i s m
L(p): LG -. LG' i s a L ie a l g e b r a . T h i s is of c o u r s e t r u e for the k e r n e l
of any Lie a l g e b r a h o m o m o r p h i s m .
We a l s o would l ike the i m a g e of a h o m o m o r p h i s m of L ie g r o u p s to
be a Lie g roup . I n d e e d we have
P R O P O S I T I O N 7, 5, 4, Le t
g r o u p s . Suppose G c o n n e c t e d ,
and L(im p) -- im L(p), where
m o r p h i s m of L ie a l g e b r a s,
p:G -. G' be a homomorphism of Lie
Then imp is a Lie subgroup of G'
L(p): LG -. LG' is the induced homo-
P r o o f : Le t H be the c o n n e c t e d Lie s u b g r o u p of G' with
LH = i m L(p) . H i s g e n e r a t e d by the e ~ e m e n t s exp (L(p )A) with
A ~ LG . Now p(G) is g e n e r a t e d by the e l e m e n t s p(exp A) with
- 1 4 8 -
A ~ L G . B u t p ( e x p A ) = e x p ( L ( p ) A ) by 6 , 1 . 6 .
p(G) = H , a s b o t h g r o u p s a r e c o n n e c t e d .
T h e r e f o r e
R e m a r k . T h e r e i s t h e q u e s t i o n , if t h e i n d u c e d m a p ~ : G -. p ( G )
i s a h o m o m o r p h i s m , i. e , a n a l y t i c . T h i s i s i n d e e d s o ( s e e 7 . 7 . 6 ) .
C o n s i d e r n o w a s e q u e n c e o f h o m o m o r p h i s m s of L i e g r o u p s
('V) G' P' " > G P > G "
a n d t h e i n d u c e d s e q u e n c e o f h o m o m o r p h i s m s o f L i e a l g e b r a s
(a) I..P~' L(p').> LG .L(p"),, LG"
P R O P O S I T I O N 7, 5 , 5 . S u p p o s e G' c o n n e c t e d , T h e n t h e
e x a c t n e s s o f (~) i m p l i e s t h e e x a c t n e s s o f (a) .
P r o o f : I f i m p ' = k e r p" , t h e n i m L ( p ' ) = L ( i m p ' )
= L ( k e r p " ) = k e r L ( p " ) by 7 . 5 . 3 a n d 7 . 5 . 4 , q . e . d .
E x a m p l e 7 . 5 . 6 . L e t G be a c o n n e c t e d L i e g r o u p . T h e e x a c t
s e q u e n c e 0 -~ G e --, T G -. G -. e o f L i e g r o u p s i n d u c e s a n e x a c t s e q u e n c e
0 -. G e -. L ( T G ) -~ L G -~ 0 o f L i e a l g e b r a s . N o t e t h a t t h e n a t u r a l s p l i t t i n g
G r T G o f t h e f i r s t s e q u e n c e d e f i n e s a s p l i t t i n g L G - ~ L ( T G ) o f t h e
s e c o n d s e q u e n c e .
O b s e r v e t h a t t h e c o n v e r s e o f p r o p o s i t i o n 7 . 5 . 5 i s n o t t r u e , e v e n
i f a l l g r o u p s a r e c o n n e c t e d .
E x a m p l e 7 . 5 . 7 . L e t p: G -~ G' be a h o m o m o r p h i s m a n d l o c a l
i s o m o r p h i s m , T h e n 0 :> L G L ( P / > LG ' > 0 i s a n e x a c t s e q u e n c e .
-149-
But e -* G -~ G' -* e' i s not n e c e s s a r i l y exac t , i . e . G
a r e not n e c e s s a r i l y i s o m o r p h i c ,
The fo l lowing p a r t i a l r e s u l t is s o m e t i m e s use fu l .
and G'
PROPOSITION 7 .5 , 8. Le t p: G -~ G' be a h o m o m o r p h i s m of
Lie g roups . Suppose G and G' c o n n e c t e d . T h e n p is s u r j e c t i v e
if and only if L(p): LG -~ LG' is s u r j e c t i v e .
P roof : If p is s u r j e c t i v e ,
m u s t c o i n c i d e with the Lie a l g e b r a
L(p} is s u r j e c t i v e .
Suppose c o n v e r s e l y L(p}
is s u r j e c t i v e by 6 . 2 . 6 for e v e r y
(and h e n c e c losed} s u b g r o u p of
the Lie a l g e b r a L(p}LG of p(G)
LG' of G' , which shows that
s u r j e c t i v e . T h e n p , g : Gg -. G' p(g)
g ~ G. p(G) is t h e r e f o r e an open
G ' , i . e . p ( G ) : O' . |
The cond i t i on that G' is c o n n e c t e d canno t be o m i t t e d , as shown
by the e x a m p l e of the i n c l u s i o n of the c o n n e c t e d c o m p o n e n t of the iden t i ty
into a n o n - c o n n e c t e d Lie g roup , which i n d u c e s an i s o m o r p h i s m of Lie
a l g e b r a s .
The c o r r e s p o n d i n g s t a t e m e n t for i n j ec t i ons is not t r ue , We have
s e e n in 7. I. I tha t an i n j e c t i o n p : G -. G' i n d u c e s an i n j e c t i o n
L(p ): LG -+ LG'
injectivity of
R - ~ .
�9 But the i n j e c t i v i t y of L(p) does not i m p l y the
p , as shown by the e x a m p l e of the c a n o n i c a l h o m o m o r p h i s m
-15 0-
7 .6 . C l o s e d s u b g r o u p s of the ful l l i n e a r g roup . Let V be a f in i te
d i m e n s i o n a l ~ - v e c t o r s p a c e and GL(V) the g roup of l i n e a r a u t o m o r p h i s m s .
We s h a l l c o n s i d e r s o m e c l o s e d s u b g r o u p s of GL(V) .
Le t ~: VX V -~ 1R be a b i l i n e a r and n o n - d e g e n e r a t e d f o r m on V .
Let H be the subgroup of GL(V) l e av ing r i nva r i an t :
H = {g ~ GL(V) / r gw) = r w) for any v, w ~ V] .
C o n s i d e r for f ixed v, w ~ V the map GL(V) -* G L ( V ) X G L ( V ) -* VXV-*
d e f i n e d by g --~-~>(g, g) --~-,~>(gv, gw) ~----->r gw) . As ~ is con t inuous ,
th i s m a p is con t inuous , T h e r e f o r e the se t
i s c l o s e d in GL(V) . Now
H = N v , w ~ V
S(v, w) = [g ~ GL(V) /# (gv , gw) - r w)}
S(v, w)
T h i s shows that H is a c l o s e d subgroup of GL(V).
We iden t i fy the L ie a l g e b r a of GL(V) with s
4 . 3 . 8 ) . T h e n we have the fo l lowing c h a r a c t e r i z a t i o n of
( s e e p r o p o s i t i o n
LH.
PROPOSITION 7 .6 , 1.
LH = {A C s162 w) + r Aw) = 0 for v ,w ~ V}
P roo f : Let A ~ LH . T h e n e x p t A E H for any t ~ l~ and
r v, exp(tA)w) = r w) for v, w ~ V. D i f f e r e n t i a t i n g with
r e s p e c t to t we obta in for t = 0 , r w ) + r Aw) = 0 .
by
by
by
t E ~ t , which m e a n s
Suppose conversely that A ~ s satisfies this condition. Denote
A* the adjoint linear map of A with respect to ~ , characterized
#(Av, w) - r A* w) = 0. The hypothesis can therefore be expressed
A* = -A . We s h a l l show that (exp tA)* = (exp tA) -1 for e v e r y
e x p t A ~ H for e v e r y t C~ IR . T h i s i m p l i e s
-15 i-
A E LH .
