Introduction to Lie Groups and Transformation Groups

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 Transformation Groups 1965

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

These

notes

were

and transformation and Zurich. map

groups,

The notions

and a vectorfield

categories

In sections [11] .

are

7, S. H e l g a s o n

[ 3] w a s c o n s t a n t l y

on t h e v a r i o u s

sources.

avoidance

allows

the use

groups

over

1964

supposed

known.

There

on Lie groups

of B u e n o s

manifold,

influenced

Aires

a differentiable

is an appendix

by a paper

5. Z a n d 5. 3, a l o t i s t a k e n f r o m

Chevalley

June

are

lectures

held at the Universities of a d i f f e r e n t i a b l e

two chapters

In chapter

systematic

for introductory

on

and functors.

The first

C.

written

of t h e u s e

manifolds.

was

feature

of l o c a l theory

S. K o b a y a s h i

often used.

consulted.

A special

of t h e p r e s e n t e d

Banach

[61

of R .

Palais

a n d K. N o m i z u

Of course,

The bibliography

orients

of t h i s p r e s e n t a t i o n

is the

coordinates

on a manifold.

with slight modifications

See e.g.B.

[lg] .

Maissen

This

for Lie

[10].

Philippe

Tondeur

CONTENTS

.

G -Objects.

1.1.

Definition and examples.

1

1. Z.

Equivariant

7

1.3.

Orbits.

"1.4. .

.

.

morphisms.

13

Particular

G-sets.

23

G-Spaces. Z. 1.

Definition and examples.

28

Z.Z.

Orbitspace.

30

G-Manifolds. 3.1.

Definition and examples

of L i e g r o u p s .

34

3. Z.

Definition and examples

of G - m a n i f o l d s .

37

Vector fields. 4. 1.

Real functions.

40

4. Z.

Operators

4Z

4.3.

The Lie algebra

4.4.

E f f e c t of m a p s

4. 5.

T h e f u n c t o r L.

4. 6.

Applications

4. 7.

T h e adjoint representation of a Lie group.

and vectorfields. of a L i e g r o u p . on o p e r a t o r s

46

and vectorfields.

of t h e f u n c t o r a l i t y

50 5Z

of L.

59 64

T h e * indicates a section, the lecture of w h i c h is not n e c e s s a r y for the understanding of the subsequent developments.

.

.

Vectorfields

groups

~roups

of t r a n s f o r m a t i o n s .

66

5.1.

1-parameter

5. Z.

1-parameter groups equivariant maps.

5. 3.

The bracket

of t w o v e c t o r f i e l d s .

74

5.4.

1-parameter

subgroups

77

5.5.

Killing vectorfields.

*5.6.

The homomorphism

*5. 7.

Killing vectorfields

The exponential

of t r a n s f o r m a t i o n s . of t r a n s f o r m a t i o n s

and

of a L i e g r o u p .

70

84

aV: R G

> DX f o r a G - m a n i f o l d .

and equivariant

maps.

89 96

m a p of a L i e g r o u p . 103

6.1.

Definition and naturality

6. Z.

e x p is a l o c a l d i f f e o m o r p h i s m

6.3.

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 .

1 iZ

Application

116

*6.4. 6.5.

.

and 1-parameter

of e x p . at t h e i d e n t i t y .

to fixed points on G-manifolds.

IZO

Taylor' s formula.

Subgroups

108

a n d s u b a l g e b r a , s.

7.1.

Lie subgroups.

128

7. Z.

Existence

132

7. 3.

Discrete

7.4.

Open subgroups;

7.5.

Closed

subgroups.

7.6.

Closed

subgroups

of l o c a l h o m o m o r p h i s m s .

138

subgroups.

142

connectedness.

144

of t h e f u l l l i n e a r

7. 7. Cosetspaces and factor groups.

group.

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

8. Z.

The adjoint representation

8. 3.

The automorphism

Appendix:

g r o u p of an a l g e b r a .

g r o u p of a L i e g r o u p ,

C a t e g o r i e s a nd f u n c t o r s .

Bibliography

of a L i e a l g e b r a ,

160 16Z 167 170 175

-1-

Chapterl.

G-OBJECTS

T h e first two p a r a g r a p h s of this chapter are essential for all that follows, w h e r e a s p a r a g r a p h s i. 3 and i. 4 are only required for the lecture of g. g and shall not be used otherwise.

F o r the notion of

category and functor, see appendix. i. i

Definition and examples. If X

is an object of a category

~ , w e denote by Aut X the group

of equivalences of X with itself. Let G DEFINITION r: G

>Aut

I. i. I

X.

X

be a group.

A n operation of G

on X

is a h o m o m o r p h i s r n

is called a G-object with respect to

A n o p e r a t i o n of G on X is a representation of G

T.

by automorphisrns

of X . Example

1.1.2

AG-object

X in the category

set X

equipped with a homomorphism

of X .

Such a homomorphism

is equivalently defined by a m a p

GxX

satisfying

Ens

is a

7 of G i n t o t h e g r o u p of b i j e c t i o n s

by the same letter)

(g, x)

of s e t s

>

~ - ~ ~

X

v (x) g

(denoted

-Z-

a)

(x)

=

v

Tglgz

b)

(v gl

T (x)

=

e

(x))

for

gz C G

gz

gl'

x

for

xC

X

'

e C G, x C X

T h e last conditions in the e x a m p l e i. i. Z suggest calling an operation in the sense of definition i. i. i m o r e precisely a leftoperation

of G

on X .

A right-operatio

be a h o m o m o r p h i s m

~" : G ~

opposite

i.e.

g r o u p of G ,

multiplication

> A u t X, w h e r e

the underlying

( g l g g ) o = g z g 1.

shall generally

n of G

use the word

on X w i l l t h e n G ~ is the

s e t of G w i t h t h e

X is then a G o -object.

operation

as synonymous

We for

left-operation and only be m o r e precise w h e n right-operations also occur.

Example

1.1. 3

Let G

be a group.

If to any g C G w e

assign the corresponding left translation L L (~/) = g

gv

for

V g G,

the underlying set of G . translation R g

of groups.

Let

defined by on

Similarly, the assignment of the right

of G , Rg(V)

1. I. 4

of G

w e obtain a left-operation of G

=

defines a right-operation of G Example

g

Vg for

on the underlying set of G .

p :G

> G' be a h o m o m o r p h i s m

It defines an operation

set of G' in the following way:

V C G , to any g 6 G

set

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

O-g

= Rp

(g)

Example

for

a right operation

.

g g G

1.1. 5

L e t G be a g r o u p .

assign the inner automorphism for

~ C G. Example

consider

H on t h e s e t Example by

T : G

To any g C G

we

i n d u c e d b y g, ~I ( ~ ) = g~/g g g

-1

T h i s d e f i n e s a n o p e r a t i o n of G on i t s e l f . 1.1.6

the map

multiplication

o- b y t h e definition

L e t H b e a s u b g r o u p of t h e g r o u p G a n d

G x H

O x O

> G defined by restricting > O.

the

It d e f i n e s a r i g h t - o p e r a t i o n

of

O. Let the group G operate

1.1. 7

> Aut G'.

on t h e g r o u p G '

On t h e s e t G' x G t h e m u l t i p l i c a t i o n

law

(gl" gl)(gz" gz ) = (gl ''r gl (gz ')' glg2 ) for

defines a group structure, O' x v G.

Consider

j :G'

>G'

p : G' x ~

G

s :G

> G' x

the semi-direct

gig

G (i = ~, Z)

product denoted

the homomorphisms

x~: G > G

T

gi' g G',

G

j(g')

= ( g ' , e)

f o r g' 6 G ' ,

e neutralinG g E G

p ( g ' , g) =

g

f o r g' g G ' ,

s(g)

(e' J g)

for e' neutral in G'

=

g GG

-4-

The sequence

(*)

with

P

e

> G'

= G'x

G

T

P

is e x a c t a n d

an exact sequence pos

J > P

> G

> e

s satisfies p oS = 1G

(*) a n d a h o m o m o r p h i s m

s : G

> P with

= 1G (a s p l i t t i n g of (*)) d e f i n e s a n o p e r a t i o n

the automorphism

-r o f G ' g

inner automorphism normalsubgroup

of P

G'.

1.1. 8

is as follows: and GL(V) operates

Let

G-groups

A typical case

on V .

The semi-direct of V .

exists

a functor

forgetting a morphism

V :~

More precisely > Ens

about the additional

of V.

Then

product

GL(V)

V xGL(V)

of a f f i n e m o t i o n s .

categories

set and whose morphisms sets.

IR-vectorspace

Note that the multiplication

composition

We shall only have to consider

of t h e u n d e r l y i n g

of t h e s i t u a t i o n j u s t m e n t i o n e d

automorphisrns

to the natural

have an underlying

to the

a r e in ( 1 - 1 ) - c o r r e s p o n d e n c e

V be a finite-dimensional

i s t h e g r o u p of a f f i n e m o t i o n s just corresponds

restricted

(*).

t h e g r o u p of l i n e a r

naturally

T of G on G v :

to g ~ G is the

defined by s(g),

with splitting exact sequences

Example

corresponding

Therefore

Conversely

~ whose

objects

are applications

this means

that there

w h i c h c a n b e t h o u g h t of a s

structure

just as an application.

on X in

R

To avoid endless

and taking repetitions

-5-

we make

t h e following c o n v e n t i o n :

only consider notation

categories

X for an object

An operation the underlying

of t h e g r o u p More

PROPOSITION

1.1.9

from

of t h e group ~

G

on X

VX.

an operation

on

we have the

Let F : R - - >

g' b e a c ovariant

to the category

C~

set

G on X d e f i n e s

generally

t h e c a_tegor_~y

n o w on w e s h a l l

and shall use the same

X and its underlying

set.

functor

FX

of t h a t s o r t

From

R'.

A n operation

induces a well-defined operation on

~.' Proof:

F

defines a h o m o m o r p h i s m

Aut X

B y c o m p o s i t i o n with the given h o m o m o r p h i s m obtain a h o m o m o r p h i s m operation of G Remark

> Aut F X ,

G

> Aut X

we

w h i c h is the desired

on F X .

I. I. I0

equivalences

G

> Aut F X .

If in a given category

as m o r p h i s m s ,

~

w e only consider

w e obtain a n e w category

~. Iso

Evidently proposition I. I. 9 is still valid if w e are only given a functor

>

F : ~.

~I.

ISO

If F :~ of G

on X

>

~' is a contravariant functor,

induces a right-operation of G

operation on X Example P : Ens

ISO

on F X ,

a left-operation and a right-

is turned into a left-operation on F X . 1.1.11

> Ens,

C o n s i d e r the covariant functor

making

c o r r e s p o n d to each set X

the set P X

-6-

of i t s s u b s e t s ,

to each map

PX

> PX'

of s u b s e t s .

G-set

by proposition

@: X

1.1.9.

>X'

> X' t h e i n d u c e d m a p

Let X be a G-set.

having the same effect as to a map

X

The functor P

P

-1

Then : Ens

o n o b j e c t s of E n s ,

the map

-1

r

: PX'

P X is a > Ens,

but a s s i g n i n g

> PX (inverse

i m a g e s of subsets), transforms the G-set X

into the G ~ -set

PX. Example

1.1.12

L e t R b e a f i x e d o b j e c t of t h e c a t e g o r y

The contravariant

hR(X) = [X, R],

functor

= f o

gives for any left-operation on t h e s e t [ X , R] . phism,

we write

If T

left-

G on X b y

T : G

Let

f o r f E ; [ X t, R] , ~ : X

of G

> Aut X is the g i v e n h o m o m o r -

A

be a r i n g ,

2l

of G ~

the category

of

X is defined by an operation

maps; i.e.

a representation

By proposition

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 s e n s e to s p e a k of a n e l e m e n t

of

of G i n X

1.1.9 such a representation

i n d u c e s a n o p e r a t i o n of G on t h e s e t of s u b m o d u l e s

makes

> X',

[X, R].

A G-module

A-linear

in the usual sense.

defined by

for the induced homomorphism

1.1.13

A-modules.

>Ens

of G on X a r i g h t - o p e r a t i o n

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 Example

hR : ~

of X .

to c o n s i d e r ,

of a n o b j e c t X .

it

-7-

DEFINITION called

invariant

m

A subset M element of P X if

respect to

O-g(~) =

g

c

x in the G-object

if x i s f i x e d u n d e r

: Tg(X) = x f o r a l l X

is called

X

is

every

g ~ G.

invariant

if it is an invariant

under the induced G-operation (example i. I. Ii),

T (M) c g

Exercise

An element

or G-invariant

transformation

i.e.

i. i. 14

M

for all

1.1.15

T: G

X and

>Aut

T'g o ~

operation of G

Let

o

g C G.

X

and

X'

be G-objects

I-' : G

~" -I for g G G , g

>Aut ~: X

on the set of m o r p h i s m s

from

of

X'. >X'

X

~ with

defines an

to X'.

( E x a m p l e i. 1. IZ is a special case of this situation, if w e considert trivial G-operation on X'.) S h o w that there is a suitable functor inducing this operation according to proposition I. I. 9.

1.2

Equivariant Let

morphisms.

G and

G'

to be a G-object

of

T: G

>Aut

DEFINITION

groups

and

with respect

X, X'

homomorphism

~: X - - >

be

a G'-object

R a category.

Suppose

X

to a homomorphism of

~ with respect

to a

!

v : O' I. 2. i

X' w i t h r e s p e c t

> Aut X'. A

p-equivariant m o r p h i s m to a homomorphism

P:G

>G'

-8-

is a morphism

~0: X

following diagram

> X'

of g s u c h t h a t f o r a l l g ~ G t h e

commutes

X

~o

> X' I

T

g

d T

>X

X

If G = G '

and

P = IG ,

Example

1. Z. Z

w e just speak of anequivariant

map.

operations is

If X i s a G - s e t

given as in example

P -equivariant

G'

1. Z. 3

I d

~

x

with the

: X

> X'

commutes

> X

' T

,

X t

.

Let

P : G -->

If G

translation

as in example

>

X t

G v be a homomorphism

and G' are operating

on i t s e l f b y l e f t -

1.1. 3, t h e n a m a p

if a n d o n l y if

fore p itself is an example respect

1.1. ~, t h e n a m a p

T

of g r o u p s .

p -equivariant

a G'-set

if and only if the following diagram

GxX

Example

a n d X'

~(glg2) = of a

to t h e l e f t - o p e r a t i o n s .

q~ : G

p(gl) ~(gz).

p-equivariant

> G' is There-

map with

-9-

If w e c o n s i d e r inner

the operations

automorphisms,

of G a n d

then for all

G

P

Example subgroup

is

I. Z. 4

> G'

P

> G'

p -equivariant. If w e c o n s i d e r

the right-operation

H of G o n G a s i n e x a m p l e

P : G-->G equivariant

sending map,

by

I

G

i.e. p

on itself

g C G the diagram

I

commutes,

G'

H into

where

1.1.6,

then a homomorphism

H can be considered

p/H

denotes

of the

as a

the restriction

p /Hof

p

to H. Example equivariant This

1. Z. 5

map

of t h e G - s e t

is just the associativity

Example of G g

Any right-translation

on X ,

i. Z. 6

If

automorphism

p(g)=

of G i. Z. 7

G is an

by the left-translation.

law in G.

T: G

then for any g ~ G

- equivariant, w h e r e

Example

G defined

of a g r o u p

> Aut X defines

the m a p

[~ : G - - > G g

T

g

an operation

:X

denotes

is

>X the inner

defined by g. Let X

"r g(X o) de fines a m a p

be a G-set. p :G

> X.

For

fixed

x

o

C X

If w e c o n s i d e r

the

-10 -

operation

of G on G b y 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.