T h e r e r e m a i n s to s h o w t h a t A*
But t h i s f o l l o w s f r o m t h e e x p r e s s i o n
6 . 1 . 5 .
= - A
exp tA
i m p l i e s ( exp tA)* = (exp tA) -1
oo ( tA) n = Eh=0 n,--Hl--- g i v e n in
E x e r c i s e . D e d u c e a l s o p r o p o s i t i o n 7 . 6 . 1 f r o m 6 . 4 . 5 .
if r
g r o u p of V wi th r e s p e c t to r
a l g e b r a c o n s i s t s of t h e o p e r a t o r s
w i th r e s p e c t to r .
i s m o r e o v e r s y m m e t r i c , t h e g r o u p H i s t h e o r t h o g o n a l
a n d denoter O ( V , ~ ) . T h e L i e
of V w h i c h a r e a n t i s e l f a d j o i n t
E x a m p l e 7 . 6 . Z.
w i th a E u c l i d e a n m e t r i c ~ .
b a s e w i th a p o s i t i v e o r i e n t a t i o n .
a l g e b r a by the d e f i n i t i o n [e l , e 2]
C o n s i d e r a 3 - d i m e n s i o n a 1 l R - v e c t o r s p a c e V
L e t e l , e 2 , e 3 be a n o r t h o n o r m a l
T h e n V c a n be t u r n e d in to a L i e
= e 3 , [ e l , e 3] = - e 2 , [ ez , e 3] = e 1 �9
We i d e n t i f y s w i th the L i e a l g e b r a of G L ( V ) ( s e e 4 . 3 . 8 ) . L e t
a ~ V a n d c o n s i d e r A ~ ~(V) d e f i n e d by A v = [ a , v ] f o r v ~ V .
T h e n ~(Av, w ) + ~(v, Aw) = ~ ( [ a , v ] , w ) + ~ ( v , [ a , w ] ) = 0 , a s i s s e e n
by the i n t e r p r e t a t i o n of ([a, v ] , w) a s t he o r i e n t e d v o l u m e of t he p a r a l l e l e p i p e d
d e f i n e d by a , v, w . T h e r e f o r e
L O ( v , 9) of t h e o r t h o g o n a l g r o u p
m a p ~IW-~ LO(V, ~) d e f i n e d by
f o r e v e r y v ~ V i m p l i e s a = 0
d e f i n e d on V . T h i s m e a n s t ha t 3" is i n j e c t i v e . Bu t bo th
LO(V, ~) h a v e d i m e n s i o n 3 , so ~ is a l i n e a r i s o m o r p h i s m .
v e r i f y t h a t 3: V -~ LO(V, ~) i s a n i s o m o r p h i s m of L i e a l g e b r a s .
A i s c o n t a i n e d in the L i e a l g e b r a
O(V, 9) w i t h r e s p e c t to r . T h e
a --,---,> A i s l i n e a r . But [ a, v] = 0
f o r t h i s p a r t i c u l a r L i e a l g e b r a s t r u c t u r e
V a n d
We f i n a l l y
L e t
-152-
A iv = [a i , v ] for v ~ V , i
[A1, A 2 ] v = A 1 A z v - A z A l V -
u s i n g the J a c o b i iden t i ty .
We have t h e r e f o r e s e e n that
the g roup O(V, r by the map ~: V -. LO(V, r , de f ined by
w h e r e Av = [ a , v ] for e v e r y v E V .
T o s ee the g e o m e t r i c s i g n i f i c a n c e of the c o r r e s p o n d e n c e
c o n s i d e r the 1 - p a r a m e t e r s u b g r o u p a of O(V , r de f i ned by
= 1, 2 . T h e n
[a 1,[a 2 . v ] ] - [a 2 , [ a 1 ,v ] ] : [ [a 1,a 2 ] , v ]
V is i s o m o r p h i c to the Lie a l g e b r a of
a ~,--,,~> A ,
s a t i s f y i n g ~t = Aat by 5 . 4 . 5 . Let v ~ V
~r t = &t v = A a t v " A v t can be w r i t t e n as ~z t
that the 1 - p a r a m e t e r s u b g r o u p a t of O(V, 9)
and v t =
= [ a , v t] �9
de f ined by
a ~ A ,
A ,
a tv . T h e n
T h i s show s
A is the
= gAg 1 and t h e r e f o r e o-gA
Le t V be a g a i n of a r b i t r a r y f in i te d i m e n s i o n and ~ a n o n - d e g e n e r a t e d
b i l i n e a r and s y m m e t r i c f o r m on V . Suppose ~ p o s i t i v e de f in i t e . By
the s a m e a r g u m e n t as fo r GL(V) ( s e e e x a m p l e 7 .4 .1 ) , one shows tha t the
c o n n e c t e d c o m p o n e n t of the i d e n t i t y is the k e r n e l of the h o m o m o r p h i s m
1 - p a r a m e t e r g roup of r o t a t i o n s of V with r o t a t i o n ax i s a .
O(V, r o p e r a t e s in V by a u t o m o r p h i s m s of the Lie a l g e b r a s t r u c t u r e
d e f i n e d in V . The i s o m o r p h i s m s ~ : V -*LO(V, r d e f i n e s t h e r e f o r e a
r e p r e s e n t a t i o n of O(V , r in LO(V, r . If o-: O(V, r -~ Aut LO(V, r
d e n o t e s th i s r e p r e s e n t a t i o n , t hen o- is d e f i n e d by (O-gA)(v) = [ga, v] for
g ~ O ( V , r a ~ V , A = J(a) and e v e r y v ~ V . T h i s r e p r e s e n t a t i o n
o- is j u s t the ad jo in t r e p r e s e n t a t i o n of O(V, r , b e c a u s e
(O-gA)(v) = [ga, v] = g[a, g ' l v ]
= g ( A ( g ' l v ) ) = (gAg-1)(v) for e v e r y v ~ V
-15 3-
det: O(V, 4)-~]R* . T h i s g roup is deno ted by SO(V, 4) �9
P R O P O S I T I O N 7 . 6 . 3 .
space , ~ a p o s i t i v e de f in i t e ,
the o r t h o g o n a l g roup of V with r e s p e c t to
of o r t h o ~ o n a l o p e r a t o r s wi th d e t e r m i n a n t 1
SO(V, 47 a r e c o m p a c t .
Le t V be a f in i t e d i m e n s i o n a l l R v e c t o r -
s~ -mmet r i c b i l i n e a r f o r m on V , �9 r
and SO(V, 4) the g roup
. T h e n O(V, ~) and
P roo f : SO(V, 4) i s an open s u b g r o u p of
c l o s e d . Hence it i s s u f f i c i e n t to p r o v e O(V, ~b 7
i s a c l o s e d subg roup of GL(V) .
O(V, ~b) i s a l s o c l o s e d in s �9
O(V, ~b 7 i s bounded in s (V) �9
Now let I I: s be the n o r m
to ~b by
O(V, 4) and t h e r e f o r e
c o m p a c t . Now O(V, 4)
GL(V) be ing an open s u b s e t of ~VT,
It s u f f i c e s t h e r e f o r e to show tha t
on s
= ~(A v,,,, Av) I/z {AI v~oSUp~ V ~(v. v)i/2.
def ined with r e s p e c t
Then any g ~ O(V,r s a t i s f i e s [g[ = 1 and O(V,r i s b o u n d e d i n
s
Let now 4: V ~ V - ~ be a s k e w - s y m m e t r i c b i l i n e a r and n o n - d e g e n e r a t e d
f o r m on V (V of even d i m e n s i o n ) . The s u b g r o u p of GL(V) l e a v i n g
i n v a r i a n t i s the s y m p l e c t i c g roup of V with r e s p e c t to ~ , deno ted
Sp(V, 47 �9 A s t h e r e i s e s s e n t i a l l y a un ique ~ of tha t type , a n y two
s y m p l e c t i c g r o u p s of V a r e i s o m o r p h i c . The L ie a l g e b r a of Sp(V, 47
c o n s i s t s , a c c o r d i n g to 7 .6 .1 , of the o p e r a t o r s of V which a r e a n t t s e l f a d j o i n t
with r e s p e c t to ~ .
- 1 5 4 -
C o n s i d e r now the h o m o m o r p h i s m
is d e n o t e d by SL(V) .
P R O P O S I T I O N 7, 6 . 4 . Le t V
s p a c e . The se t of o p e r a t o r s wi th t r a c e
Lie a l g e b r a of SL(V) ,
P r o o f : B y 4 . 5 . 1 1 , L ( d e t ) = t r .
L ( k e r det) = L(SL(V)) = k e r t r , q . e . d .
det: GL(V) -~ JR* , The k e r n e l
be a f i n i t e - d i m e n s i o n a l l R - v e c t o r -
0 is a Lie a l g e b r a . It is the
Now 7 . 5 . 3 s h o w s
7 . 7 . C o s e t s p a c e s , f a c t o r g r o u p s .
T H E O R E M 7 . 7 . 1 . Le~t G be a L ie g r o u p and H a c l o s e d s u b -
g roup . Le t G / H be the o r b i t s p a c e of the o p e r a t i o n of H on G by
r i g h t - t r a n s l a t i o n s ( s e e Z. Z. 3). C o n s i d e r the n a t u r a l o p e r a t i o n of G on
G / H ( s e c t i o n 1. 4). T h e n t h e r e e x i s t s a un ique s t r u c t u r e of anal~rtic m a n i -
fold on G / H , i n d u c i n g the quo t i en t topo logy and m a k i n g it a G - m a n i f o l d .
Le t H be equ ipped with the s t r u c t u r e of Lie g r o u p of 7 .5 .1 . Le t
M be a v e c t o r - s u b s p a c e of LG such tha t LG = M @LH . D e n o t e by
p: G -~ G / H the c a n o n i c a l p r o j e c t i o n . T h e n t h e o r e m 7 . 7 . 1 is b a s e d on the
fo l l owing l e m m a , which p r o o f we omi t ( s ee S. H e l g a s o n , [ 6 ] , p . 113) .
LEMMA 7 .7 , 2. T h e r e e x i s t s a n e i g h b o r h o o d U
such tha t e x p / U : U - . exp (U) is a h o m e o m o r p h i s m and
- .p (exp U) is a h o m e o m o r p h t s m onto a n e i g h b o r h o o d of
of 0 in M ,
p/exp(U):exp(U)
p(e) in G / H .
- 155 -
T h e s t r u c t u r e of a n a l y t i c m a n i f o l d on G / H is t h e n d e f i n e d a s
o f o l l o w s . If N O d e n o t e s the i n t e r i o r of p (exp U) a n d U the
i n t e r i o r of U, t h e n ( e x p / ~ ) -1 o ( p / e x p (~))-1 a :No-* U c M is a c h a r t
at p(e) E G / H . Now G o p e r a t e s by h o m e o m o r p h i s m s on G / H ,
so t h a t t h i s d e f i n e s a l s o c h a r t s a t a n y p o i n t o f G / H . It i s to s h o w
tha t t h e s e c h a r t s a r e c o m p a t i b l e , i . e . d e f i n e a n a n a l y t i c s t r u c t u r e on
G / H , T h e n , by c o n s t r u c t i o n , G o p e r a t e s by a n a l y t i c m a p s on G / H ,
T h e u n i c i t y of t h e a n a l y t i c s t r u c t u r e on G / H a s a n n o u n c e d in 7 . 7 . 1
f o l l o w s f r o m t h e l a s t s t a t e m e n t in
P R O P O S I T I O N 7 . 7 . 3 . L e t X be a G - m a n i f o l d wi th r e s p e c t to
a t r a n s i t i v e o p e r a t i o n T : G ~ A u t X . S e l e c t x 0 ~ X a n d le t H
be t h e i s o t r o p y group of_ x 0 . C o n s i d e r , t h e m a p (p: G / H -. X d e f i n e d
b_~ ~(gH) = Vg(X0) . L e t G / H h a v e t h e a n a l y t i c m a n i f o l d s t r u c t u r e
d e f i n e d a b o v e . T h e n (p is d i f f e r e n t t a b l e . I f ~ i s a h o m e o m o r p h i s m ,
t h e n it i s a d i f f e o m o r p h i s m .
0 P r o o f : We u s e N O a n d U wi th the s a m e m e a n i n g a s b e f o r e
a n d w r i t e B - e x p ( ~ ) . T h e n the h o m e o m o r p h i s m p / B : B -- N O
p e r m i t s d e f i n i n g B a s a s u b m a n i f o l d of G , m a k i n g the i n j e c t i o n
: B r d i f f e r e n t i a b l e o D e n o t e by
Then ~0/N 0 = qlo L o (p/B) -I and ~0
4 : G - ~ X the m a p q~(g )= l "g (X0) ,
i s t h e r e f o r e d i f f e r e n t i a b l e .
Now s u p p o s e ~ to be a h o m e o m o r p h i s m ( s e e r e m a r k b e l o w ) ,
w i l l be a d i f f e o m o r p h i s m i f t he t a n g e n t l i n e a r m a p of r a t a n y po in t
i s an i s o m o r p h i s m , (p b e i n g a n e q u i v a r i a n c e ( s e e 1 .4 .10) , it i s s u f f i c i e n t
-156-
to p rove th i s for the point x 0 . Now the d e c o m p o s i t i o n r 0 = @oI. o (p /B) "1
s h o w s that it is su f f i c i en t to p r o v e ~ e: Ge -" Tx0(X) to be s u r j e c t i v e . We
s h a l l p r o v e k e r ~*e = He " T h e n r a n k ~ = d i m G - d im k e r ~ e
= d im G - d im H = d im G/H = dim X (the las t equa l i t y b e c a u s e a
h o m e o m o r p h i s m ) , and th i s wi l l f i n i sh the p roof .
T h e r e r e m a i n s to show that k e r @*e = He " H
g roup of x 0 , c l e a r l y HeC ke r @*e " Let c o n v e r s e l y
C o n s i d e r t h e K i l l t n g v e c t o r f i e l d A* on X de f ined by
the c o r r e s p o n d i n g
5 . 6 . 2 . T h e n A
which shows A ~*e
every t ~ ~ .