9:G

>G',

~0: X

>X',

G"-objects

are G, G ,

If X, X', X "

respectively,

> G" homornorphisms

P' :G'

> X"

~01 : X'

p, p'-equivariant m o r p h i s m s

!

~0

respectively, then clearly morphism.

!

o r

is a

For fixed G the G-objects

fore form a category morphisms

~

G

and

P

o

p-equivariant

of a c a t e g o r y

with the equivariant

~ there-

morphisms

as

(Definition 1.2.8).

As a complement

to proposition

P R O P O S I T I O N 1. Z. 9

1.1.9 we have

Let F : K

>

K'

be

functor, X, X' respectively G, G'-objects of ~ , a homomorphism

and

q~: X

> X' a

a covariant

p : G~>

G'

p-equivariant m o r p h i s m . !

Consider F(~)

the natural

: FX

operations.

> FX'

operations is

i n d u c e d on F X

Then

and FX

P-equivariant with respect

to t h e s e

F o r a f i x e d g r o u p G this d e f i n e s in p a r t i c u l a r

extension of the m a p into G-objects of

an

of proposition 1.1.9, sending G-objects of K' , to a functor F G :

~G

>

~,G .

-ii-

Proof:

The commutative

diagram

X

,>

~

X'

r % r

[ -- ~o '

X

is t r a n s f o r m e d

f

> X ~

by F i n t h e c o m m u t a t i v e

FX

F(~)

> FX'

r

T

F(Tg)i

i F('~

FX

showing the

F(~)

P-equivariance

induced operation

p(g)

> FX'

of F ( ~ ) w i t h r e s p e c t

on F X a n d F X ' .

If a n e q u i v a r i a n t

diagram

morphisrn

to the

T h e r e s t is c l e a r .

~0 : X

>X'

in

~G h a s a n

!

inverse then

~: X

>X

~ is necessarily

equivalence

in

~

G

in

R , i.e.

equivariant,

a n d ~0 t h e r e f o r e

~0 o ~ = I x ,

,

an

.

There is a canonical functor consists

% o ~0 = l x ,

V :

RG

in forgetting about the G-operation.

> ~

which

On t h e o t h e r h a n d ,

-12 -

we define a functor object

X of

mapping

~

DEFINITION o b j e c t of invar!ant

~ .

More

1.2.11

Let

A morphism

if for all

> R,

object

morphism.

7: G

it m a k e s

~0: X

R an arbitrary

an invariant

~G by considering

G-operation

Therefore

morphisms

and

>

~ the trivial

G o n 1x .

equivariant of

I : ~

of

> Aut X

sense

to speak

where e .

precisely

>R

of

X is a G-object

We call sucha

map

we have the

X be a G-object

~0 : X

on e v e r y

in

~

g G G the following diagram

of

~ , R an

is called commutes

X

X

I

PROPOSITION and

~9 : X

homornorphism (x) i s

1. Z. 12

> X' a 9:G

Let

X be a G-set,

9-equivariant >G'.

X'

a G

map with respect

-set

t o _a

If x 6 X i s G - i n v a r i a n t ,

then

p(G)-invariant.

!

Proof:

Tg(X) = x i m p l i e s

Tp(g)(~(x)) = ~(Tg(X)) = ~(x).

-13-

As a consequencej p(G)-invariant

subsets

Exercise functors

G-invariant

and morphisms

~G

can be considered

G as a category

Equivariant

morphisms

The functor

F G of p r o p o s i t i o n

1. Z . 1 4

s e t of m o r p h i s m s exercise

1.1.15.

from

morphisms

morphisms

, are the invariant example

1.3

X to X'

The invariant

are the equivariant the invariant

If X a n d X'

X

elements

functor

elements X--> > R,

functor

categories.

are G-objects

is a G-set

law).

transformations.

is the c a n o n i c a l

induced by F between the corresponding

of

according

~ , the to

under this operation

X'. where

As a special

case,

R i s a n o b j e c t of

under the operation

d e f i n e d in

1.1.1Z.

Orbits. Let

X be a G-set

DEFINITION the g i v e n o p e r a t i o n

1. 3.1

with respect

to

T:

G-

The orbit or G-orbit

is the s e t

of

of a s i n g l e o b j e c t

composition

are then just natural 1.2.9

as a category

consisting

g with g C G with natural

Exercise

of X g o i n t o

of X t .

1.2.13

(interpret

subsets

>Aut

X.

of x C X u n d e r

~(x) = [ "rg(X)/g. G G } .

-14-

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

Proof:

As

x

only have to show: secting orbits

~(x) c

More

X

~(x)

X

.

We

have inter-

= ~(x' ) .

Vg(X), y = Vg, (x')

.

= (-rye -rg_ 1 . - r g , ) ( x ' )

Let

For ~

z

~{x'),

~ ~2(x):z i.e.

~(x) = ~2(x' ) . ~x:X

-. X / G

the

A n orbit is the orbit of any of its points. ~(Vg(X)) = =(x), i.e.

~

is an invariant map.

generally w e have

LEMMA 1.3.3.

Let

X

be a

m

the canonical arbitrary,

set.

Proof:

For any invariant

If

~,

@: X / G -. R

4:X/G

"rg = r

is constant on each orbit

G-set,

~X: X -- X / G

,,

map onto its orbit set

is one and only one map

map

g

be the set of orbits,

canonical map. This implies

x,x'

then

This s h o w s

X/G

the different

into disjoint sets.

if t w o p o i n t s

z

~(x' ), Let

X

[2(x), ~(x' ) ,

we h a v e

is a G-set,

g ~(x), t h e o r b i t s c o v e r

y ~ ~2(x) (] ~2(x'): y = = "rX(x )

X

~(x),

X/G

map -. R

r X -. R such that

for all

g ~: G

and therefore

with the desired

R

and

property.

an there =

~,Ir x

, then defines a

-15 -

On the other hand, Tr x : X

> X/G

a map

x~ : X / G

gives an invariant

> R composed

~o= ~

map

o~

X"

with

We

have proved

PROPOSITION the canonical

map

1. 3 . 4

R

into invariant

Remark. property

X be aG-set,

o n t o i t s s e t of o r b i t s

The correspondence to

Let

X/G

% ~-~-> maps

and

WX : X R an arbitrary

% o ~ , sendin~ maps

from

>X/G

from

set. X/G

X t_~o R i s b i j e c t i v e .

is characterized by this

universal

up to a canonical bijection by a standard argument.

This property allows therefore the definition of X / G arbitrary category.

in an

Of course, there r e m a i n s to s h o w the

existence of such an orbit-object in a given category.

PROPOSITION p :G

>G'

9-equivariant

~: X/G

I. 3.5

Let X be a G-set, X' a G'-set,

a homomorphism map.

and

Then there

q~ : X

exists

>X' a

one and only one map

> X'/G' , such that the following d i a g r a m c o m m u t e s

X

Q0

I X/G

>X

I ~'

> x ~/ G t

-16 -

Proof:

By the universal

it is sufficient to s h o w map.

property

that w x,

stated in proposition

o ~ :X

1. 3 . 4 ,

> X t / G t is a n i n v a r i a n t

But

(~x'

~ ~)

~

Tg = = x '

= ~x'

~ ( ~ ~ *g)

o(T;(g)- 0 )

!

o Tp(g))

= (~r x,

an invariant ,rx,

o ~

map.

through

Example

~

o~ =

w x,

o ~2

is n o w d e f i n e d as the f a c t o r i z a t i o n

of

X/G.

1. 3. 6

L e t G be a g r o u p a n d H

operating by right translations d e n o t e s t h e s e t of o r b i t s ,

a subgroup,

on G ( e x a m p l e 1. 1 . 6 ) .

>G v be a map such that

= ~0(g}~0(h) f o r

g ~ G,

homomorphism

and ~

h~ is

H.

Then

~o/H : H

~/H-equivariant.

G

r

G/H

~

>

G I

l f-G , > G /H'

Let

~(gh) > H' is a

By proposition

~0 : G / H

such that the diagram

H.

Let further

~(H) C H v and

1. 3 . 5 t h e r e e x i s t s o n e a n d o n l y o n e m a p

Gf

Then G/H

t h e s e t of l e f t c o s e t s m o d u l o

G v be a n o t h e r g r o u p a n d H v a s u b g r o u p of G v : G

being

WXV

> GV/H t

-17-

commutes.

In the c a s e w h e r e

of G a n d G'

respectively

induced homomorphism Consider

H a n d H' a r e n o r m a l

subgroups

a n d ~0 i s a h o m o m o r p h i s m ,

(p i s t h e

of t h e q u o t i e n t g r o u p s .

now a fixed group G.

defined the orbit set X/G. equivariant r

X/G

map

(p: X -

> X'/G.

B : Ens G ~

Ens

by proposition

f r o m G - s e t s to sets:

>B

is that a n equivalence ~0: X / G

~0: X

> X'

If w e c o n s i d e r the "forget-functor" V : E n s G

is a natural t r a n s f o r m a t i o n

: X

introduced the m a p

X/G

wx :X

c a n be e x t e n d e d to a m a p as the m a p

B(~0) : "~ (p.

.... > X ' / G .

of V

T o the beginning of this p a r a g r a p h j f o r

b y the m a p

1. 3 . 5 a n y

B(X) = X/G,

defined as forgetting about the G - s e t structure, w e w :V

X we have

> X' i n d u c e s o n e a n d o n l y o n e m a p

induces a bijection

Remark.

any G-set

In this way we obtain a covariant functor

A standard consequence in E n s G

Moreover

For

PX

-i w x , w x : PX

just the orbit of M

> PX,

into B .

a G-set

-1 ~ = ~X ~

-->

PX

a n d w e interpret n o w For

M

Explicitly

(-,- (x) l g c G , g

X 2 we have

The right side

c

X

u n d e r the i n d u c e d G - o p e r a t i o n

-

see that

w h i c h c a n a l s o be d e s c r i b e d

as

> PX.

> Ens

xe;M}.

Q(M) on P X .

is

-18-

2(M) i.e.

is therefore

the saturation

t h e u n i o n of a l l G - o r b i t s The invariance (M) = M .

of M w i t h r e s p e c t

of X i n t e r s e c t i n g

of M c X c a n

F o r an arbitrary M

now

M.

be expressed

c X the set

is the

The orbits

invariant sets.

Let X

Isotropy groups.

CfK-" {g

by

~(M)

intersection of all invariant sets containing M . are the m i n i m a l

to G,

e G /

= x }.

G

be a G - s e t and x C X .

Consider

is a s u b g r o u p of G . x

DEFINITION

i. 3.7.

G

is called the isotropy g r o u p of x. x

PROPOSITI

ON

I. 3.8.

Gg x = g Gxg-I

F o r simplicity w e write

Proof:

w e have g h g -I . gx = g h x

h6G

Vg(X) = gx.

T h e n for

= g x , w h i c h implies

x

g G x g -1 c

G

gx

g-1 G g x g c G x

As or

g-1.

gx = x,

Gg x c g G x g - 1 ,

we have by the same which proves

the proposition.

This can also be e x p r e s s e d in the following way. the m a p

by

> SG

~0 : X

qg(x) = G x .

G

I. 3.8 the d i a g r a m

into the set of subgroupsof G ,

operates by inner a u t o m o r p h i s m s

orbit being a c o n j u g a c y

class

of subgroups.

argument

Consider defined on S G , .an

B y proposition .

-19 -

X

r

> SG

.l

[ g X ~

commutes fore

for all g ~ G,

>

SG

i.e.

~0 i s a n e q u i v a r i a n t

~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

map.

There-

a map

N

~0: X / G

>SG/G,

a well-defined orbit-type

conjugacy

Let x

there

to every class

of s u b g r o u p s

orbit-types

be in

0

~ ( X o ) of o r b i t - t y p e

glXo = gzXo

or

If Xo i s i n

~] (x o) of o r b i t - t y p e Therefore

of o r b i t - t y p e

Example consisting

on n~n .

corresponds called the

{e} a n d

Then for x 6 gx ~ = x.

g~lg 1 = e

and

~(x)

Because gl = g z "

t h e n Xo i s G - i n v a r i a n t ,

G.

1. 3 . 9 .

Consider

different

The origin

G,

classes

the fixed points are exactly the

of t h e r e a l q u a d r a t i c

a determinant

{e}.

such that

g~ l g l x o = Xo i m p l i e s

(Xo) - x o .

orbits

of G ,

are the conjugacy

is one and only one g C G

and

o r b i t of X t h e r e

of the orbit.

Particular G.

i.e.

from

the full linear matrices

zero,

with the natural

The orbit-type

G L ( n , JR),

with n entries

O and its complement

o r b i t s of t h i s o p e r a t i o n .

group

n~n - { o }

of

having

operation are the

O is GL(n,

]1%).

-20-

Example metric

1. 3 . 1 0 .

Consider

and the corresponding

o r b i t s of t h e n a t u r a l as center.

origin is isomorphic 1. 3.11.

+b cz +d az

operates

on X .

Example

G-operation

JR).

The

with the origin

O ( n - 1 , JR).

Let X denote the complex plane with a T h e g r o u p of t r a n s f o r m a t i o n s

w i t h a, b, c

of t h e

d 6 ~; a n d a d - b c ~ O

X i s t h e o r b i t of a n y p o i n t x ~ X . 1. 3.1Z.

Consider

itself by inner automorphisms. of t h e c e n t e r

O(n,

euclidean

g r o u p of a p o i n t d i f f e r e n t f r o m t h e

to the orthogonal group

point at i n f i n i t y a d j o i n e d . type z "">

orthogonal group

operation are the spheres

The isotropy

Example

IRn w i t h t h e s t a n d a r d

CG.

the operation

The fixpoints are the elements

We h a v e a l r e a d y

considered

on t h e s e t SG of s u b g r o u p s

subgroup is its conjugacy class.

of a g r o u p G on

of G .

Therefore

the induced T h e o r b i t of a

the invariant

sub-

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 . Moreover

it follows that the different conjugac 7 classes

a partition

of S G .

T h e e f f e c t of a n e q u i v a r i a n t is described

form

by the

m a p on t h e i s o t r o p y

groups

-21-

PROPOSITION

P: G

>G'

a homomorphism

p-equivariant m a p .

Proof:

Let X be aG-set,

i. 3.13.

Then

Let g ~ G

and

~0 : X

Xv

>X'

a

G' -set,

a

p (G x) c G ~ ( x ) .

, i.e. g x =

x.

Then

x

p (g)~0(x) =

~0 (gx) =

Exercise p: G

> G'

1. 3 . 1 4 .

~0 : X

TrX

Let

p(g) ~ G'

(p(x)

a G-set,

X be

> X t in the sense

o irX : P X

> PX

G !-set,

> X'/G'

is induced by a

of p r o p o s i t i o n

For a G-set

"

Xv a

a n d "~ ~o : X / G

under which ~

E x e r c i s e I. 3. 15. =

i.e.

'

a homomorphism

Study the conditions map

~(x)

X

p-equivariant

1.3.5.

c o n s i d e r the m a p

defined above.

S h o w that

has the following properties: a)

b)

~ (~)) = (~ fo~.the e m p t y

Mc

~ (M) f o r

set r of X

M c X

for a family

a map.

( M x ) x 6 A of M X c

X.

-ZZ -

Therefore

Q

is a "Kuratowski-operator"

on X a n d 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 t o t h e d e f i n i t i o n : if a n d o n l y if an arbitrary

~(M) = M. equivalence

This remains relation

M c X is c l o s e d

t r u e if we c o n s i d e r

R on X ( n o t n e c e s s a r i l y

~=

d e f i n e d b y a g r o u p G) a n d the m a p

~-I

o

where

lr

x

set X/R. R

> X/R

:X

>PX,

X

is the canonical map onto the quotient

Show that more

on a set X

: PX

~r

x

generally for an arbitrary

the " s a t u r a t i o n - o p e r a t o r

"

~: P X

relation > PX

defined by

(M) = [ y ~ X / x R y

for Mc

X

satisfies

the properties

is a r e f l e x i v e and t r a n s i t i v e

Exercise 1. 3.15

1.3.16.

for some

x~

M]

1) t o 4) if a n d o n l y i f R

r e l a t i o n on X .

Consider

the topology defined in exercise

on a s e t X e q u i p p e d w i t h a n e q u i v a l e n c e

relation

Show the following properties: 1) M c

X i s c l o s e d i f a n d o n l y if M i s a u n i o n

equivalence classes;

2)

M

c X

is closed if and only if M

is open.

of

R.

-Z3-

What are the conditions

on X / R

for the topology in question

i) t h e s e c o n d

countability

to satisfy

axiom,

z) t o b e c o m p a c t ,

3)

to be connected?

E x e r c i s e i. 3.17. Let X a map. R

Suppose

X

be aG-set,

R an arbitrary

set and ~ : X

'>R

e q u i p p e d with the topology defined in exercise I. 3.15 and

topologized by the discrete topology.

T h e n (~ is invariant if and only if

(~ is continuous.

I. 4

Particular G-sets. Let X

be a G-set,

define s o m e

> Bij X.

7 :G

We

particular properties an operation c an have.

DEFINITION

Ker

defined by a h o m o m o r p h i s m

7

1.4. 1

is an effective operation if 7

is injective, i.e.,

T = {e}. We observe

that

Ker

exactly an e l e m e n t of G effective,

then there

T=

N xCX

Gx

exists

a factorization

Example

i. 4. Z

has the center C G

of K e r T being

effectively

~

G/Ker

T

T on X .

T h e operation

= Ker

T through

If T is n o t

Bij x

G/Ker T operates

an e l e m e n t

contained in e v e r y isotropygroup.

i and G/Ker

,

[~ of a g r o u p

as kernel.

O

by inner a u t o m o r p h i s m s

-24-

DEFINITION x6X

implies This

Free

neutral

element:

of f i x p o i n t s " . G

x

set".

Example

1.4. 4

exists

the element

every

of G,

for

x~X.

for some

g ~ e has no fixpoint. group

is reduced

X is also called

operation

The operation

1.4. 5 T

a g ~ G

to the a

is effective.

of G o n G b y l e f t - t r a n s l a t i o n s

operating

is free.

on G by right-translations.

This

is a transitive

such that

operation,

if for

x 1, x z ~ X

7g(Xl) = x z , simpl 7 transitive,

if, moreover,

g is unique.

A simply is simply

Tg(X) = x

is free.

DEFINITION there

= {e} for

g

The isotropy

Note that a free

H be a subgroup

operation

T

that a transformation

"free

"G-principal

Let

if

g = e.

means

means

T is a free operation,

1.4. 3

transitive

transitive

operation

on each

orbit.

is free. Because

Conversely,

a free

operation

i f x = g i X o (i = 1, Z) , t h e n

Xo = gz-Ix = gz-i glXo and therefore gz-igl C G x o = { e } ,

i.e.

gl = gz"

T h e definition of a transitive operation can also be put in the following form:

there exists an element x

C X

s u c h that

~(Xo) = X .

X is then

O

the orbit of each point x C X . a point.

This s h o w s that the set of orbits X / G

is

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 transitively on X .

i. 4. 6

A G-set X is called h o m o g e n e o u s ,

if G

operate s

-25-

Example

1.4. 7

on t h e u n i t s p h e r e

The orthogonal group

O ( n , JR) o p e r a t e s

transitively

S n-1 i n IRn .

More generally,

a n 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

operation

on e a c h G - o r b i t . Example

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

complex plane operates A fundamental following way.

G/H.

transitively.

example

Consider

right-translations.

L

g

: G

r (vH) = g vH g

operation,

Remark.

making

The isotropy

G/H

> G satisfies a map

exists a subgroup G/H First define

g r o u p of

~ : G/H

>

LEMMA 1.4.9

( q~ o -ry)(gH)

~(Xo)

@ V ~ G

= ~0 ("vgH) =

the equivariance

on G b y

H

of ~ .

Lg(vH)

= g vH

>G/H.

r

and is

is H. homogeneous

G-set

~: G/H

X

there

> X of G - s e t s ,

as a G-set in the sense indicated.

let X be an a r b i t r a r y

For

H operating

which is evidently homogeneous.

H of G a n d a n e q u i v a l e n c e

is c o n s i d e r e d

Proof:

is o b t a i n e d in the

9 : G/H g

a G-set,

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

where

G-set

T h e n w e c a n d e f i n e a n o p e r a t i o n of G o n t h e o r b i t s e t

d e f i n e s by

the desired

of a h o m o g e n e o u s

a group G and a subgroup

The left translation

therefore

of t h e u n i t d i s k i n t h e

G-set

X by

and x

r

is equivariant

O

~ X.

We put H = G

x o

= gx o.

and injective.

one h a s ( T ~ o ~ ) ( g H )

= v v ( g x o) =

Vgx o , a n d t h e r e f o r e To show the injectivity,

v V9

~

=

consider

Vgx o and ~D o 0 "

V'

i.e.

g l ' gz ~ G

and

-Z6-

s u c h t h a t q~ ( g l H) = O ( g z H ) . -1 gz gl E; H .

and therefore

B u t gl E; gz H implies

If X i s h o m o g e n e o u s ,

Q ( x o) = X

then

-i o r gz glXo = Xo

glXo = gZXo

This m e a n s

gl H = gz H ,

q.e.d.

a n d cp i s a n e q u i v a l e n c e .

We h a v e p r o v e d PROPOSITION x

0

6 X.

Let X

Le___ttH b e t h e i s o t r o p y

on G / H

~

(gH)

The group

= gx o

H depends

operations

on s e t s .

by t h e o p e r a t i o n

on G / H

G-sets

is the intersection

an invariant s u b g r o u p of G =

gl'H

signifies

for some L c

K.

remarks

=

il

~ L

Therefore

with

proposition,

groups,

in view we have

H, of

where

we can consider H is a subgroup of G

therefore

.

c ontained in H. L c

>X

but the conjugacy

defining the operation

gHg I

N gCG

@ : G/H

on effective and transitive

of t h e t y p e G / H ,

of t h e i s o t r o p y

is an invariant s u b g r o u p of G

igH

Then the map

i n v i e w of t h e t r a n s i t i v i t y .

K of t h e h o m o m o r p h i s m

K

K

by some

the G-operation

of G - s e t s .

I n v i e w of t h e p r e c e d i n g

w i t h o u t l o s s of g e n e r a l i t y The kernel

of G .

on t h e c h o i c e of x o 6 X ,

We c o n c l u d e t h i s c h a p t e r

G -set and select

g r o u p of x o a n d c o n s i d e r

is an equivalence

of H i s w e l l - d e f i n e d

of G .

be a homogeneous

induced by the left-translation

defined by

class

1.4.10

then

Conversely,

L c

Lg = gL

and

K,

if L i s

because lgH = gH which

-27-

PROPOSITION

I. 4. Ii

the G-operation on G / H K

Let G be a sroup, H a subgroup and_consider

induced by the left-translations of G .

of the h o m o m o r p h i s m

~: G

> Bij { G / H )

T h e kernel

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 a n d c a n be d e s c r i b e d as

K

C O R O L L A R Y 1.4. 12

=

fl g~G

-I gHg

G o p e r a t e s e f f e c t i v e l y on G / H

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 Exercise I. 4.13 in proposition I. 4. i0.

if an d o n l y if

{e}.

Study the effect of the choice of the point x o G X

-28-

Chapter

Z.

G-SPACES

Z. 1 D e f i n i t i o n a n d 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

topological space such that the maps G

x

G

>G

,

G

> G

-1 (gl'

gz ) ~

glg2

are c ontinuous. D E F I N I T I O N Z. 1. Z

Let G be a topological group.

is a t o p o l o g i c a l s p a c e w h i c h is a G - s e t w i t h r e s p e c t Moreover

t h i s m a p is s u p p o s e d to be c o n t i n u o u s .

called a topological transformation

A G-space

to a map

G x X

group.

so that X is a G-object in the category moreover,

Let G and

on X ,

of t o p o l o g i c a l s p a c e s .

t h e c o n t i n u i t y of t h e m a p G x X

We

> X.

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 groups is a homomorphism L e t X be a G - s p a c e , homomorphism.

>X.

T h e p a i r (G,X) i s a l s o

It i s c l e a r t h a t t h e g r o u p G i s a c t i n g b y h o m e o m o r p h i s m s

require,

X

A hornomorphism of g r o u p s ,

9: G

> G w of t o p o l o g i c a l

which is continuous.

X' a G' - s p a c e a n d

P: G

> G'

a

-Z9 -

DEFINITION

Z. i. 4

p -equivariant m a p The map

r

A

9-equivariant m a p

~: X

in the sense of definition I. 2.1 w h i c h is continuous.

makes

the following d i a g r a m c o m m u t a t i v e

G

x

X

>

X

[

[

px ]

$ X'

G' x ~0 i s c o n t i n u o u s a n d t h e r e f o r e the universal

property

An equivalence of G - s e t s

> X' is a

also

X'

>

P x @, as follows i m m e d i a t e l y by

of t h e p r o d u c t t o p o l o g y .

of G - s p a c e s

X, X t i s a n e q u i v a l e n c e

~:X

>X'

which is a homeomorphism.

Example

Z. 1 . 5

G on G b y l e f t o r r i g h t - t r a n s l a t i o n s The operation

The operation

L e t G be a t o p o l o g i c a l g r o u p . makes

the space

of G on G b y i n n e r a u t o m o r p h i s m s

of

G a G-space.

also makes

G a

G-space. Remark.

L e t X b e a t o p o l o g i c a l s p a c e a n d G t h e g r o u p of

homeomorphisms

of X .

The discrete

t o p o l o g y on G c e r t a i n l y

makes

X a G-space. Let X be a of h o m e o m o r p h i s m s

compact G-space. with the compact-open

that Aut X is a topological group, continuous .

Consider topology.

and that the map

the group Aut X It c a n b e p r o v e d

G x X

> X is

- 30-

2. Z

Orbitspace. Let G

be a topological g r o u p and X

of orbits X / G

and the canonical m a p

topology on X / G

w

x

a G-space. :X

> X/G.

is the strongest topology on X / G

T h e o p e n s e t s of X / G DEFINITION

C o n s i d e r the set T h e quotient

making

wx

are the sets having an open saturation

Z. Z. 1

The orbit space

X/G

continuous.

in X.

of t h e G - s p a c e

X is

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 . PROPOSITION

Z. Z. Z

w

: X

: > X/G

is an open map.

The

X

t o p o l o g y on X / G the map w

Proof:

~(M) means

is characterized

continuous

X

Let

as bein8 the unique topology making

and open.

MC

7g(M)

X be open.

(~rxl o Wx)(lVi ) that

....

'

is open and therefore

b e i n g t h e u n i o n of a l l s e t s

Tg ( M ) .

But this

~rx(M) i s o p e n b y 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 .

prove the second

statement,

from

Y.

X to a set

Two topologies

and open necessarily one topology,

consider

coincide.

~-1(0)

more

a map

on Y making both

Because

is open in X

generally

~

also

To

~ : X

> Y

continuous

if O i s a n o p e n s e t of Y i n t h e

and ~ (~-1(O)) = O is also open in

the other topology. Example

Let

Z. Z. 3

G

be a t o p o l o g i c a l

of G w i t h t h e r e l a t i v e

topology.

translations m a k e s

an H - s p a c e .

G

The operation

group and H a subgroup of H on G

T h e canonical m a p

onto the orbitspace is continuous and open.

by right

wG : G

> G/H

-31-

The quotient following

topology

property.

Let

on X/G

R be an arbitrary

m a p x~--------> ~ o lr , s e n d i n g continuous therefore

maps

X

~r

> X/G

:X

x

space.

from

continuous

by the

space.

x~ : X / G - - >

The proposition

The

R into 1. 3. 4 c a n

by

Z. Z. 4

Let

the canonical

be a topological

G map

The correspondence

~roup,

onto the orbitspace

$ ~

to R onto invariant

X/G

topological

maps

> R is injective.

now be completed

PROPOSITION

can also be characterized

qt o ~ , s e n d i n 8

continuous

maps

from

X aG-space,

and

R an arbitrary

continuous

maps

X to R is

bijective. PROPOSITION a homomorphism map

4: X

Let G, G' b e topological groups,

Z. Z. 5

and X, X' respectively G, G'-spaces.

A

> X' induces one and only one continuous m a p

> X'/G'

p: G - - > G ' p-equ/variant

~: X / G

such that the loll owing d i a g r a m c o m m u t e s

X

0

I

X'

i

~x [

7fX!I

X/G Proof:

>

- ~

> X'/G'

T h e r e is only t o s h o w 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 of G on X

9

Select

Consider a G - s p a c e x

0

6 X

and let

X with a transitive operation

H be the isotropy

group

of x

0

-32-

Define,

1 . 4 . 10, a m a p

as in proposition

equivalence

of G - s e t s

and continuous,

The following counter-example TZ

= IR~-/~Z

by Z

( x 1, x 2) ~ 3" irrational 1e m i~a X =

number.

Fixing

( x 1, xz) 6

but not a homeomorphism.

Exercise group.

Z. Z. 7

T z we define

Exercise neighborhood U,

L e t ]R o p e r a t e o n where

~:]R Consider

and 0 a n >T z

as in

the image

X is dense in

T Z and

s p a c e ]R.

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 open s u b (Consider the partition

of G d e f i n e d

of G / H . ) Z. Z. 8

of e .

V "I = V .

Let G be a connected topological group and U a The neighborhood

inner point of V O~

V = U N U

-1

has the properties:

C o n s i d e r the sets V n ={gl .... gn/gi C V ,

T h e union V ~176= U V n

is a group, the group generated by V .

as e ~

V cV

c~

V ~176is an open s u b g r o u p of G

is connected, this s h o w s an arbitrary neighborhood

V ~176 = G. U

of e.

i = l,...,n}. e is an

A n y point of V ~176 is therefore an

inner point, the left-translations being h o m e o m o r p h i s m s invariant.

X C IR,

> X is a continuous

~ : ]R

Because

to the c o m p l e t e

Then H is closed in G .

by the elements

a homeomorphism.

the canonical h o m o m o r p h i s m

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 .

c a n n o t be h o m e o m o r p h i c

Vc

is taken from Bourbaki.

> ]R/Z

This map is an

but not necessarily

~ (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 .

bijection,

> X.

7x(x I, x 2) = (x I + a(X), x 2 + a(8•)) a:]R

,

~ : G/H

leaving V ~176

and therefore closed.

This proves that G

As

G

is generated by

-33-

Exercise

2.2.9

Let G

be a topological group and G

the connected o

component

of the neutral element

e C G,

the identity c o m p o n e n t

S h o w that G o is a closed invariant s u b g r o u p of G

of G .

-34-

Chapter

3.