A ~ R G , i . e . A* = 0-(a)
and A* a r e
= A * = 0 . e x 0
Thus A E H e e
is a
be ing the i so t ropy
A e E k e r @. . e
A e , r e s p e c t i v e l y
in the no t a t i on of t h e o r e m
- r e l a t e d ( s ee the p r o o f of 5 . 6 . 2 ) ,
Then by 6 . 3 . 1 , e x p t A e ~ H for
in view of 7 .5 , 3, q. e. d.
R e m a r k . The map r -~ X de f ined in 7 . 7 . 3
h o m e o m o r p h i s m , if G h a s coun tab ly m a n y c o m p o n e n t s .
cond i t ion , the a r g u m e n t in the p r o o f above shows , for an a r b i t r a r y
G - m a n i f o l d X (with a not n e c e s s a r i l y t r a n s i t i v e o p e r a t i o n ) and
that r --. X is a d i f f e o m o r p h i s m onto the o rb i t of x 0 .
i s in fact a
U n d e r th i s
x 0 ~ X ,
B e f o r e t u r n i n g to the c a s e w h e r e H is a n o r m a l s u b g r o u p of G ,
we give a de f in i t ion .
DEFINITION 7 . 7 . 4 . Let ~ be a Lie a l g e b r a o v e r a r i n g ,~ .
An idea l ~ v of ~ " is a v e c t o r s u b s p a c e of ~ s a t i s f y i n g [A, B] ~: ~ ' ,
for e v e r y A ~ , B E ~ .
If ~ is an i dea l of ~ , the quo t i en t v e c t o r s p a c e ~ / ~ t l is
c a n o n i c a l l y e q u i p p e d with a Lie a l g e b r a s t r u c t u r e , and ~ is the k e r n e l
-15 7-
of the canonical homomorphism ~ -. ~/~ . Conversely, the kernel
of a Lie algebra homomorphism, with domain ~t~ is an ideal of
~-, and the vectorspace isomorphism of ~/~ with the image is a
Lie algebra isomorphism.
We p r o v e now
P R O P O S I T I O N 7 . 7 . 5 , Le t H be a c l o s e d n o r m a l s u b g r o u p of
t h e L i e g r o u p G . The f a c t o r g r o u p G / H with the m a n i f o l d s t r u c t u r e
d e f i n e d in 7 .7 , 1 i s a L ie {~roup. The c a n o n i c a l h o m o m o r p h i s m
p : G -* G / H i n d u c e s L(p): LG -~ L(G/H) with k e r n e l LH , such that
L G / L H "~- L(G/H)
P r o o f : The f a c t o r g r o u p G / H is a t o p o l o g i c a l g r o u p with r e s p e c t
to the quo t i en t topo logy . C o n s i d e r the un ique m a n i f o l d s t r u c t u r e of
7 . 7 . 1 o n G / H such tha t the m a p
(g, xH) ~ g x H is a n a l y t i c .
o p e r a t i o n s in G / H a r e a n a l y t i c ,
of the m a n i f o l d s t r u c t u r e on G / H ,
a h o m o m o r p h i s m of Lie g r o u p s .
Consider L(p): LG -~ L(G/H) , By
Therefore L(p) induces an isomorphism
Note that if H is a normal subgroup of
the factorgroup is not Hausdorff.
G~g G/H -* G/H given by
There remains to show that the group
which i s i m m e d i a t e . By c o n s t r u c t i o n
p : G -~ G / H is a n a l y t i c , and t h e r e f o r e
7 . 5 . 3 k e r L(p) = L ( k e r p) = LH .
LG/LH "= L(G/H)
G which is not closed,
A s a c o n s e q u e n c e we ob ta in
-15 8-
PROPOSITION 7 . 7 . 6 . Let p : G -* G' be a h o m o m o r p h i s m of
Lie g r o u p s . Suppose G connec t ed , C o n s i d e r the _canonical m.ap
p : G / k e r p - 'p(G) induced by p , T h e n p ,is an i s .omorphis .m,of
Lie g r o u p s , whe r e G / k e r p is equipped with the Lie g roup s t r u c t u r e
of 7 . 7 . 5 and p(G) with tha t of 7 . 5 . 4 . T h i s show s in p a r t i c u l a r tha t
the map ~ : G - ~ p ( G ) induced by p is a.nal~-tic.
P roo f : C o n s i d e r the c o m m u t a t i v e d i a g r a m
L(p)
L ( G / k e r p)
L G / L ( k e r p )
G / k e r p P > p (G) C----------------> G '
L(p (G)) r > LG'
L(p )LO
T h e r e is at m o s t one map y: L ( G / k e r p) -, L(p(G)) f i l l i ng in, a s L(p)
is s u r j e c t i v e and L(p(G)) -~ LG' is i n j e c t i v e . C o n s i d e r the c a n o n i c a l
i s o m o r p h i s m y: L G / L ( k e r p) -~ L(p)LG induced by L(p): LG -. LG' .
m a k e s the d i a g r a m c o m m u t a t i v e . T h i s p r o v e s tha t ~ is a n a l y t i c at
e . be ing jus t y in c a n o n i c a l c h a r t s . Hence "~ is e v e r y w h e r e a n a l y t i c
M o r e o v e r L(~ " y is an i s o m o r p h i s m and t h e r e f o r e ~ an i s o m o r p h i s m
-15 9-
of Lie groups. The map ~: G -*p(G) is the compos i t ion "po p of
ana ly t ic h o m o m o r p h i s m s and hence analy t ic ,
-160-
C H A P T E R 8, G R O U P S O F A U T O M O R P H I S M S
8, 1,
d i m e n s i o n a l
~I x ~I -~ ~I . GL(!il}
v e c t o r s p a c e , Au t tl
T h e n A u t ~I c GL(~I)
T h e a u t o m o r p h i s m g r o u p of a n a l g e b r a . L e t ~I be a f i n i t e
~ - a l g e b r a , i, e. a v e c t o r s p a c e w i th a b i l i n e a r m a p
i s t he g r o u p of a u t o m o r p h i s m s of t h e u n d e r l y i n g
i s t h e g r o u p of a u t o m o r p h i s m s of t he a l g e b r a ~I ,
E x a m p l e 8, 1, 1. ~I a ~ - L i e a l g e b r a .
L E M M A 8 . 1 . 2 . A u t ~I i s a c l o s e d s u b g r o u p of GL(~I) .
P r o o f : L e t A, B ~ !ll a n d c o n s i d e r t he m a p
GL(Ill) -~ GL(~I) x GL(92)-~ ~I X ~I -~ ~1
d e f i n e d by q~---~-~(~,~p)-------->(cpA,~B}--~-~--> ~ A . ~pB , T h e m u l t i p l i c a t i o n
a • ~I-~ 92 b e i n g c o n t i n u o u s (~I i s f i n i t e d i m e n s i o n a l } , t h i s m a p is c o n t i n u o u s .
T h e s e t S ( A , B ) = {r ~ G L ( ~ I ) / ~ A . (pB = ~0(A,B)} i s t h e i n v e r s e i m a g e
of r B) u n d e r t h i s m a p a n d t h e r e f o r e c l o s e d in GL(~I) . Now
Aut(~l) = A, BN~ ~I SCA, B) a n d t h e r e f o r e A u t ill c l o s e d in GL(t l )
B y 7 . 5 . 2 we h a v e t h e r e f o r e
P R O P O S I T I O N 8, 1. 3, L e t 92 be a f in i t e d i m e n s i o n a l R - a l g e b r a ,
T h e n A u t ~I i s a c l o s e d L i e s u b g r o u p of GL( ' l l ) . I t s L i e a l g e b r a
i s c h a r a c t e r i z e d by
b(~I) = [ D E s Au t~ I f o r e v e r y t ~: IR} .