G-MANIFOLDS

This chapter introduces the fundamental notions of these lectures. In the following chapters, w e p r o c e e d to a detailed study of G - m a n i f o l d s and Lie groups. 3. i

Definition and e x a m p l e s Manifold will m e a n

of Lie ~roups.

a Hausdorff,

but not necessarily connected

manifold. DEFINITION

3. I. I

A Li e group is a group G

w h i c h is an analytic

manifold such that the m a p s G

x

G

>

G

(gl' gz ) ~

glgz

G

> G

g ~

g-1

are analytic. Differentiable shall always m e a n

C ~ . 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). F o r a great part of the theory, w e shall only m a k e

explicit use

of d i f f e r e n t i a b i l i t y . In t h e d e f i n i t i o n a b o v e ,

manifold.

analytic manifold m e a n s

Replacing it by c o m p l e x analytic manifold,

notion of a c o m p l e x Lie group.

real analytic one obtains the

-35-

Two arbitrary are

connectedness

analytically

diffeomorphic.

g ~-~-~> g 2 g l l g

is an example

connectedness makes

sense

to speak

Example GL(n,

3. I. 2

IR ) - t h e g r o u p

different

from

Example Then

components

TG

product

B. 1. 3

G1 x G 2

> G Remark.

gl ~ O l '

g2 6 O 2

have the same

All the

dimension

a n d it

of a L i e g r o u p .

The additive

group

]Rn

of q u a d r a t i c

matrices

o r (~ n . , with

Let

G b e a Lie group and This

follows

from

3rn

=

n rows

]Rn/Z n

;

and determinant

TG

the tangent

the fact that

Let

G 1 and

G 2 be Lie groups.

bundle.

T is a functor

Then the direct

is a Lie group. 3. 1 . 5

Let

G and G'

be

Lie groups.

of Lie groups is a h o m o m o r p h i s m

A hornomorphism

of groups w h i c h is analytic.

It is to be noted that in the literature the t e r m h o m o m o r p h i s m

is often r e s e r v e d for analytic h o m o m o r p h i s m s p: G

the map

of s u c h a d i f f e o m o r p h i s m .

therefore

O

products.

S. 1 . 4

DEFINITION p :G

G 1, 0 2 of a L i e g r o u p

zero.

direct

Example

For

of t h e d i m e n s i o n

is a Lie group.

conserving

components

of groups such that the m E p

> 9 (G) is open. Example

The choice

3. i. 6

of a b a s e

GL(V)

> GL(n,

IR)

structure

on the group

Let V be an n - d i m e n s i o n a l vector space over JR. e 1. . . .

,

of g r o u p s , of l i n e a r

en

of V d e f i n e s permitting

automorphisms

an isomorphism

us to define GL(V)

a Lie of

V.

group This

-36-

structure

is independent

of t h e c h o i c e of t h e b a s e .

of t h e b a s e o f V c o r r e s p o n d t o t w o

isomorphisms

which differ by an inner automorphism Example

3. 1. 7

Let

G e of G a t t h e i d e n t i t y

GL(V)

of G L ( n ,

G be a Lie group and

with its Lie group structure

(example

e of G

3. 1. 3).

and its natural

assigning

of L i e g r o u p s .

to each tangent vector

Lie groups.

The natural its origin,

two choices > G L ( n , JR)

]1%). TG the tangent bundle

Consider

the tangent

injection

If G e i s e q u i p p e d w i t h t h e L i e g r o u p s t r u c t u r e a homomorphism

Because

j : G

space >TG.

e

defined by addition, projection

p : TG

j is > G,

is also a homomorphism

of

The sequence

O

>G

~

>

TG

P

>G

>

e

e

is exact. s : G

Moreover, >TG,

Exercise

there

satisfying 3.1.8

exists

of a n E u c l i d e a n

countable base.

Let G be a locally Euclidean

3.1.9

Exercise

3.1. I0

open s u b g r o u p of G .

of t h e i d e n t i t y

space.

Therefore

Exercise

the natural

injection

p o s = 1G.

i. e. , h a v i n g a n e i g h b o r h o o d subset

a splitting,

topological

e homeomorphic

The identity component

group,

to an open

G O of G h a s

a

G is paracompact.

A Lie group is locally connected. The identity component

G o of a L i e g r o u p i s a n

-37-

3.2

Definition

and examples

DEFINITION differentiable G x X pair

3. 2 . 1

Let

manifold

> X.

differentiability

group.

by diffeomorphisms

of d i f f e r e n t i a b l e

of t h e m a p

Example

3. 2. ~

B. 2. 3

parameter

groups

manifold

Example GL(V)

3.2.4

are

the

p : G

> GI a

3. Z. 5

of t r a n s f o r m a t i o n s

defines

We

of f u n d a m e n t a l

call

Let G

is differentiable.

importance

a special

in chapter

for

name:

one-

also defines

of

5. G on the underlying

G as a G-manifold.

V be a finite-dimensional

is then a Lie group.

> X ~ is a

1. Z. 1 w h i c h

of a L i e g r o u p

automorphisms Let

~: X

We shall take up the study

The operation

by inner

map

of d e f i n i t i o n

of transformations.

groups

a homomorphism. V,

9 -equ/variant

They have received

by left-translations

of G on i t s e l f

X is a

Moreover

G v -manifold, a n d

a

IR-manifolds

of G - m a n i f o l d s .

Example

A

map in the sense

the theory

one-parameter

manifolds.

so that

of L i e g r o u p s .

DEFINITION 9 -equivariant

X

on X,

The

> X is required.

G x X

X be a G-manifold,

homomorphism

to a map

a Lie transformation

!

Let

with respect

X is a

to be differentiable.

G is acting

in the category

is a G-set

A G-manifold

this map is supposed

( G , X) i s a l s o c a l l e d

G-object

G be a Lie group.

X which

Moreover

The group

of G-manifolds.

The operation

G as a G -manifold. ]R-vectorspace.

be a Lie g r o u p and

7: G

>GL(V)

7 a representation of the Lie g r o u p G

in

-38-

A s o b s e r v e d at the end of section 2. I, for a locally

Remark.

compact G-space

X

the continuity of the m a p

e x p r e s s e d by the continuity of the h o m o m o r p h i s m the operation, if Aut X

>X

G x X

canbe

> Aut X defining

G

is equipped with the c o m p a c t - o p e n

topology.

O n e would like to describe similarly the differentiability of the m a p G x X-->

X

morphisms

for a G - s p a c e of X

X.

But for this the group Aut X

should first be turned into a manifold ( m o d e l e d over

a suffi'ciently general topological vectorspace), difficulties.

of diffeo-

w h i c h presents serious

Nevertheless w e shall use this viewpoint for heuristical

remarks. Example

3.2.6

Let X

be a G - m a n i f o l d and T the functor assigning

to each differentiable manifold its tangent bundle. TG-manifold, of T G

because

(example

T

Then

conserves direct products.

3. i. 7), T X

is also a G-manifold.

G

TX

is a

being a s u b g r o u p

This justifies m a n y

classical notations in the theory of transformation groups, w h i c h at first sight s e e m Example with respect product structure

abusively short.

3. Z. 7

Let G

to an operation

and G' be Lie groups and G' T : G

G" x w G d e f i n e d i n e x a m p l e of t h e p r o d u c t - m a n i f o l d .

which corresponds

Then the semi-direct

1.1.7 is a Lie group with the analytic This generalizes

to the trivial operation

L e t V be a f i n i t e d i m e n s i o n a l motions

> Aut G' .

a G-manifold

3. 1.4~

of G on G ' .

]R-vectorspace.

of V , w h i c h i s t h e s e m i - d i r e c t

example

product

T h e g r o u p of a f f i n e

V x GL(V) with respect

- 39 -

to the natural

o p e r a t i o n of G L ( V )

Example

3. 2 . 8

on V, i s a L i e g r o u p b y 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

consider

the exact

sequence

O

>

J

G

>TG

P

s : G

> TG defined by the natural

>

G

>e

e

of e x a m p l e

3. 1. 7.

The splitting

i n j e c t i o n of G g i v e s

r i s e t o a n 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

G e defined by

gs(g)/Ge

Wg =

of G i n G e p l a y s a n i m p o r t a n t representation). with respect

(example 1.1.7).

r o l e i n t h e t h e o r y of L i e g r o u p s ( a d j o i n t

, TG is i s o m o r p h i c

to t h i s o p e r a t i o n

This representation

r .

to the semi-direct

product

C,e xTG

-40 -

C h a p t e r 4.

VECTORFIEI.nS

In this chapter w e begin with the detailed theory of G - m a n i f o l d s and Lie groups.

T h e Lie algebra of a Lie g r o u p is defined and the f o r m a l

properties of this c o r r e s p o n d e n c e are studied. 4. i.

Realfunctions. T h e adjective "differentiable" shall be omitted f r o m n o w on, it being

understood that all manifolds and m a p s Let X on X .

CX

are differentiable.

be a manifold and denote by C X is a c o m m u t a t i v e

being defined pointwise.

the set of real-valued functions

ring w i t h identity, the operations on functions

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. g)*: C X '

> CX

homomorphism

defined by

and C X ' ,

identities.

~

~*

defines a contravariant

~/

of m a n i f o l d s

to t h e c a t e g o r y

If

T:G

s

If w e consider the ]R-algebra

CX is a G O - r i n g ,

C :~

i~ of c o m m u t a t i v e

R-algebras

Now let X be a G - m a n i f o l d .

right.

~0" is a ring

of ]R -algebras

This shows that the correspondence

commutative

1.1.10,

> X' induces a m a p

is a h o m o m o r p h i s m

then ~ *

respecting

remark

~0 : X

g)*(f')= f' o ~ for f' 6 C X ' .

respecting identities.

structure on C X

respectively

A map

CX,

> e from the category rings with identity,

with identity.

According i.e.

X ~

to p r o p o s i t i o n

1.1.9 and the

a r i n g on w h i c h G o p e r a t e s

> Aut X is the g i v e n o p e r a t i o n ,

"r* : G

from the

> 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

for f ~ CX. Exercise 4.1. i.

Let X

p o n d i n g s e t s of r e a l - v a l u e d morphism

CX'

Exercise > X

q~i:X

> CX 4.1. Z.

and X' be manifolds, functions.

CX

and C X '

Show that an arbitrary

is a h o m o m o r p h i s m

Exercise 4. i. 3.

ring homo-

of ]R-algebras.

L e t the s i t u a t i o n be as in e x e r c i s e

(i = 1, 2) be m a p s

the corres-

s u c h t h a t ~1

= ~Z"

4. 1.1 a n d

Show t h a t t h e n

~1 = Og"

Let the situation be as in exercise 4. I. i and consider

the map

[x, x'] from maps @~ - . >

~

X

> X'

to r i n g h o m o m o r p h i s m s

Exercise

for p a r a c o m p a c t

> [cx', cx] > CX d e f i n e d by

4. 1. g s h o w s t h a t t h i s m a p i s i n j e c t i v e .

manifolds

X, X' this m a p

imitate the theory of duality for CX

CX'

as the dual space of X .

A-modules

is bijective.

(Hint:

Show that T r y to

over a ring A , considering

T h e study of the bidual space will then give the

desired result. ) This result should allow on principle a cornplete algebraisation of the theory of differentiable manifolds. E x e r c i s e 4. i. 4. is not d e c o m p o s a b l e

A manifold X

is connected if and only if the ring C X

in a direct product of non-trivial rings.

-4Z

4. 2.

Operators

-

and vectorfields.

L e t X b e a m a n i f o l d a n d C X t h e s e t of r e a l - v a l u e d sidered

functions,

con-

a s a n ]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. A : CX

An operator

A on X

is an ]R-linear m a p

> CX.

Example

4. 2.2.

An automorphism

f i e l d on X i s a n o p e r a t o r .

of C X

More generally,

is an operator.

a differential

A vector-

operator

on X

is an operator. L e t OX d e n o t e t h e ] R - a l g e b r a manifold and phism

~

~ :X

: OX

of o p e r a t o r s

> X' a diffeomorphism,

> OX' b y t h e d e f i n i t i o n

~A

definition means that the following diagram

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 functor phisms

0 : ~ iso ~> to the category

~

v :G

> Aut X .

then ~ induces an i s o m o r = @~-1 o A o ~ .

I r
OI of t h e

of operators

-44-

PROPOSITION operators on X. OK

a G-set.

4. Z. 4.

Let X be a G - m a n i f o l d and O X the set of

T h e definition ( T g ) , ~ ( A ) =Tg

~-I

oA

oT

g

for A C O X

makes

This operation conserves the ]R-algebra structure on O X

well as the associated

structure

of a n ] R - L i e

algebra.

In particular,

a___s

the

invariant elements under this operation f o r m a ]R-algebra and _a JR-Lie algebra respectively. PROPOSITION and

~:X

4. Z. 5.

> X' .a

morphism

p :G

with respect

X' a G' -manifold

Let X be a G-manifold,

p-equivariant d.i f f e o m o r p.h i s m with respect to a h o m o . > G'.

to the operations

sends G-invariant

> O X ' is a p -equivariance

Then @~ : OX

defined in proposition

operators

4. 2 . 4 .

p(G)-invariant

on X into

Moreover, operators

on

X'. This follows f r o m r e m a r k We

n o w apply this to vectorfields.

vectorfield on X.

Then A

A(f I + fz) = Af I + ~

(ii)

A(flf2)

Therefore

A COX.

Let X b e a manifold and A

A : CX

for

for

In fact,

[ , ] : OX x O X

w h i c h satisfies

for

fl' f2 ~ C X

X C ]R these

properties

of v e c t o r f i e l d s

of v e c t o r f i e l d s

> CX

a

fl' fz C C X

= Afl-f z + fI.AIz

The composition

composition structure

is a m a p

(i)

(iii) A(?~) = O

fields.

I. i. I0 and propositions i. Z. 9 and i. g. IZ.

with respect

are characteristic

for vector-

in OX is not a vectorfield, to the associated

~ OX gives a vectorfield.

]R-Lie Here

but the algebra

(ii) i s e s s e n t i a l .

-45-

Thus the vectorfields

form

denote the JR-Lie algebra If X

a subalgebra

of a l l v e c t o r f i e l d s

and X' are manifolds and

the i s o m o r p h i s m

@# : OX

certainly sends D X

of t h i s

> OX'

~: X

]R-Lie algebra.

L e t DX

on X . > X

a diffeomorphism,

then

defined at the beginning of this section

into D X ' . Applying proposition 4. 2.4 w e therefore

obtain COROLLARY

4. Z. 6.

Let X be a G - m a n i f o l d and D X

the ]R-Lie

,-i T h e definition ( 7 g ) , ( A) = 7g

algebra of vectorfields on X. A ~ DE

makes

In particular,

DE

a G - L i e algebra with respect to 7: G

the invariant

elements

o A 9 7g f o r > Aut D E .

of DX u n d e r t h i s o p e r a t i o n

form a

]R-Lie algebra. And proposition COROLLARY

4. Z. 5 g i v e s

4. Z. 7.

Let

X I

X be a G-manifold,

a G ! -manifold

!

> X

and ~0 : X morphism respect

a

p-equivariant > G' 9 T h e n

p: G

to the operations

diffeomorphism

~, : DX

definedin

> DX'

with respect

to a homo-

is a P-equivariance with

c o : r o l l a r y 4. Z. 6.

Moreover,

g)$ s e n d s

G-invariant vectorfields on X into p (G)-invariant vectorfields on X' . For

later

use,

we make

e x p l i c i t t h e e f f e c t of q g , . !

LEMMA

4. 2 . 8 .

L e t ~0: X

the induced isomorphism x6

X

and

f' C C X ' .

> X

be a d i f f e o m o r p h i s m and

on vectorfields, Then

defined by

cpsA = r

,-1

~$ : D X

o A o q~ 9 L e t

( ~ A ) ~ (x) f' = Ax(cp#f' ). I__fcP.x : T x (X)

!

> T (x)(X)

(~,A)~(x)

denotes, the linear m a p

= ~,

A X

. X

> DX'

of tangent spaces induced by r

then

-46 -

Proof:

((~,A) f')($(x)) = ~*((r

definition of ~0,. This m e a n s

f'))(x) = ((A~*)f')(x) by

( ~0,A)~(x)f' = Ax({p*f').

The right side is

exactly the definition of ( ~ x Ax)f' and therefore also (~A)~(x) = ~ , x A X .

4. 3. The Lie algebra of a Lie group. Let

G be a Lie group 9

DEFINITION of i n v a r i a n t

vectorfields

Explicitly

stated,

for all g E G . remind

4. 3.1.

The Lie alsebra under

the operation

this means

that

LG is a Lie algebra

us of left invariant

LG

of G i s t h e IR - L i e a l g e b r a

of G

t% 6 L G i f

by corollary

insures

the existence

and only if (Lg).A 4. Z . 6 .

as well as the founder

The following lemma

on G b y l e f t - t r a n s l a t i o n s .

The letter

of t h e t h e o r y , of many

= A L shall

Sophus Lie.

left invariant

vectorfields on a Lie group. LEMMA tangent

space

4. 3. Z.

G be a Lie group, _

of G a t t h e i d e n t i t y

and onl F one A C LG Proof:

Let

such that

If A e x i s t s ,

Ag = ((Lg). A)g .

then

by this formula.

A

e

~

condition As

for

Ag

= A

~ G

Then there

e

G e the exists

one

e

for

g C (3 a n d

in particular

4. Z. 8 t h i s m e a n s

= (Lg).eAe

X shows

"

the uniqueness.

L e = 1G , w e c e r t a i n l y

invariance of A is seen from

e

= (Lg),A

I n v i e w of l e m m a

(1)

This necessary

e and A

LG its Lie algebra,

have

A

e

We now define = A

e

.

The left

A

-47-

((Lg),A)g~ =

(Lg),

A

= (Lg),

A e

There remains

= (Lg),

= A e

(L),eA

. g'~

to s h o w t h a t t h e 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 differentiable vectorfield), By lemma

w h i c h m e a n s t h a t A(CG) c C G .

Let f ~ CG.

4.2.8

{(Lg),A)gf

=

Ae(L$f)g

and therefore {Af)(g)

Let

N: I

=

Ae(Lg f).

> G , I an interval of IR o-ontaining O ,

d Nt/t = o = A e G wi t h ~-~

A e ( L g f)

a differentiable curve in

Then

=

~

L

f) ( ~ )

t=o

=

~'[ f( g ~t) t -0

which shows ~f

6 CG. |

The correspondence

A

e

"~>

~:G

A

of t h e l e m m a

>

defines a bijective map

LG

e

w h i c h i s s e e n to b e a n i s o m o r p h i s m proved

of I R - v e c t o r s p a c e s

by (1).

We h a v e

-48-

THEOREM

G

e

4. 3. 3.

the tangent

space

G be a L i e g r o u p ,

of G at t h e i d e n t i t y

(~(Ae))g

defines a m a p

Let

= (Lg).eA e

e.

for

LG its Lie algebra The formula

g C G,

A e C Ge

----> 113, which is an i s o m o r p h i s m

~: G

and

of ]R-vectorspaces.

e

This map LG to G

~

allows transporting

In this sense,

e

COROLLARY Lie algebra

all g6; G.

4. 3 . 4 .

is often referred

e

Let

the map (Lg).

e

of d i m e n s i o n

: G

e

M o r e generally, the m a p s

> G

g

of ] R - L i e

algebra

from

to as the Lie algebra

G be a Lie group

is a L i e a l g e b r a

LG

Consider

G

the structure

of d i m e n s i o n

n.

o f G.

The

n.

, which is an isomorphism

for

P(gl' gz ) = (Lgz)*e(Lgl):1~e : Gg I

>Ggz

have the properties:

1)

P ( g z ' g3 ) p ( g l '

Z)

P(g, g) = IGg

DEFINITION an JR-linear is called

a parallelizable

to speak Let

for

g l ' g z ' g3 6; G

for

g 6; G

Let X be a manifold and P(gz' gl ) : T 1(x)

m a p f o r a l l ( g l ' g z ) 6; X x X ,

The m a p s sense

4. 3.5.

gz ) = P ( g l ' g3 )

satisfying

1) a n d Z).

> Tgz (x)

Then

X

manifold.

P(gz' gl ) are then necessarily isomorphisms of t h e d i m e n s i o n

and it m a k e s

of X.

e b e a f i x e d p o i n t of X a n d

Ai

(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. e

A. (i = 1, 9 . . , 1

Then

P ( e , g) A.

1e

n) on X s u c h t h a t t h e x ~ e c t o r s

= A.

lg

defines

A. (i = 1, . - - , lg

vectorfields

n) f o r m

a base

-49 -

of T X for all g C G . g COROLLARY Example Then

LIR ~

4. 3. 6.

4. 3. 7.

T h e manifold of a Lie group G

is parallelizable.

Consider

IR with its additive Lie group structure.

IR as vectorspace,

because the tangent space of IR at O is

]R . T h e r e is only one possible Lie algebra structure on 11% , defined by =

Ofor

c

B y the s a m e Now

L 11~ = IR for the additive group

that considering

endomorphiams

map

argument,

let V be n-dimensional

first r e m a r k

g G G.

m.

IR-vectorspace

GL(V)

c s

s (V) x s (V)

shows that (Lg)~

> A,~

s

g

We

is identified to s (V) for all

is the restriction of the bilinear

defining the multiplication in

= gay

= GL(V).

= algebra of ]R-linear

of V, the tangent space G

T h e multiplication in G L ( V )

and G

~Ir= IR/~.

for g ~ GL(V),

Ay

s

This

C G y identified to s

Y We show now PROPOSITION with the tangent algebras,

where

in the sense Proof:

4. 3 . 8 .

space

Let

at the identity,

on s

of s e c t i o n

After the canonical

we consider

we have

[AI'Az] g

=

Z

of L { G L ( V ) )

L ( G L ( V ) ) = s (V) a s L i e

the Lie algebra

4. 2 t o t h e n a t u r a l

A 1, A 2 C L ( G L ( V } } .

identification

algebra

structure

associated

structure.

We use the formula

-

w h i c h is valid for the global chart given by the e m b e d d i n g

GL(V)c

s

-50 -

In view of A i = g A i w e h a v e g e

I~

Aigl(g) Ajg

which s h o w s

[ A I, A z] g

But the right

= AlgAZg

side is just the commutator

g = e this gives the desired We

= AjgAig

[Alg,

A z ] in g

s

on G

LG

of a Lie group G

by left translation.

i s the Lie algebra of the right invariant vectorfields.

4. 3. 3, w e can define an i s o m o r p h i s m obtaining therefore an i s o m o r p h i s m

O

e

> RG

LG ~ RG

by considera-

Doing the s a m e for

the right translations, w e obtain another Lie algebra R G . RG

For

result.

have defined the Lie algebra

tion of the operation of G

- AZgAlg.

Explicitly: A s in t h e o r e m

of ]R-vectorspaces,

of ]R-vectorspaces.

shall see in section 4. 6 that there is also a natural i s o m o r p h i s m

We LG

~ RG

of the ]R-Lie algebra structure. Exercise 4. 3.9.

Let G

be a Lie group, C G

of real-valued functions on G , D G on G

and L G

4. 4.

Effect

In section

the ]R-Lie algebra of all vectorfields

the Lie algebra of G .

of m a p s

on operators

the ]R-vectorspace

S h o w that D G

= CG

|

LG.

and vectorfields.

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

W e w a n t t o 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.

differentiable)

on operators. maps.

-51 -

Let X, X

v

be manifolds and A ,

A

!

operators on X, X

!

respectively.

!

DEFINITION map

&0: X

4. 4.1.

A

and A

are

~0-related with respect to a

> X' , if the following d i a g r a m c o m m u t e s . CX'

CX

t AI I

(0"

CX If ~ is a diffeomorphism,

A

and

CX'

~0~A are

@-related operators.

But in the general case, A does neither determine an A ! such that A

and

A' are ~0-related, nor is A' unique, if it exists. I

LEMMA

4. 4. g.

Let

~0: X

> X

be a m a p .

(i) mIf A i and A v.1 (i = i, 2) a r e X

and X' respectively,

~-related operators on

then the following operators

are ~-related: !

and

AI + A 2 ,

AIA g

and

AIA 2

[ A I, A2]

and

[ AVl , A'Z]

I

(ii) If A

I

AI+ A 2

and A' are

!

,

~0-related operators on X

res]~ectively, then for

k C IR ~IA and

and X'

XA' are

~0 -related. Proof:

(i) Let f' ~ CX'.

Then

(D*((A I, + A'z)f') = @$(A'I f' + A'zf') = ~ 9 (All) + ~0~(AV2f') .

,_,

= AI(~f') + AZ(~*f') = (A I + AZ)(~f'),

-SZ-

!

!

showing that A I + A z and A 1 + A z are

W-related.

The

of A I A g and A I'A 2' is seen by c o m p a r i n g the d i a g r a m s @-relatedness,

~-relatedness serving to define

and the third assertion is a consequence

(ii) ~*((kA')f')

= (D *(k(A f )) = @ = k. A(~0*f')

The lemma

applies in particular

of this and (ii).

k. ~*(A'f')

= ( XA)(@*f')

, q. e. d.

to v e c t o r f i e l d s .

For that we make

explicit the notion of (~-relatedness in PROPOSITION

4.4. 3.

Let X, X' b e manifolds, @ : X

map

and A , A' vectorfields on X, X'

are

~ -related

Proof:

if a n d o n l y if

Let f' g C X ' (4. Ax)f'

~xAx

respectively. = A~( x)

Then

for every

> X' a A a n d A'

x g X.

Then

= Ax(~f')

= (A(~f'))(x)

x

b y d e f i n i t i o n of ~)~

.

On t h e o t h e r h a n d

x

A @(x)f' !

Comparison

4. 5.

proves

= (A'f')(@(x))

= (@*(A'f'))(x)

the lemma.

T h e f u n c t o r L. We h a v e d e f i n e d t h e L i e a l g e b r a

to extend this correspondence L E M M A 4 . 5 . i. morphism

L G f o r a n y L i e g r o u p G.

to a functor from Lie groups to Lie algebras.

L e t G , G' be L i e g r o u p s , P : G

of L i e g r o u p s a n d A ~ L G .

one A ' C L G '

We w a n t

s u c h t h a t A and A' a r e

>G'

a homo-

Then there exists one and only O-related.

-53-

Proof:

A' e x i s t s w i t h t h e d e s i r e d

Suppose

properties.

By propo-

sition 4. 4. 3 w e obtain

(i) where

A'e

p ,eAe

=

)

e, e is t h e i d e n t i t y of G, G'

is o n l y one A' 6 IX]' s u c h t h a t

Now we define conversely (1) 9 T h e r e p o Lg

-

remains

respectively.

~',

= A' , .

e

e

By lemma

This p r o v e s uniqueness.

A' a s t h e u n i q u e e l e m e n t

to show that A and

4. 3. Z there

A' a r e

of L G '

s ati s lying

~-related.

Now

Lp(g) o p i m p l i e s !

p,gAg

= 9,g(Lg),eAe

which is the desired

= (Lg(g))*e'9*e Ae

r e s u l t i n v i e w of p r o p o s i t i o n

I n t h e p r o o f of t h e l e m m a to a n e i g h b o r h o o d of e C G .

= Ap(g)

4. 4. 3.

4. 5 . 1 , w e u s e d o n l y t h e r e s t r i c t i o n It i s u s e f u l t o i n t r o d u c e

of

a corresponding

notion. DEFINITION neighborhood

4. 5. Z.

A local homomorphism

of e ~ G . T

U

is a differerLtiable m a p

p(glgz)

p :U

G

> G' defined on

> G' w h i c h satisfies

= 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 glg z e U .

The restriction neighborhood

L e t G , G' be L i e g r o u p s a n d U an open

of a h o m o m o r p h i s m

p : G

of e 6 G i s a l o c a l h o m o m o r p h i s m

a map with its restrictions local homomorphisms,

to open subsets

> G' t o a n o p e n G ~>

G'.

of t h e d o m a i n ,

obtaining thus the category

If w e i d e n t i f y

we can compose

of L i e g r o u p s a n d

-54-

An equivalence

local homomorphisms. isomorphism.

Explicitly

stated we have

D E F I N I T I O N 4. 5. 3. isomorphic

Two Lie groups,

if a n d o n l y i f t h e r e e x i s t s

and a diffeomorphism both P and

P' a r e

THEOREM

p: U

> U

l

G and G' , a r e l o c a l l y

open neighborhoods l

with inverse

p :

U I

U, U t Of e, e'

> U such that

local homomorphisms.

4. 5 . 4 .

h o o d of e i n G a n d

in t h i s c a t e g o r y i s c a l l e d a l o c a l

L e t G, G ' b e L i e g r o u p s ,

P: U

a local h o m o m o r p h i s m .

~ G'

(L(p)A)e,

defines a h o m o m o r p h i s m

U a n open nei~hbor-

=

p~ A e e

of Lie algebras

for

The f o r m u l a

A 6 LG

L(p) : LG

> LG' .

The

following diagram is commutative, P e

G

t

> G,

e

e

I

I

LG

where

L(P )

~ denotes the isomorphism

> LG'

of t h e o r e m

4. 3. 3.

Moreover

for

!

A C LG

the v e c t o r f i e l d s

Proof: that

L(p)

A/U

a n d L( p)A ~ L G

I n t h e p r o o f of l e m m a

is a h o m o m o r p h i s m .

We observe

right invariant vectorfields.

p-related.

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 i s is a c o n s e q u e n c e

that a homomorphism

same way a homomorphism

are

R(p ) : R G

p : G > RG

of l e m m a

> G' defines I

4. 4. Z.

in the

of the Lie algebras

of

-55-

Complement L(p ) = P ~ / L G

to 4. 5.4.

, where

Proof:

If

p ~ : DG

P :G

> G' is an i s o m o r p h i s m ,

9 > DG'

is the m a p defined in section 4. Z.

We have to show the commutativity LG

L(P)

then

of t h e d i a g r a m

; LG'

N

A

p~' DG Let A 6 LG.

> DG

Then on one hand (L(p)A)p (g)

= (Lp(g))~e p , e A e

=

p,(Lg),eAe

and on the other h a n d

(P~A)p(g)

by lemrna 4. Z. 8.

: p. A g g

This shows the desired property.

O n e cannot define U(p) in general by P~, because this m a p makes

sense for a d i f f e o m o r p h i s m

THEOREM

4. 5.5.

Let ~

of Lie ~roups, s

homomorphisms

of Lie algebras.

L(p)

defines a covariant

This is clear

by theorem

We can also consider

p .

be the cate~or}r of Lie groups and local

homgmorphisms

9 ~"">

only

the category

of ] R - L i e

The correspondence functor

L :s

algebras

G~-~>

LG,

> s

4.5.4.

the functor

g i v e n by G " ' " > G

, P ~ e

T h e commutativity of the d i a g r a m in t h e o r e m 4. 5.4 expresses that is a natural transformation of this functor into L, in fact a natural equivalence.

and

P e

M

-56-

COROLLARY

4.5.6.

T h e Lie algebras of locally i s o m o r p h i c

groups are isomorphic. Proof: We

L

sends equivalences in ~

into equivalences in =s

apply this to the natural injection G o ~

component

G

of the connected

of the identity G o into G , w h i c h is a local i s o m o r p h i s m .

This s h o w s

LG o ~

LG.

T h e Lie algebra is therefore a property of

G o , in fact, of an arbitrary neighborhood of the identity. Example

4. 5. 7.

T h e canonical h o m o m o r p h i s m Therefore

is a local i s o m o r p h i s m . LEMIVIA

Suppose

p: "]["

If

"Jr being compact,

Proof: I.

4.5.8.

tC

"IF with

LIR ~

> I t = IR/Z

]R

L ~ r , w h a t w e already know.