-161-
H e r e s deno t e s the Lie a l g e b r a of e n d o m o r p h i s m s of the
u n d e r l y i n g v e c t o r s p a c e of ~/ .
D
D E F I N I T I O N 8.1. 4. A d e r i v a t i o n D
s s a t i s f y i n g
D ( A . B ) = D A . B + A, DB
of ~I is an element
for e v e r y A , B ~
P R O P O S I T I O N 8.1. 5.
d e r i v a t i o n s of 9/ ,
The Lie a l g e b r a b(~/) is the se t of
P r o o f : Le t D E ~(~/) . By 8, 1. 3
exp t D ( A . B ) = (exp tD .A) , ( e x p t D . B)
D i f f e r e n t i a t i n g with r e s p e c t to t
D(A. B)
Conversely, let D
Dn(A, B) i+~--n
we ob ta in for
= DA, B + A . DB and
be a d e r i v a t i o n of ~ .
nl DiA. Dj B i; j;
for e v e r y A, B E ~/ ,
t ~ JR,
t = 0
D is a d e r i v a t i o n of ~/ .
B y induc t ion we get
i > 0 , j > 0
( F o r n = 0 t h i s i s t r u e ,
Now, by 6 .1 .5 we h a v e
T h e r e f o r e
exp tD(A. B)
D ~ b e i n g the i den t i t y . )
O0
exp tD =n=~ (tD)nn ~.
CO
= ~ (tD)n (A .B)
n--0
(exp tD. A) . (exp tD. B)
-162 -
a n d e x p tD E A u t ~ l f o r e v e r y t ~ ~ . B y 8, I. 3 t h i s s h o w s
R e m a r k . T h e f a c t t h a t the s e t o f d e r i v a t i o n s o f ~ is a s u b -
a l g e b r a of t h e L i e a l g e b r a s f o l l o w s a l s o d i r e c t l y a n d i s t r u e
w i t h o u t a n y r e s t r i c t i o n on t h e d i m e n s i o n of ~l . P r o p o s i t i o n 8.1. 5
s u g g e s t s , in t h i s c a s e a l s o , v i e w i n g h e t ~ r i s t i c a l l y t h e L i e a l g e b r a
of d e r i v a t i o n s a s t h e L i e a l g e b r a of t h e g r o u p of a u t o m o r p h i s m s of ~I .
In p a r t i c u l a r , l e t X be a m a n i f o l d a n d CX the 1 R - a l g e b r a
of f u n c t i o n s X-~ ~ . T h e L i e a l g e b r a DX of v e c t o r f i e l d s on X
i s t he L i e a l g e b r a of d e r i v a t i o n s of CX . Now by 4 .1 . 3, A u t X
c a n be i d e n t i f i e d wi th A u t CX . So DX c a n be t h o u g h t of a s t he L i e
a l g e b r a of Au t X , a s we h a v e i n d i c a t e d a t s e v e r a l p l a c e s b e f o r e .
L e t now X be a G - m a n i f o l d wi th r e s p e c t to a n o p e r a t i o n
v : G -. A u t X . It i n d u c e s an o p e r a t i o n 1"* : G -~ A u t CX . C o n s i d e r
t he h o m o m o r p h i s m 0":RG -. DX of 5 . 6 . Z . It c a n be t h o u g h t of b e i n g
t h e h o m o m o r p h i s m of L i e a l g e b r a s i n d u c e d by t h e h o m o m o r p h i s m ~-* ,
r e m a r k s f o r a L i e a l g e b r a ~ , o v e r a r i n g ~ .
A n y e l e m e n t A E ~ ' g i v e s r i s e to a l i n e a r m a p
by the d e f i n i t i o n (ad A) (B) = [A, B] .
T h e ad~oint r e p r e s e n t a t i o n of a L i e a l g e b r a . We b e g i n wi th s o m e
ad A: ~ t -~
L E M M A B. 2.1. ad A i s a d e r i v a t i o n of ~t~ .
- 16 3 -
Proof : The Jacob i identi ty can be wr i t t en in the fo rm
[A, [Bl, ]32]] = [[A, Bl] , B2] + [B V [.4., B2]]
which proves the desired result.
DEFINITION 8 . 2 . 2 . Let ~ be a Lie a lgeb ra . The i n n e r
der iva t ion of ~ defined by A r ~," is the map ad A : ~ - ~ ~t~ .
Cons ide r the map ad:
of e n d o m o r p h i s m s of ~ - .
s into the Lie a l g e b r a
LEMMA 8 . 2 . 3 .
a l g e b r a s .
ad: ~ -~ s is a h o m o m o r p h i s m of Lie
Proof:
name ly
This is again a consequence of the J acob ian iden t i ty ,
(ad [A1, A2])(B ) = [[-4,1, A 2 ] , B ]
= [ A I , [ A 2 , B ] ] - [ A z , [ A 1 , B ] ]
= ( a d A 1 ~ a d A z ) ( B ) - ( a d A 2
= [ a d A 1 , a d A g ] ( B ) ,
o ad AI)(B )
q.e.d.
We have seen before that a d ( 7 ) c 3(~') , where 3 (~.) is
the Lie a l g e b r a of de r iva t ions of o~ , s u b a l g e b r a of the Lie a l g e b r a
s . We shal l a lso wr i te a d : ~ -. 3(~) for the h o m o m o r p h i s m
induced by ad: g<~-, s .
DEFINITION 8 . 2 . 4 . Let ~ be a Lie a lgeb ra .
a d : ~ - ~ ~(~) is ca l led the adjoint r e p r e s e n t a t i o n of ~
r e p r e s e n t a t i o n of g in r
The h o m o m o r p h i s m
, It is a
-164-
of
i d e a l of ~ ,
Now let G be a Lie group.
group. The Lie algebra is by 8. I, 4
LG ,
Consider the adjoint representat ion of
By the preceding it induces a homomorphism
T h e i m a g e of t h i s h o m o m o r p h i s m is the s e t of i n n e r d e r i v a t i o n s
, wh ich t h e r e f o r e f o r m s a Lie a l g e b r a .
Le t ~(~'~) deno te the k e r n e l of t h i s h o m o m o r p h i s m . It is an
c a l l e d the c e n t e r of ~ , and is c h a r a c t e r i z e d by
if and on ly if [ A , B ] = 0 fo r a n y B g ~" .
By 8 . 1 . 3 Aut LG is a Lie
the s e t of d e r i v a t i o n s b(LG) of
G in LG, Ad:G -.Aut LG .
L(Ad): LG -~ ~(LG) .
T H E O R E M 8 . 2 . 5 . L(Ad) = ad
P r o o f : Le t A ~ L G . T h e n
L(Ad)A = ~ d {Ad exp tA}t = 0
by de f i n i t i on of L(Ad) .
a d A , q . e . d .
But by 5 . 5 . 8 , the s e c o n d m e m b e r i s j u s t
C O R O L L A R Y 8 . 2 . 6 .
L ( A d G ) = ad (LG) .
Le t G be a c o n n e c t e d Lie g roup . T h e n
Proof: By 7 .5 .4 we have L(AdG) = L( im Ad) = im L(Ad)
= im ad = ad LG, q . e . d .
C O R O L L A R Y 8 . 2 . 7 . Le t G be a c o n n e c t e d Lie g roup . T h e n
the c e n t e r ZG is a L ie s u b g r o u p of G . I t s L ie a l g e b r a is the c e n t e r
of LG .
- 1 6 5 -
P r o o f : B y 6 . 2 . 1 0 we k n o w t h a t Z G
Z G i s t h e r e f o r e 2 c l o s e d L i e s u b g r o u p o f
L ( Z G ) = L ( k e r A d ) = k e r L (Ad) = k e r a d
c e n t e r o f L G .
N o t e t h a t A d : G ~ A u t L G
G / Z G ~ A d G
= k e r A d . B y 7 . 5 . 3 ,
G w i t h L i e a l g e b r a
. B u t k e r a d i s t h e
i n d u c e s a n i s o m o r p h i s m
o f L i e g r o u p s ( s e e 7, 7 . 6 ) .
C O R O L L A R Y 8 . ~ . B . e x p a d A = A d e x p A f o r A @ LG.
P r o o f : T h i s i s t h e n a t u r a l i t y of e x p .
We s h a l l m a k e u s e o f t h e f o l l o w i n g t w o l e m m a s .
G
V
A v e c t g r s p a c e
P r e c i s e l y : TgW c
f o r e v e r y A @ L G .
L E M M A 8. a. 9. L e t V b e a f i n i t e d i m e n s i o n a l ~ : v e c t o r s p a c e ,
a c o n n e c t e d L i e g r o u p , T: G -. G L ( V ) a r e p r e s e n t a t i o n o f G in
a n d L(v) : L G -. s t h e i n d u c e d r e p r e s e n t a t i o n of L G ha V .
W ~ V i s G - i n v a r i a n t i f a n d o n l y i f i t i s L G - i n v a r i a n t .
W f o r ever~r g E G i f a n d o n l ~ r i f ( L ( v ) A ) W c W
P r o o f : S u p p o s e W c V to be G - i n v a r i a n t a n d l e t A E LG ,
w @ W ,
(L(-r)A)w - ~ T e x p tA
w h i c h t s t h e t a n g e n t v e c t o r o f t h e c u r v e
t ; 0 , a n d t h e r e f o r e (L( ' r )A)w ~? W ,
I t=O w = d {T , e x p tA w I It=0
t ~ T W in W f o r e x p tA
S u p p o s e c o n v e r s e l y W (- V to be L G - i n v a r i a n t , i . e . f o r e v e r y
A ~ L G we h a v e ( L ( T ) A ) W c W o N o w by 6 . 1 . 5 f o l l o w s i m m e d i a t e l y
-166 -
( exp L(a-)A)W c W . B y t h e n a t u r a l i t y 6 . 1 . 6 of exp t h i s i s e q u i v a l e n t
to Tex p AW ~ W for e v e r y A ~ m . Let g - [g ~ G ] ~ g W ~ W } .
T h e n ~ i s a s u b g r o u p of G . B y the p r e c e d i n g , ~ c o n t a i n s a
n e i g h b o r h o o d of e in G a n d t h e r e f o r e ~ = G .
L E M M A 8. Z, i0. L e t G, G' be L i e ~ r o u p s a n d H, H ' c o n n e c t e d
L ie s u b g r o u p s of G, G' r e s p e c t i v e l y . L e t p: G -. G ' be a h o m o -
morl~h_i_sm. T h e n p(H) c H' i f a n d onl~r i f L ( p ) L H c LH ' .
P r o o f : C l e a r f r o m 7 . 5 . 4 .
L e t ~ be a L i e a l g e b r a . T h e d e f i n i t i o n 7 . 7 . 4 of a n i d e a l o f
c a n be r e s t a t e d by s a y i n g t h a t a v e c t o r s p a c e ~ c ~ is a n i d e a l if a n d
if ~ is a d ~ - i n v a r i a n t . o n l y
L e t G be a L i e g r o u p . We h a v e s e e n in 7 . 7 . 5 t h a t t h e L i e
a l g e b r a of a c l o s e d n o r m a l s u b g r o u p of a L i e g r o u p G is a n i d e a l
of LG . We a r e n o w a b l e to p r o v e
P R O P O S I T I O N 8 . 2 , 11. L e t G be a c o n n e c t e d L i e g r o u p a n d
H a c o n n e c t e d L i e s u b g r o u p of G . T h e n H i s a n o r m a l s u b g r o u p
o f G if a n d o n l y if LH i s a n i d e a l of LG ,
P r o o f : LH is a n i d e a l of LG
i n v a r i a n t . In v i e w of L(Ad) = a d
t h e A d G - i n v a r i a n c e of L B , L e. A d g LH ~ LH f o r e v e r y
Bu t A d g = L(~g) by d e f i n i t i o n , a n d u s i n g 8 . 2 ; 10 we s e e t h a t
A d g LH c LH if a n d o n l y if ~ ( H ) c H . T h e r e f o r e LH
if a n d o n l y if LH i s a d L G -
a n d 8. ~. 9, t h i s i s e q u i v a l e n t to
g E G
i s a n
ideal of
A d G
-167-
LG if and on ly if ;Yg(H) c H for e v e r y g E G , q . e . d .
COROLLARY 8 .2 .12 . Let G
is a n o r m a l Lie s u b g r o u p of
b e a c o n n e c t e d Lie group. T h e n
(Aut LG)0 .
P roo f : Ad G is c o n n e c t e d and t h e r e f o r e c o n t a i n e d in (Aut LG)0 ,
Now L(Ad) = ad LG by 8 . 2 , 6 . In v iew of 8 . 2 , 1 1 t h e r e is only to show
that ad LG is an i d e a l of L(Aut LG) = ~(LG) , Th i s is t r u e for an
a r b i t r a r y Lie a l g e b r a ~> , Le t n a m e l y D ~ ~ ) , A ~ ~ . T h e n
t h e r e i s to show [D, a d A ] ~ ad ~ . F o r B ~ ~ we have
[D, a d A ] B = D [ A , B ] - [A, DB] = iDA, B] = ( a d D A ) B , which shows
[D, a d A ] = a d D A .
R e m a r k . The g r o u p Ad G is not n e c e s s a r i l y c l o s e d in Aut L G .
8 .3 . The a u t o m o r p h i s m g roup of a Lie g roup . Let G be a Lie g roup
and Aut G the g roup of a u t o m o r p h i s m s (of the Lie g roup s t r u c t u r e ;
h o w e v e r , r e m e m b e r 6 . 3 . 4 ) . The func to r L d e f i n e s a h o m o m o r p h i s m
L : A u t G -, Aut LG into the g roup of a u t o m o r p h i s m s of LG . If G
~s c o n n e c t e d , 6. Z. 9 shows tha t th i s h o m o m o r p h i s m is i n j e c t i v e .
Aut
Aut
by
E x a m p l e 8 .3 .1 .
" l r - . Aut (L"II') = G L ( ~ ) = ~ * is i n j e c t i v e . In f a c t ,
"11" = {12r, -1,]/,} , w h e r e -1T d e n o t e s the m a p induced on
- l ~ ( s e e 8 . 3 . 4 ) .
If
L : A u t G-* Aut LG
C o n s i d e r the L ie g roup "It = ~ t / z . T h e n
-ff
G is c o n n e c t e d and s i m p l y c o n n e c t e d , the h o m o m o r p h i s m
is an i s o m o r p h i s m by 7. Z. 7.