> ]R is a h o m o m o r p h i s m ,

then

p = 0.

p (,j[n) is contained in a closed interval

p(t) # 0.

T h e n there exists a positive integer

n such that nP(t) ~ l;which is a contradiction. This proves that there is no h o m o m o r p h i s m the identity i s o m o r p h i s m isomorphism Example and

T :G

proved

= LIR.

B u t of c o u r s e

4.5.9. > GL(V)

Let

V be an ] R - v e c t o r s p a c e

a representation

the natural

local

4. 3. 8 t h a t

L(T) : L G

of f i n i t e d i m e n s i o n ,

of t h e L i e g r o u p

G i n V.

s (V) i s t h e L i e a l g e b r a

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 ,

homomorphism

> IR i n d u c i n g

has this property.

in Proposition

The map

LT

"ff

> s (V).

and induces

We

of G L ( V ) . therefore

a

-57 -

DEFINITION the a

A-Lie

4. 5 . 1 0 .

algebra

of

Let A be a ring,

A-endomorphisms

algebra

O in V is a homomorphism

then called

an

with respect

Following

example

a finite-dimensional Lie algebra

4. 5 . 9 ,

:0

s of

~> s

V

is

to v .

a representation

]R-vectorspace

and

A representation

o f V.

A-Lie

O-module

V a A-module

of a L i e g r o u p

V defines

a representation

G in of the

L G i n V.

We now consider rnultiplicative

group

the homomorphism ]R$ of t h e r e a l s .

det : GL(V) The Lie algebra

> ]I%$ i n t o t h e of ]R $ i s JR.

We p r o v e PROPOSITION algebras

induced

4. 5.11.

The homomorphism

by the homomorphism

s

GL(V)

> IR of L i e

> ]R $

is the trace

map. Proof: =

A.

Let

A ~

s (V) a n d

a t a curve

in GL(V)

with

a~ = e ,

Then

0

det, A e Now

for any n o n - d e g e n e r a t e d

n-tuple

of v e c t o r s

Vl, ' ' - ,

~0(Vl, " ' ' ' and therefore

=

d {det at}/t dt = 0

n-form

"

co on V (n = d i m V) and any

Vn of V w e h a v e

Vn ) " det

at

=

r

tvl,

''',

a t v )n

-58-

d = d-t-{U~(0~tVl' " ' " 'atVn)}/t=O

~(Vl, - . . , Vn) 9 d e t , e A

= Z ~(atVl'"''atVi-l' i

~tvi ' atVi+l' '

'~tVn)f/=O

= ~, ~(Vl,''',Av i, --., v n) i = ~(Vl,''',

showing

det~

A = tr A .

Vn) - %r A

This is the desired

result

in view

of theorem

e

4.5.4. COROLLARY Proof: we have for

4. 5.1Z.

tr : s

tr(AB) = tr(BA)

for

A, B Cs

> 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)

tr(AB - BA) the latter bracket

Example

= tr[A,B]

= [tr A, tr B]

being the trivial one in

4.5. 13.

Let G

0 ----> G

e

= O,

1R .

be a Lie group and

> TG ----> G

>e

the sequence of example 3. I. 7. It induces a sequence of Lie algebra homomorphisms O ~ Here

LG e -'G e

of this sequence.

L(Ge)

(see example

> L(TG) ~

LG

4. 6. 4 b e l o w ) .

See

The inclusion

G ~

TG

> O 7.5.6

for the exactness

induces a homomorphism

- 59-

4. 6.

Applications 4. 6 . 1 .

groups

The Lie algebra

> G

Lie algebra

l

(i = 1, 2 ) a r e

homomorphisms

identities

of G 1, G 2 .

of L.

of a p r o d u c t

and G 1 xG Z the product

Pi : G1 x G 2

the

of t h e f u n c t o r a l i t y

group.

group.

G 1, G 2 be Lie

The canonical

homomorphisms

L ( P i ) : L ( G 1 x GZ) By theorem

Let

4.5.4,

projections

of L i e g r o u p s > LG.I "

Let

and induce e l, e 2 be

we have the commutative

diagram (expressing the naturalit 7 of ~ )

X

1t G 2

t

LG 2

the vertical

arrows

being isomorphisms. (G 1 x G 2 ) e l ' ez

implies

therefore

the isomorphisms L ( G 1 x G2)

If qi : L G I x L G 2

~--

The

IR-linear

isomorphism

G1 x el G2e 2

of I R - v e c t o r s p a c e s =

LG 1 x LG 2

.

> LG. (i = I, 2) denotes the canonical projection, 1

this i s o m o r p h i s m is given b y the commutative d i a g r a m

-60 -

(%

L(G 1 x G z)

> LX31 x L G z

LG 1 We w a n t t o t r a n s p o r t

L G 1 x L G z.

the Lie algebra structure

= (A_[, A'Z) with

of L(G 1 x G z) t o

L ( G 1 x GZ) we h a v e a(A) = (L(PI)A, L(pz)A) = (A I, A Z)

For A~

S i m i l a r l y f o r A' C L(G 1 x G 2) w e have

with A.I = L(Pi)A (i = I, 2).

a(A')

LG 2

A~:

a[A, A']

= L ( P i ) A ' (i = 1, Z).

Then

= (L(PI)[A, A'] , L(pz)[A, A'] ) = ( [ 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 . [ a(A), a ( A ' ) ]

=

We d e f i n e

a[A, A']

which means

(1) [(A 1, AZ), (A'1, AZ) ] = ([A 1, A'I], [A Z, A'Z]) for

A. , A'. ~ LG. 1

1

(i = 1, Z) .

1

W i t h t h i s d e f i n i t i o n , a is a n 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 b e a r i n g a n d O 1, O z Consider the product module >O 1 xO2

d e f i n e d by (1).

D E F I N I T I O N 4. 6. Z. O1, 0 2 is t h e L i e a l g e b r a

W e can n o w state

A-Lie algebras.

O1 x OZ with the map

[ , ]~O 1 x Oz) x ( O 1 x O7)

Then O1 x O z is a A-Lie T h e d i r e c t p r o d u c t of t w o

algebra.

A - Lie algebras

O 1 x 0 2 w i t h t h e m u l t i p l i c a t i o n d e f i n e d b y (1).

-61-

4. 6. 3.

PROPOSITION direct

product.

to the direct

product

that

product

that

Lie algebras

A 1, A z .

4. 6. 5.

Lie algebra

T h e relation b e t w e e n

structure

(example

Lie algebra

LG

G ~ the opposite group and I : G

that the

But we have already

IRn for the additive g r o u p

=

in a Lie algebra

It i s t h e n c l e a r

x L]R.

L(IR n) = ]Rn w i t h t h e t r i v i a l

Similarly L ( T n)

isomorphic

is commutative.

L(IR n) = LIR x . . .

LIR = IR w i t h t h e t r i v i a l

G 1 xG 2 the

LG 1 and LG 2.

for the multiplication

= 0 for any pair

4.6.4.

and

L ( G 1 x GZ) i s c a n o n i c a l l y

that commutativity

of c o m m u t a t i v e

Therefore

Lie groups

L G 1 x L G 2 of t h e L i e a l g e b r a s

[ A 1, AZ]

Example

G 1, G z b e

The Lie algebra

We remark means

Let

and R G .

seen

4. 3. 7).

structure. ~.n

Let G

> G ~ the i s o m o r p h i s m

=

~n/~Tn " be a Lie group, defined by

l(g) = g-i for g C G .

r

LEMMA

4. 6 . 6 .

Proof:

C o n s i d e r the m a p

being constant.

I, e -

= O:G

r

r

g

1Ge q~: G

>G

g

= (Rg_l) ,

g

>G

e

.

defined by

~(g) = gg-l.

But

+ (Lg),g=lI,

g

and therefore -I

I,g

For

=

- (i~),g

-1

~ (Rg=l)*g

=

_ (Lg_1), e o (Rg_1), g

g = e we obtain

l,e

=

- IGe

,

q.e.d.

-62-

Remark.

We h a v e s h o w n t h e f o r m u l a I,

This means already

g

=

- (L

g

_l),e ~ (Rg_l) ,

that the tangent map to the map

given by the tangent maps

already

implies

Let A be a ring and ~ algebra

is the

by the bracket

[ A 1, A 2 ]

PROPOSITION group.

Identify

isomorphism

LG

O

4.6.7.

of t h e o r e m

over

This can be used G x G of I : G

A 9

with the Lie algebra

>G

ona

>G.

The opposite

Lie

structure

defined

L e t G be a L i e g r o u p a n d G ~ t h e

opposite

[A I,Az] for A I, A z e

=

with G

the differentiability

~

g g G

> G in each point is

of t h e m u l t i p l i c a t i o n

a Lie algebra

A-module

I : G

of t h e t r a n s l a t i o n s .

to prove that the differentiability group manifold

for g

e

and L{G ~ 4. 3. 3.

with G ~

by the canonical

Then

L(G ~

: (LG) ~

(LG) ~ being the opposite Lie al~ebra of L G . Proof:

After the indicated identification w e have L(1) = l•e for

the isomorphism

I, A i = - A i 6

I : G

L(G~

>G ~ .

To

T o A. ~ L G (i = 1, 2) c o r r e s p o n d s 1

[ A I, A2]

C

LG

there corresponds on one

e

hand

- [ A I, A Z ] L G

I is an isomorphism.

and on the other hand also [ - A I, -Ag]L(GO) , as Therefore

[At' AZ]L{GO )

-

[AI, A T ] L G

=

[A I, AZ](LG)O

q.e.d.

-63-

COROLLARY

4. 6 . 8 .

L e t G be a L i e g r o u p ,

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 vectorfi elds. of t h e o r e m

and RG the Lie a l g e b r a

I d e n t i f y L G a n d RG w i t h

4. 3. 3.

We

G e by the canonical isomorphism

=

(LG) ~

observe that left translations of G

of G ~ a n d vice versa,

so that IX] = R ( G ~

and R G

m e n t i o n e d identifications, by proposition 4. 6. 7 shows

RG

of r i g h t i n v a r i a n t

Then aO

Proof:

LG the Lie a l g e b r a

are right translations =

L(G ~

L(G~

After

= (LG) ~

the

which

= (LG) ~

This shows,

of course, the existence of a natural i s o m o r p h i s m

e~

RG =

LG.

Moreover

COROLLARY

then LG

Let G

be a Lie group.

If G

is c o m m u t a t i v e ,

is c o m m u t a t i v e . T h e c o m m u t a t i v i t y of G

Proof: and therefore We

4.6.9.

LG

implies R G

= LG.

But R G

= (LG) ~

= (LG) ~ , q.e.d.

shall see in chapter 6 that, for connected G, the c o n v e r s e is

als o true. Example

4. 6.10.

Let V

be a finite-dimensional ~ t - v e c t o r s p a c e

and consider the natural representation of G L ( V ) identifying L(GL(V)) w e obtain

s (V).

in V.

We

have seen that

canonically with the tangent space at the identity,

B y corollary 4. 6. 8, R G

identified with the tangent i

space at the identity is ( s (V)) ~

-64-

4. 7.

The adjoint representation Consider

~Y: G

t h e o p e r a t i o n of G on G b y i n n e r a u t o m o r p h i s m s

> Aut G (see example

G-group

of a L i e g r o u p .

G into a G-Lie

1.1.5).

algebra

The functor L transforms

LG

according

to proposition

We r e p e a t t h e d e f i n i t i o n of t h e i n d u c e d G - o p e r a t i o n composed

on L G :

the 1.1.9.

it i s t h e

homomorphism G

~

D E F I N I T I O N 4. 7.1. is t h e r e p r e s e n t a t i o n

> Aut G

L

> Aut L G

The adjoint representation

of a L i e G r o u p G

of G i n L G i n d u c e d b y t h e o p e r a t i o n of G on G b y

inner automorphisms:

Adg

= L([[g).

4. 7. Z.

L e t G, G '

!

PROPOSITION

homomorphism. respect

Then

> LG

L(p) : LG

to the adjoint representations

Proof:

The commutativity

be L i e ~ r o u p s a n d !

is a

p: G

p-equivarianee

>G

a

with

of G, G ' i n L G , L G

of t h e d i a g r a m !

G

iI

Consider permitting

>G

! G -

(see also example

P

I p

> G

1. Z. 3) a n d t h e f u n c t o r a l i t y

the canonical isomorphism

to i n t e r p r e t

of L p r o v e ~l : G

G e as the Lie algebra

it f o l l o w s t h a t t h e e f f e c t of A d g : L G

I

the statement.

> LG

of t h e o r e m 4.3. 3

e

of G .

From

theorem

4.5.4

> L G on G e i s g i v e n b y t h e m a p

-65-

(

gg)*e

:G

e

in e x a m p l e

--->G e.

T h i s p r o v e s t h a t t h e o p e r a t i o n of G on G e d e f i n e d

3. Z. 8 is j u s t t h e 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

G e and LG. Another description of the adjoint representation is given in

P R O P O S I T I O N 4. 7. 3. the adjoint representation

L e t G be a L i e g r o u p ,

and A G LG. Ad g A

Proof: ([[g),A

Adg A = L(gg) A

= (R _ I ) , ( L g ) , A g

=

> Aut L G

Then (Rg_l) CA

= (gg),A

= (Rg_I),A

Ad : G

for

by complement

4. 5 . 4 .

A G LG.

T h i s s h o w s t h a t t h e o p e r a t i o n of G i n G b y r i g h t t r a n s l a t i o n s a r i g h t o p e r a t i o n of G i n LG a n d t h e a d j o i n t r e p r e s e n t a t i o n e f f e c t of t h i s o p e r a t i o n .

But

defines

describes

the

-66-

Chapter

5.

VECTORFIELDS GROUPS OF

The Lie algebra

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

The key for the understanding and ordinary 5,1.

AND 1-PARAMETER TRANSFORMATIONS.

differential

of t h i s is t h e r e l a t i o n

equations,

on the group.

between vectorfields

which is studied in this chapter.

1-parameter

groul~of transformations.

DEFINITION

5.1.1.

An IR-manifold

X is called a 1-parameter

group of transformations. Let X

Here ~t

_

be a manifold and

I denotes d

dt ~ t

an open interval

the tangent vector ~t f

which characterizes Now let

q~:IR xX

A ; (A x ) x C X If

t ~

Ax

t ~-'~> $ t ~st

= @t"

respectively,

then

, t --,--~

of ]R c o n t a i n i n g

0.

of ~ i n t h e p o i n t

d f(•t ) dt

~/t"

for every

>X,

~t' a curve on X . We We

denote by then have

f ~ CX

(t, x) " - " ~ > (P t(x) , be a 1 - p a r a m e t e r

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 r o u p of t r a n s f o r m a t i o n s

(1) Then

>X

~t"

of t r a n s f o r m a t i o n s . 1-parameter

_

y:I

- d-i~Ot(x)

/t = 0

is a vectorfield is a 1-parameter If

of X . "

-

~ t Ix)

group

'~g~ i s a

Define for every

x ~ X

/t = 0

on X. g r o u p of t r a n s f o r m a t i o n s ,

A, B a r e t h e v e c t o r f i e l d s

then also

i n d u c e d b y (Pt ' ~ t

-67-

d ~-: { ~ ( x ) }

=

Bx

In this definition group

t=0

of t h e v e c t o r f i e l d

of t r a n s f o r m a t i o n s ,

globally

d :[ Gst(X)/t

:

=-

=0

corresponding

we have not made

sA

,

x

to a 1-parameter

use of the fact that

St

is

defined.

DEFINITION IR a n d

5 . 1 . Z.