- 1 6 8 -
E x a m p l e 8 . 3 , 2 . G = ~ . T h e n A u t ~t = ~{* .
M o r e g e n e r a l l y , l e t G
c o n n e c t e d L i e g r o u p . T h e n
A u t G -. Au t LG = G L ( L G )
be a c o m m u t a t i v e c o n n e c t e d a n d s i m p l y
e x p : L G -, G i s an i s o m o r p h i s m by 7 . 2 . 8 .
i s an i s o m o r p h i s m .
L e t G be a c o m m u t a t i v e and c o n n e c t e d L i e g r o u p , A n a u t o -
m o r p h i s m r of G d e f i n e s an a u t o m o r p h i s m L(r of LG . C o n s i d e r
t he h o m o m o r p h i s m exp : LG -* G . T h e n L(r k e r exp c k e r exp
in v i e w of t h e c o m m u t a t i v e d i a g r a m
LG _L(@) > LG
exp i l e x p
G ~ > G
We h a v e p r o v e d h a l f of
P R O P O S I T I O N 8 . 3 . 3 . L e t G be a c o m m u t a t i v e c o n n e c t e d L i e
g r o u p . T h e n the i m a g e of t h e h o m o _ m o r p h i s m L: Au t G -~ G L ( L G )
c o n s i s t s o f the a u t o m o r p h i s m ~ o f LG wi th ~ k e r exp c k e r exp .
P r o o f : We h a v e to s h o w t h a t g i v e n ~ 6 G L ( L G ) wi th
~ k e r exp c k e r exp , t h e r e e x i s t s r 6 Au t G wi th L(~o) = cp
( e x p , ~ ) k e r exp = e i m p l i e s t h a t t h e r e e x i s t s a f a c t o r i z a t i o n
of exp , r t h r o u g h exp a n d c l e a r l y L(cp) = $ .
�9 But
~ : G - * G
-169-
R e m a r k . T h e r e is a s i m i l a r c h a r a c t e r i z a t i o n of Aut G for
an a r b i t r a r y c o n n e c t e d L ie g roup �9 One h a s on ly to c o n s i d e r the
u n i v e r s a l c o v e r i n g g r o u p ~ and the cove rLng h o m o m o r p h i s m ~ - . G .
P r o p o s i t i o n 8 . 3 . 3 a l l o w s u s to d e t e r m i n e Aut G,
G ~-- L G / k e r exp . We show
a s
P R O P O S I T I O N 8 . 3 . 4 . Le t G = 'it n T h e n A u t G ~" A u t Z n
P r o o f :
Z n T h e n
T h i s s h o w s
We d e n o t e by T.. n a s u b g r o u p of LG i s o m o r p h i c to
T n ~ L G / Z n Now Aut T n = [ ~ G L ( L G ) / ~ ( Z n) c z n ~
Aut"Jr n ~ Aut Z n 0 q . e . d .
Appendix. Cate_~pries and fu.nctors
-170-
Definition.
(i)
(ii)
A ca t ego ry e cons i s t s of
a c l a s s of objec ts A, B, C, . . . ;
for each pa i r (A,B) of objec ts a set
a r e ca l led m o r p h i s m s f rom A to B
range B (we wri te a : A - . B or
these se t s being pa i rw i se d is jo in t :
[A,B] r] [A',B'] : r
(iii) for each triple (A, B, C)
[A,B] x [B,C]
A
(A,B)
of ob jec t s a map
> [A,C]
(a, 8) ~ ~a
ca l led compos i t ion of m o r p h i s m s ;
(iv) for each object A an e l emen t 1 A
m o rph i s m s;
these data being subjec t to the two ax ioms
(I) If ae [A,B], Be
(Z) If a e[/%,B], then
R e m a r k . The m o r p h i s m 1 A
is un ique ly defined by condi t ion Z.
with thee same p r o p e r t i e s , then 1 A' 1 A
e [A,A],
[B,C], ~ e[C,D], then
al A =a , IBa = a .
[A, B ] , which e l e m e n t s
or with domain A and
a >B for a ~ [ A , B ] )1
( A ' , B ' ) imp l i e s
ca l led ident i ty
whose ex i s t ence is r e q u i r e d by (iv)
Because if l~ is a second m o r p h i s m
=l A, =I A .
Examples . The c a t e g o r y Ens
m o r p h i s m s the maps between se t s with the u sua l compos i t i ons . The
c a t e g o r y ~ of groups is def ined by the groups as objec ts , group
whose ob jec t s a r e the se t s and
-171-
h o m o m o r p h i s m s as m o r p h i s m s and the u s u a l c o m p o s i t i o n of h o m o m o r p h i s m s .
T a k i n g the t o p o l o g i c a l s p a c e s a s o b j e c t s and the c o n t i n u o u s m a p s
a s m o r p h i s m s wi th the u s u a l c o m p o s i t i o n , we ob t a in the c a t e g o r y
of t o p o l o g i c a l s p a c e s . S i m i l a r l y the c a t e g o r y ~l of d i f f e r e n t i a b l e
m a n i f o l d s is d e f i n e d by t ak ing the d i f f e r e n t i a b l e m a n i f o l d s a s o b j e c t s and
d i f f e r e n t i a b l e m a p s a s m o r p h i s m s .
Le t
a:A -B
6: B ~A
va lenc e
isomorphic :
a: A ~A .
be a c a t e g o r y and A, B o b j e c t s of ~ . A m o r p h i s m
is c a l l e d an e q u i v a l e n c e o r an i s o m o r p h i s m , if t h e r e e x i s t s
wi th ~a = 1 A and a~ = 1 B . If t h e r e e x i s t s a n e q u i -
a : A - ~ B , t hen A and B a r e s a i d to be e q u i v a l e n t o r
A ~ B . An a u t o m o r p h i s m of A is an e q u i v a l e n c e
Def in i t i on : Le t
F:e - e' f r o m ~ to
(i) of an o b j e c t
(ii) of a m o r p h i s m
a:A ~B of ~ ;
and ~' be c a t e g o r i e s .
~' is the a s s i g n m e n t
FA of ~' to e a c h o b j e c t
Fa : FA - . F B of ~'
s u b j e c t to the two c o n d i t i o n s
(i) F(I A) = IFA
(Z) F(~a) = F(~)F(a)
If the c o n d i t i o n (Z) is r e p l a c e d by
(Z~ F(~a) = F(a)F(~),
we s p e a k of a c o n t r a v a r i a n t f u n c t o r F: e - e'
A c o v a r i a n t f u n c t o r
A of ~ ;
to e a c h m o r p h i s m
-172 -
E x a m p l e s . L e t e be a c a t e g o r y a n d A
O n e c a n d e f i n e a c o v a r i a n t f u n c t o r hA: e -. E n s
hA(X) = ~%,X] f o r a n y o b j e c t X of ~ , h A(cp)(a) = Ca f o r
~p:X-~ X ' , a : A - . X , H e r e we h a v e hA(~) : [A, X] -~ [ A , X ' ] .