U an open set of X.

formations

of X d e f i n e d

( > 0,

Let

I

be an open interval

E

Alocall-parameter

on U is a map

group

qg: I (

x U

(-(,E)

of

of l o c a l t r a n s -

> X , (t, x ) ~

Ot(x)

such that I)

f o r a l l t 6; I ( , ~ t

2)

if t, s,t

Equation A on U , properties

+ s 6; I

(1) s t i l l m a k e s

which

is called

of a l o c a l

used for this fact,

curve

t ~"~>~

with the initial

and t(x)

is a diffeomorphism

x , S s ( X ) E; U , t h e n ~ t + s ( X )

sense

for

x 6; U a n d d e f i n e s

the vectorfield

1-parameter

5 . 1 . 3.

~_roup of t r a n s f o r m a t i o n s of X ,

and

>st(U)

induced

group

,

= opt ( ~ s ( X ) ) .

a vectorfield

o n U b y q9t .

The

of l o c a l t r a n s f o r m a t i o n s

are not

butthey allowto prove

PROPOSITION

subset

(

: U

A

Let

of X ,

> X

be a local..l-parameter

E > O, I E = ( - E , E ) C

where

the induced

satisfies

~: IE x U

vectorfield

the differential

on U . equation

Ot(x)

-- Aot(x )

Oo(X)

--

condition •

Then for

]R

U

an o p e n

x 6; U , the

-68-

Proof:

% t (x)f

Let

f C CX.

d

--

Then for fixed

=

Lira

x C U

_/_Is{ f(~t+s (x)) - f(~t(x))}

s-->O Lim s-->0 =

=

__Is{f( Ss(~t (x))) - f( ~~

d

/

ds

/s =0

Acpt(x) f

COROLLARY ..1-parameter induce

Proof:

q.e.d.

5.1.4.

groups

the same

,

Let

x U

of l o c a l t r a n s f o r m a t i o n s

vectorfields

This

~p, qJ : I (

> X be

defined

two

on U .

local If they

on U , t h e y c o i n c i d e .

is just the uniqueness

theorem

for ordinary

differential

equations. Applying we

prove

the existence

theorem

for ordinary

differential

equations,

now

PROPOSITION an

5.1.5.

E > 0,

Let

there

exists

group

of l o c a l t r a n s f o r m a t i o n s

A be a vectorfield

an open subset cp : I

U of x U

X

on X.

For

any

x ~ X

and a local 1-parameter

> X , which

induces

on U

E

the given vectorfield. Proof:

For

a fixed

x C X

we define

t "'>

the differential equation ~ot(x)

=

A~t(x )

~Pt(x) a s t h e s o l u t i o n

of

-69

with initial value

r

= x.

~ t (~ps(x))

-

W e n o w prove

r

=

for

(x)

such that both sides are defined.

Write

s , t, t + s

C I

E

cat+s(X) = Ctl(t) , ~at(~as(X)) = az(t).

Then

al(t)

= cbt+s(X)

~(t)

= ~ot(~s(x))

= A t+s(X)

=

Aal(t )

and

1

(i = 1, Z) a r e s o l u t i o n s

al(O ) = a Z ( O ) : It r e m a i n s : U that

of t h e s a m e

therefore

Certainly

--

on x.

= ~ _ t ({Pt(x))

For

Aa2(t )

equation.

x----'~ ~ t(x)

~Po i s t h e i d e n t i t y

differentiably opt(r

t (Os(x))

differential

to show that the map

> cPt(U ) .

A

c1 = a 2 , p r o v i n g

q)s(X), w e o b t a i n

oPt(x) d e p e n d s

x C U

=

As

the desired

result.

is a diffeomorphism

transformation. sufficiently

We know

small

= (Po(X) = X , p r o v i n g

t and that

opt i s a d i f f e o m o r p h i s m . DEFINITION

5.1.6.

A vectorfi~ld

A on X is

complete,

if it i s

induced by a 1 - p a r a m e t e r group of transformations. Example

5. I. 7.

C o n s i d e r the vectorfield A

on ]R w h i c h has in

e v e r y point a positively oriented vector of length one. submanifold

(0, l)c JR.

Then

A

C o n s i d e r the

r.~strict'ed to (0, I) is not complete.

A criteria for c o m p l e t e n e s s is given in

-70-

LEMMA

5.1. 7.

Let A

be a vectorfield on X .

Suppose there exists

r > 0 a n d a local 1 - p a r a m e t e r group of local transformations T h e n ~ has an extension to a 1 - p a r a m e t e r

inducing A . tions and A

> X

group of t r a n s f o r m a -

is therefore complete.

Proof: ~t for

~0: I x X

~t

is a d i f f e o m o r p h i s m

It l > f.

(

Write

t = k. ~

for

It I ~

E.

T h e r e is only to define

+ r with an integer k

and

Irl < ~2 -k

If k > 0, define __(~ = (r

~

. If k < 0 , define ~t = ((D__E)

Z

Now

z

~t satisfies all conditions. ~ Example

on X

~ ~r

5. I. 8.

Let X

be a c o m p a c t manifold.

A n y vectorfield A

is complete. We r e m a r k

that the relation b e t w e e n vectorfields and local l - p a r a m e t e r

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 - p a r a m e t e r groups of transformations and equivariant m a p s . Convention on notations.

transformations on X vectorfield A

on X ,

G i v e n a local 1 - p a r a m e t e r group of local

w e denote it ~t and just speak of the induced without specifying the d o m a i n s of definition.

for a given vectorfield A

on X ,

w e just write

group of local transformations inducing A formulas are valid as soon as they m a k e particular if only

l-parameter

%Dt for a local 1 - p a r a m e t e r

on s o m e sense.

Also,

subset of X .

They make

group of transformations

The

sense in

occur.

-71-

PROPOSITION transformations

5.2.1.

!

Let

~t' ~t be local 1-parameter groups of

on X, X' , A, A ' the induced vectorfields and !

a map.

_If @ o ~ t

Proof:

~: X

>X'

!

= ~t

o ~ for all t,

then A and A

are

~-related.

W e have !

Ix) A~;(x) = Ax~t(q~(x)) by differentiating

with respect

to t .

Or !

,~tlx) AtiYtlx) which shows the proposition

A

(~t(x))

i n v i e w of l e m m a

It is convenient to call a m a p an equivariance

=

with respect

4 . 4 . 3.

~ : X ---->X' satisfying

to the given local 1-parameter

q~ ~ ~t = ~!t " ~ g r o u p s of

!

local transformations induced vectorfields for equivariant

~t and A

maps.

PROPOSITION

kDt .

a n d A' a r e t h e n Precisely

5.2.2.

on X, X' r ~ e s p e c t i v e l y ,

The proposition

$' '

T h i s is c h a r a c t e r i s t i c

we have

L e t X, X' be m a n i f o l d s ,

St

- -

~0-related.

says that the

corresponding

A, A' v e c t o r f i e l d s

local 1-parameter

t

groups

-!

of l o c a l t r a n s f o r m a t i o n s ~0-related,

and

~0: X

> X

a map.

If A a n d A! a r e

then O

0

t

~

,

t I

Proof: Then

For

x C X write

al(O ) = a2(O)

= ~0(x).

satisfy the same differential

al (t)

= @ (q~t(x)) , a z ( t )

We p r o v e equation.

=

qit(~O(x)) .

el = aZ b y s h o w i n g t h a t But

al'

ag

-7Z~l(t) = (~q,t(x)AqSt(x) = ~l(t) !

by lemma

4.4. 3,

as

COROLLARY

A and

A

are

E2.(t)

=

A a'

5. Z. 3.

diffeomorphism.

~-related

z(t)

and

, q.e.d.

Let X, X' be manifolds and

If A i s a v e c t o r f i e l d

on X,

~: X

seneratin$

> X' a

a local

1-parameter !

~roup

of local transformations

~enerates

the local

Proof:

>X

{~ ~ q/t

=

5.2.4.

Let

q/t

~ (~ f o r a l l

i.e.

~0 : U

A and

~ o ~t

~0,A are

~ q~

r

A a vectorfield

of local transformations

Then

~#A

= A

is still true

> X defined

being

a diffeomorphism.

a vectorfield

on a convenient 4. 2. 8.

that

X

on X ~t

and

if and only if

t.

a restriction

of l e m m a

on

we obtain

group

that the preceding

a map

~.A

of l o c a l t r a n s f o r m a t i o n

X be a manifold,

adiffeomorphism.

We observe

formula

to observe

a local 1-parameter

~0 : X

group

this to an automorphism

COROLLARY

of X,

1-parameter

It i s s u f f i c i e n t

Applying

generating

q't ' t h e n t h e v e c t o r f i e l d

on an open subset ~o#A

subset

of X

Consider

for a local

automorphism

of X a n d h a v i n g

is then to be interpreted

and can be defined

in particular

as

by the

, w h i c h is a local S

automorphism

of X i n t h i s

sense.

As

qJ S

we see that

( ~s),A

velocity field A

= A by corollary

of the flow

property of a stationary flow.

5.2.4.

~ ~t

=

qJt ~ ~ s

This just m e a n s

for

all t,

that the

St is invariant by the flow, the characteristic

-1

"

-73-

A n application of corollary 5. Z. 4 is the following: LEMMA

5. Z. 5.

Let G be a Lie group, A 6 L G

and

t

l - p a r a m e t e r group of local transformations generated by A . L

g

o 9

=

t

r

o L

t

for all t,

g

a local Then

g C G .

W e can n o w prove PROPOSITION vectorfield

~t

Let

We consider

generated

a local 1-parameter

by A,

and show that

group

of l o c a l t r a n s f o r m a -

~ :I x G (

T h e n the proposition follows by l e r n m a 5. I. 7.

: IE x U

> G for an

E > 0 and a neighborhood U

n e c e s s a r y condition for an extension

= (~t ~ Lg)(e)

~(g) by l e m m a desired

5. ~. 5. ~: IE x G

PROPOSITION

~: IE x G

> G

,~: G

> G

and only if

for an

Suppose of e ~ G .

A

is

> G.

5. Z. 7.

By lemma

5.2.8.

> G

@ by this formula, w e obtain the

Let G

5.2.5 is be a Lie sroup, A ~ L G

1 - p a r a m e t e r group of transformations generated by A . Proof:

to a local

: (Lg o ~t)(e) : Lg(~(e))

Defining conversely

Another consequence of l e m m a

I~MMA

left invariant

St h a s a n e x t e n s i o n

1 - p a r a m e t e r group of local transformations > 0.

Every

G be a Lie ~roup.

on G is complete.

Proof: tions

5. Z. 6.

Let G

and

Then

q~t the

i t = R~(e).

5. ~. 5 a particular case of be a group (in the sense of algebra).

is a right translation (and then necessarily R ~ )

A map = y ) i_~f

-74-

L

Proof: Suppose g~(V)

g

~ ~

= %

o L

The associativity

conversely = qJ(gv)

L

g

shows

o qJ =

and in particular

The bracket

R~

K. N o r n i z u a n d

We use the notation automorphism

r

(e

g 6 G.

is necessary. For

V C G

= ~ (g) , i . e .

;g

,

q.e.d.

of the bracket

S. K o b a y a s h i ~.A

5.3.1.

[11], p.

of t w o v e c t o r f i e l d s

15).

for a vectorfield

of X a s explained

PROPOSITION

Let

X , and ~ t a l o c a l 1 - p a r a m e t e r by A.

G .

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 (taken from

for all

g

g~(e)

=

g 6

that the condition

~ o L

~e(g)

5. 3.

for all

g

A on X and a local

in 5.2.

A and group

B be vectorfields of l o c a l

on the manifold

transformations

generated

Then

[A, B]

= x

LEMMA

5. 3. Z.

1 Lirn ~ - [ B x t--~O

- ((~t).B)x]

Le___~tE > 0 ,

IE

for

= (-E, ( ) c ] R

with f(o, x) = 0 for x C X .

Then

f ( t , x) = t g ( t , x ) .

8f g ( o , x) = ~ ( o , x) .

Moreover

there exists

1 Proof:

Define

g ( t , x)

=

-~- ( t s , x ) d s o

x S X

a n d f : IE x X

g :I

x X

> ]R

> ]R with

-75-

LEMMA gt G C X

5. 3. 3.

with f o ~t

Let A

generate

= f + t gt and

opt.

F o r any f G C X

there exists

go = A/.

T h e function g(t, x) = gt(x) is defined, for each fixed x G X , It[ < r

for

some

Proof: Then

~ > O.

Consider

h(t, x) = f(~0t(x)) - f(x) and apply l e m m a

f o~)t = f + tgt"

(A.f)(x) =

We 1

Lira

-~ [f((Pt(x)) - f(x)] =

lemma

5. 3. 3.

Set

x t =

1

Lira

T f(t, x)

t-->0 =

of p r o p o s i t i o n

5. 3. Z.

have

t--->0

Proof

in

5. 3 . 1 .

Let

(p:l(x).

L i m g(t, x) = go(X) t--->0

f G CX.

Take

gt 6 C X as in

Then

((~t)$B)x f = (B(f 9 qgt))(xt) = (Bf)(xt)+

t(Bgt)(xt)

and I

L i m -[ [ B x - ((~t), B)x ] f = t--->0

i

Lira -~ [(Bf)(x) - (Bf)(xt) ] t-->0

=

Ax(Bf) - B x g o

=

[A, B]

f

= A

,

x (Bf)

L i r a (Bgt)(xt) t-->O

- Bx(Af)

q.e.d.

X

COROLLARY

X

and ~ t

by A.

5. 3 . 4 .

Let

a local 1 - p a r a m e t e r

Then for any value

A and

B be vectorfields

on the manifold

group of local transformations

of s ~ I R ,

x~

X

generated

-76 -

((Os).[A,

Let s em.

Proof: aS

(~s),A

Lira 71 [((~0s),B) x t-->0

=

B] )

= A

(~s),[A,

Then

by the remark

- ((~t+ s), B)x ]

B]

[(~s).A,(C~s).B]

=

= [A,(~s),B

at t h e e n d of 5. Z.

Applying proposition 5. 3. I. , w e obtain

[A, (~)s),B]x

1 Lim -[[((~s),B)x t--->0

=

i Lim -[ t-->0 PROPOSITION o__n X g e n e r a t i n g ~t

Proof:

If

by corollary [ A , B] fore

= O.

(~pt).B

every

~

S

:

Then

~t

A and

B vectorfields

~ 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

=

~s

~~

t

for every

St and

s a n d t if a n d

O. ~0t o @

S

5.2.4.

=

q~

S

~ opt f o r e v e r y

By proposition

By corollary

5. 3 . 4 ,

= B for every

s and t,

5. 3.1, [ A , B] d

-.

-~ ((cpt).B)x

t a n d by c o r o l l a r y

= O.

(~t).B

Suppose conversely

O for any t.

5.2.4

= B

There-

~0t c o m m u t e s

with

.

PROPOSITION local 1-parameter Suppose

[((~s) * B)x _ ((~t+s ),B)x] "

L e t X be a m a n i f o l d ,

local 1-parameter

respectively.

o n l y if [ A , B]

5.3.5.

- ((~t),(~s),B)x]

[ A , B]

5. 3 . 6 .

Let the vectorfields

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 = O.

Then

X t =

A a n d D of X g e n e r a t e ~t

and

~t

respectively.

{~t o ~t = ~ t ~ @t is a l o c a l 1 - p a r a m e t e r

g r o u p of local transformations and is generated by A + B.

] ,

-77-

Proof:

Proposition 5. 3.5 s h o w s that

group of local transformations.

~t(x)

X

is i n d e e d a l o c a l 1 - p a r a m e t e r

t

Now

= ~t

( ~ t (x)) + (r

~'t (x)

= A x t ( x ) + (r 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

(r

Bq't(x)

) Bxit(x) Therefore

5. 2. 4 (Ot) * B = B .

= B(pt(~t(x) ) = BXt(x )

and kt(x)

5.4.