S i m i l a r l y we c a n d e f i n e a c o n t r a v a r i a n t f u n c t o r hA: e - . E n s b y
hA(x) = i X , A] f o r a n y o b j e c t X of e a n d hA(~ ) ( a ) = 040
a n o b j e c t o f ~ .
in t he f o l l o w i n g way :
f o r
X' a : X ' -. A H e r e we h a v e hA(~p): iX' A ] -. iX, A ] $ �9 �9 �9
D e f i n i t i o n . L e t
( c o v a r i a n t ) f u n c t o r s f r o m
r f r o m F to G
r -. GX to e a c h o b j e c t X
c o m m u t e s f o r e v e r y ~p:X -, Y
a n d e ' be c a t e g o r i e s a n d F , G:~-~ e '
to ~ ' . A n a t u r a l t r a n s f o r m a t i o n
is the a s s i g n m e n t of a m o r p h i s m
of e , s u c h t h a t the f o l l o w i n g d i a g r a m
CX F X ~ GX
q~y , FY > GY
E x a m p l e . L e t
K - v e c t o r s p a c e s a n d
( a s s i g n i n g to e a c h v e c t o r s p a c e X i t s d u a l s p a c e X '
K - l i n e a r m a p i t s d u a l m a p ) . T h e f u n c t o r D2: K ~ - . K ~
v e c t o r s u b s p a c e o v e r K i t s b i d u a l . T h e e v a l u a t i o n
~x(X)(X' ) = < x, x'~% for
formation E:IK~-. D z ,
K be a c o m m u t a t i v e f i e l d , KI~ the c a t e g o r y of
D: K ~ -. K ~ t h e f u n c t o r d e f i n e d by the d u a l i t y
a n d to e a c h
a s s i g n s to e a c h
x ~ X , x ' ~ X ' d e f i n e s a n a t u r a l t r a n s -
-173-
N a t u r a l t r a n s f o r m a t t o n s a r e c o m p o s e d in an o b v i o u s way .
A n a t u r a l t r a n s f o r m a t i o n ~: F -* G i s a n a t u r a l e q u i v a l e n c e
if t h e r e e x i s t s a n a t u r a l t r a n s f o r m a t i o n ~: G -* F s u c h tha t
@~ = 1 F , ~@ = 1 G , 1 F a n d 1 G d e n o t i n g t h e i d e n t i c a l n a t u r a l t r a n s -
f o r m a t i o n s F-* F and G -~G r e s p e c t i v e l y .
P r o d u c t a n d s u m s , L e t
i s t e r m i n a l , i f to e a c h o b j e c t
K -. T . H e n c e the o n l y m o r p h i s m T - .T i s
t e r m i n a l o b j e c t s in ~ a r e e q u i v a l e n t .
L e t (Kj) j ~ Z be a f a m i l y of o b j e c t s of ~ i n d e x e d by a s e t
Z . C o n s i d e r the c a t e g o r y P (Kj) w h o s e o b j e c t s a r e i n d e x e d
f a m i l i e s [ q j : Q - . K j / j ~ ~ } of m o r p h i s m s of K wi th a c o m m o n
be a c a t e g o r y , A n o b j e c t T of
K t h e r e i s e x a c t l y one m o r p h i s m
i T , a n d any two
d o m a i n Q , w h i l e a m o r p h i s m
fo r w h i c h q j ' a = qj f o r j
i s a p r o d u c t of t he Kj , t h u s
( q j ) - ~ ( q j ' ) in P ( K j ) i s an a : Q - * Q '
. A t e r m i n a l o b j e c t in P (Kj)
f a m i l y of m o r p h i s m s qj : Q -~ Kj
u n i q u e a : Q - ~ P , T h e p r o d u c t ,
up to an e q u i v a l e n c e in 9 (Kj)
P
D E F I N I T I O N � 9 A p r o d u c t of (Kj) j ~ [y i s an o b j e c t P of
t o g e t h e r wi th m o r p h t s m s pj: P - . Kj f o r j ~ Z , s u c h t h a t any
c a n be w r i t t e n a s qj = p ja f o r a
l ike any t e r m i n a l o b j e c t , i s u n i q u e
In p a r t i c u l a r , t h e p r o d u c t - o b j e c t
i s u n i q u e up to an e q u i v a l e n c e in ~ .
L e t R be a c a t e g o r y , A n o b j e c t S of R i s i n i t i a l , i f to
e a c h o b j e c t K t h e r e i s e x a c t l y one m o r p h i s m S -~ K . H e n c e the
-174-
only morphism
va lent.
S --. S in 1 and any two in i t ia l ob jec ts a r e equi-
Let (Kj)j s g
g . C o n s i d e r the c a t e g o r y ~ ( K j )
~9k:K j - ~ R / j s } o f m o r p h i s m s of
a m o r p h i s m ( p j ) - , ( p j ' ) in ~ ' ( K j )
apj : pjl for
o f t h e K j , thus
be a family of objects of ~ indexed by a set
whose objects are indexed families
with common range R, while
is an a:R--R' for which
j g ;~ . An initial object in this category is a sum
DEFINITION, A sum of
toge the r with m o r p h i s m s 0-j: Kj
f ami ly of m o r p h i s m s 9 j :Kj - R
unique a :S-~ R .
(Kj)j g g is an object S of
- S for j ~ ~Y o such that any
can be wr i t t en a s a~j = pj for a
The sum is unique up to an equiva lence in ~ (Kj) , in p a r t i c u l a r ,
the s u m - o b j e c t is unique up to an equiva lence in ~ .
B IB LIOGRA PHY
-175 -
[1] Bruhat , F . , A lg~bres de Lie et groupes de Lie, Textos de
Mathemat i ca , Univ, do Reci fe , Vol. 3 (1961).
EZ] Bruhat , F , , L e c t u r e s on Lie groups and r e p r e s e n t a t i o n s of
loca l ly compact groups. Tata Inst i tute of fundamenta l r e s e a r c h ,
Bombay, 1958,
[3] Cheva l ley , C . , Theory of Lie Groups, Vol, I, P r i nce ton Univ.
P r e s s , P r ince ton , N. J. (1946).
E4] Cohn, P, M. , Lie Groups, C a m b r i d g e Univ, P r e s s , C a m b r i d g e
(1957).
[5 ] Graeub, W., L ie sche Gruppen und aff in zusammenh~ngende
Mannigfa l t igkei ten , Acta Math, 106 (1961), 65-111.
[61 Helgason, S. , D i f f e r en t i a l geome t ry and s y m m e t r i c spaces ,
A c a d e m i c P r e s s (1962).
[7] Hof/man, K. H, , Einfi~hrung in die Theor ie der Liegruppen,
Te i l I. V o r l e s u n g s a u s a r b e i t u n g , Math. Jns t , Universit'~tt
Ti lbingen (1963),
ES] Koszul , J. L . , Exposes sur les spaces homog~nes sym~t r iques .
I 1 . Soc, Math, S~o Paulo ~959).
[91 L ichnerowicz , A . , G~omet r i e des groupes de t r a n s f o r m a t i o n s ,
Dunod, P a r i s (1958).
El0] Mai s sen , B . , L i e -Gruppen mi t B a n a c h r ~ u m e n a l s P a r a m e t e r r ~ u m e ,
Acta Math, (1962) 229-269.
-176-
[11] Nomizu, K. and Kobayashi, S., Foundations of differential
geometry, Vol. I . , Interscience, N. Y. (1963).
[1Z] Palais, R. , The classification of G-spaces, Memoirs AMS,
Vol. 36 ( 1 9 6 o ) .
[13J Palais, R. , A global formulation of the Lie theory of t ransformation
groups, Memoirs AMS, Vol. 2Z (1957).
[14] Pontrjagin, L. S . , Topologische Gruppen, VoL II, Teubner
Leipzig (1958).