1-parameter DEFINITION

Remark.

a: ]R Let X

formations of X .

subgroup

IR

(A + B) Xt(x)

, q.e.d.

subgroups of a Lie group. 5.4. I. A

a homomorphism

=

l-parameter

> G

subgroup

is

of Lie groups.

be a manifold and

~t a 1 - p a r a m e t e r group of trans-

O n e w o u l d like to consider

> Aut X of Aut X.

of the d i f f e r e n t i a b i l i t y

a of a Lie group G

of this m a p .

t "">

(~t as a l - p a r a m e t e r

But it does not m a k e See also the r e m a r k

sense to speak after e x a m p l e

3.2.5. T h e trivial h o m o m o r p h i s m

of G,

O : IR

,>G is a 1 - p a r a m e t e r

subgroup

-78-

a: IN

A non-trivial l-parameter subgroup

>G

is not necessarily

an injection, as s h o w n by Example

5.4.2.

T h e canonical h o m o m o r p h i s m

]R

> ] R / Z = ~r is

a l - p a r a m e t e r subgroup of "Jr . Lemma

5.4. 5 below s h o w s that a non-trivial l-parameter subgroup

is an i m m e r s i o n . Let A be a complete vectorfield on G of transformations generated by A .

at Then

a : IR

>G

s u b g r o u p of G .

and

=

P R O P O S I T I O N 5 . 4 . 3. l-parameter subgroup of G

~0t{e )

for e C G

but a is not necessarily a 1-parameter

t h i s is t h e c a s e ,

as s t a t e d in

L e t G be a L i e g r o u p , A C L G , ~0 t th___e_e generated by A

and

defined by

a t = ~t(e).

~t

an.d (Dt is completely described by a

= Rat Proof:

atl+t 2

> G

the m a p

Moreover

5.2.5, w e obtain

= ~0tl+tz(e ) = ~tl(~tz{e))

= (L~0t2(e) ~ ~0 tl)(e)

= (~tl

= ~t2(e)~0tl(e)

In view of proposition 5. g. 7 w e have @ t

The statement

a: IR

T h e n a is a l-parameter subgroup of G .

Applying l e m m a

group

Define

ao = ~0o(e) = e ,

If A g L G ,

and ~t t h e 1 - p a r a m e t e r

o L~0t2(e))(e)

= at 2atl

= Rat "|

~0 t = R a t is o f t e n p a r a p h r a s e d

in t h e l i t e r a t u r e

by:

-79 -

"the infinitesimal

transformation

generated

by a left invariant

veCtorfield

is a right translation". We call a the 1-parameter s G denote map

~:

t h e s e t of 1 - p a r a m e t e r

LG

of G defined

subgroup

subgroups

of G.

by A E LG.

Let

We have defined

a

> s

LEMMA

5.4.4.

Let A C L G ,

s

a= ~(A)

.

Then

a is th~

solution of the differential equation at with initial

condition

Proof: a

O

= ~

o

ht

(e)

This and shows

=

e ,

lemma

a

=

An t

= e .

o

= ~t(e)

= A t(e )

by proposition 5. I. 3 and

= Aat

q.e.d. gives

a direct

description

We shall

its injectivity.

of t h e m a p

see that

~

~ : LG

is bijective.

First

> s we

prove LEMM

A 5.4.5.

Let at

Proof: to

a ~ =

s

Then

(Lat)*e h~

By differentiating

at+s

=

=

ata

(Rat)*ea~

s

=

s , we obtain dt+s

=

(

)*a

s S

and for

s = 0

the desired

result.

t

s @s

a s at

with respect

-80

THEOREM s

5.4.6.

G be a Lie group,

the set of l-parameter subgroups of G .

~'(A) = Then

ct ~ s

as the solution

~. : L G Proof:

necessarily LG

s G

is bijective.

Let

a g

s

A

=

~

>

s G.

~t = A

In fact,

is described

A

g

-

Proof: by A.

Then

t

(g~t)'t

with initial condition

ct = e. o

A~

of 4 -

LG,

A

=

e

~

, we have

o

surjectivity

the tangent

precisely 5 . 4 . 7.

Defining conversely Aat

Let

= (Lct t)

~

tt

o

= tt

t

of ~ .

vectors

definition

of t h e c u r v e vectorfield

t ,-,~a

a bijection t

defining

belong a.

The

by A g LG ,

a = ~{A) 6

s G

and

g ~ G.

=0

Let ~t ~t(g )

be the 1-parameter = A~t(g ) for any

=

Rat(g)

group g ~ G.

=

of transformations Now by proposition

ga t

and ~0t(g ) t = 0 this shows A consequence

then

"

~t(g)

For

and define

5.4. 5 shows that we obtain by the same

PROPOSITION Then

its Lie algebra we

a = ~'{A) f o r s o m e

as well to the left as to the right invariant situation

LG

F o r any A C L G

which shows injectivity o

5 . 4 . 5, w h i c h s h o w s

Lemma RG

If

as the vectorfieldwith

by lemma

of

>

e

A~

Let

-

-

(gat) ~ .

o

g

= (gat)" t=0

of t h e o r e m

5.4.6

/~p ( g ) =

A

is the

,

q.e.d.

generated 5 . 4 . 3,

-81-

PROPOSITION and

a : I

a unique

5.4. 8.

> G a local 1-parameter

Proof:

Let

subgroup

to

= ( La t ) . e a o =

equation.

homomorphism

of L i e g r o u p s .

Now

a , showing

~tt .

a

=

to a 1 - p a r a m e t e r

by

a

o

= A

and

e

a/I

= a .

~=

~(A).

exists

Lemma

~tt = ( L a t ) . e cto a n d t h e r e f o r e

But a is also a solution

ct

O

Then there

> G of G w i t h

a : n~

A E LG be defined

5.4. 5 still applies Aat

Let I be an open interval of IR containing O

= e shows

I = a

of t h i s d i f f e r e n t i a l

, and

a is an extension

of a

O

s u b g r o u p of G .

T h e uniqueness follows f r o m the fact

that there exists only one 1 - p a r a m e t e r

subgroup

a of G w i t h given

d O

A s s h o w n by the e x a m p l e

of the local i s o m o r p h i s m

"IF

> IR, there !

> G

exists not necessarily an extension of a local h o m o m o r p h i s m

G

Lie groups to a h o m o m o r p h i s m

4. 5.8).

follows

from

G is simply

the theory connected

(see r e m a r k

of t o p o l o g i c a l (see

following l e m m a

groups,

also lemma

that an extension

7. Z. 5).

Proposition

of It

exists,

if

5.4. 8 is a

particularly simple case of this situation. LEMMA

5.4.9.

corresponding

elements

transformations ~t

~ q

every

s

=

Let

generated @

s

~ ~t

G be a Lie group, C

s

~t

and

by A and

for every

t and

~

A and

B 6 LG,

the 1-parameter

B respectively.

~ and groups

~ the of

Then

s if a n d o n l y if

at~ s = ~

s

a

t

for

t and a . Proof:

For

( ~ t ~ @s )(g)

g C G, = (~t

~ @s ~ L g ) ( e )

=

(Lg o ~t

~ @ s )(e) b y l e m m a

5. g 5.

-82 -

Now

@(e)s

= -~s = L ~ s ( e )

and therefore

( L g o opt o @ s ) ( e ) = ( L g

for the same reason

o ~ t ~ L~s )(e)

=

( L g o L 1~s o r

)(e)

thus proving ( ~ t ~ @ s ){g)

:

g[5 s

(~

=

ga

at

Similarly o ~t)(g )

S

t [Bs

This proves the lemma. Consider

the expression

the proof.

In p a r t i c u l a r

proposition

5. 3. 6

transformations,

X t (e)

PROPOSITION ponding 1-parameter = ~tat

corresponding

(@ t o qJt)(e ) = ~ t a t

Xt = ~ t o St = qJt o ~ t

A + B the c o r r e s p o n d i n g

at ~ t

( ~ t oK s)(g) = g ~ s a t

= ~t a t

= a t ~t

vectorfield.

5.4.10.

= O, by

is a 1-parameter a 1-parameter

in

g r o u p of

subgroup,

and

Therefore

Suppose

= Nt d e f i n e s a 1 - p a r a m e t e r left invariant

" If [ A , B]

L e t A, B C L G and

subgroups.

' which occurred

[ A , B]

a,~ C = O.

s

t h e corres-

Then

subgroup and A + B is the

vectorfield.

Together with proposition 5. 3.5 w e obtain from l e m m a 5.4.9 PROPOSITION

5.4. II.

Le__~tA, B ~ L G ,

groups of transformations generated by A

@t and ~t the 1-parameter

and B respectively and a,

the corresponding 1-parameter subgroups of G. statements

are equivalent:

Then the following

-83-

I)

[A,B]

2)

~to~

3)

at~ s

We s h a l l to

at~t

=

=

O =

S

~s

=

~ ~t

S

every

for every

r t

see in chapter

~tat

for

s and

6 that these

for every

subgroup

t

conditions

t (see proposition

N o w w e consider a h o m o m o r p h i s m 1-parameter

s and t

a ~

s G gives

are

even equivalent

6.5. 3.).

p :G

> G

of Lie groups.

by composition

with

A

p an element

!

p o a

E

s

morphism

. ]R

If

> G' , but can by proposition

to a 1-parameter map

s

theorem

:

p is a local homomorphism,

subgroup

s

>

5 . 4 . 6.

More

PROPOSITION

s

so defined

precisely

maps

of t h e o r e m

Proof: means

where

For

s

is compatible

homoextended

by p o a.

with the map

The ~

of

we have

Let G, G' b e Lie ~roups and

9: G

> G'

T h e n the followin$ diasram

LG

commutes,

5.4. 8 be uniquely

of G' , which we also denote

5.4. iZ.

a local h o m o m o r p h i s m .

p o a is alocal

--

L(p)

--:> L G '

s (0)

>,s

is the composition with p and ~ G '

~G'

the

5.4.6.

a ~

that the diagram

s G

w e have ( 9 ~ a ) t / t = 0

(without t h e d o t t e d l i n e )

= P *e ~t~

This

-84-

P.

G

e e

L(P)

LG

s commutes.

L(9)

s

m a p of t h e u n d e r l y i n g

5.5.

V : s

LG

sets.

this proves the proposition. > Ens,

set and to any Lie algebra

> s

i

~ s

(in fact bijective),

Define the forget-functor

: V o L

G

is d e f i n e d by f i l l i n g in the d o t t e d line in the u p p e r half.

G being surjective,

its underlying

>

Proposition

assigning to any Lie algebra

homomorphism

the corresponding

5 . 4 . 12 s t a t e s t h e n t h a t

is a n a t u r a l t r a n s f o r m a t i o n ,

in fact a natural

equivalence.

Killing vectorfields. In this

section,

the relation between 1-parameter

group G and 1-parameter

g r o u p s of t r a n s f o r m a t i o n s

subgroups

of a L i e

of a G - m a n i f o l d

X

is s t u d i e d . Let X be a G-manifold and

~: ]R

> G a 1-parameter ]R

defines a 1-parameter map

with respect

to a h o m o m o r p h i s m

s u b g r o u p of G .

~ ~ G

T

The composed

> Aut X

T: G

homomorphism

~ Aut X

g r o u p of t r a n s f o r m a t i o n s

@ t of X .

Indeed,

the

-85-

IR x X

>GxX

(t,x)

~

>X

(a t , x )

---~>

Ta ( x )

= ~t(x)

t is differentiable. DEFINITION a

5.5.1.

The Killin~ vectorfield

G s G is the v e c t o r f i e l d

formations

As already

the 1-parameter ]R

subgroup

induced by the 1-parameter

> Aut X

observed

indicated above.

of A u t X .

Then ~t

The differential

the relation between

~t

interpreting course,

A* only a l o n g the c u r v e

transformations

5.5.2.

If

Tt of X ,

T: IR

one would like to ~t

as a 1-parameter

difficulties,

one p r o c e e d s

equation =

A*

t (x)

~0t a n d A* c a n h e u r i s t i c a l l y b e

=

now A* as a vectorfield

Example

111% : IR

g r o u p of t r a n s -

would define a vectorfield

As this presents

~t (x) describing

in s e c t i o n 5 . 4 ,

g r o u p of t r a n s f o r m a t i o n s

on A u t X a s e x p l a i n e d i n 5. 4. as

on X d e f i n e d bY

~t"

Remark. consider

A

t ~

A~ t

on A u t X . ~t

This describes,

of

on A u t X .

> Aut X defines a 1-parameter

then the Killing vectorfield

> IR i s j u s t t h e v e c t o r f i e l d

written as

induced by

Tt .

g r o u p of

A* d e f i n e d b y

-86-

PROPOSITION

5 . 5 . 3.

The Killin$ vectorfield vectorfield

on G defined by

B 6 RG characterized

Proof: G induces

The 1-parameter by proposition

by

o

of G '

5.5.4.

> Bij G'

Suppose

a C

Let

5.4.6

vectorfield

p: G

the operation

a : ~

vectorf~eld

=

t

> G'

~t

= La t

(respectively

B on G .

on

the

It i s c h a r a c t e r i z e d

be a homomorphism

of G o n G ' d e f i n e d b y the 1-parameter =

Lp ( a t )

B' on G' characterized

by

B',

=

=

(p o a)(t)~

t=O

the correspondence

7 g

and = L

P (g)

9

g r o u p of t r a n s f o r m a t i o n s

L(p oa)t .

e

Composing

= Be "

5.4. 3 and theorem

s G and consider

defined by

invariant

ao

is the right invar.iant

s

.

Example 7: G

a ~

g r o u p of t r a n s f o r m a t i o n s

analogue for RG) a right invariant by B e = h

o n G by l e f t t r a n s l a t i o n s .

Let G operate

It i n d u c e s t h e r i g h t

p, ao e

a ~v.~> B ' w i t h t h e c a n o n i c a l

we obtain obviously just the homomorphism

R(p) : R G

map

>RG'

RG

> s

defined by

!

p:G

>G Example

. 5.5.5.

Consider a finite dimensional

and the natural representation of G L ( V ) s u b g r o u p of G L ( V )

by lemrna 1-parameter

5 . 4 . 5.

A*

v

Therefore

subgroup

=

O

, where

v t = atv.

the Killing vectorfield

a satisfies

A*

V

--

h v, O

V

Let a be a l - p a r a m e t e r

T h e n the Killing vectorfield A *

and v G V .

defined by a satisfies

in V.

IR -vectorspace

i.e.

on V

But ~'t = ~ttv = &oat v

A* defined by the is the vectorfield

-87-

canonically defined by the e n d o m o r p h i s m We now apply the results

of s e c t i o n

~

O

C

s

5. 3 t o K i l l i n g v e c t o r f i e l d s

and

prove

PROPOSITION homomor}~hism

5.5.6. 7: G

> Aut X ,

A # the Killing vectorfield vectorfield

on X ,

[A* , c ] x

generated

a a 1-parameter subgroup of G

on X d e f i n e d b y

a .

If C is

and

an arbitrary

then

: tL-i>m0

( T o a)t

Proof:

Let X be a G-manifold with respect to a

1 cx

is the 1-parameter

The formula

by A ~.

for

t)~C)x ]

-

group

is therefore

x C X .

of t r a n s f o r m a t i o n s

a particular

case

of X

of p r o p o s i t i o n

5.3.1. COROLLARY the corresponding field on G,

5.5. 3

right invariant

vectorfield.

=

and B

~ RG

If C is an arbitrary vector-

1

Lira ? [ C g t-->0

Let G

- ((L%),C)g]

operate on G

Now

for

by left translations.

B is then the Killing vectorfield defined by

to this operation.

a ~

g C G

.

B y proposition s G

with respect

w e are in the situation of proposition 5.5.6. i

Note that in particular for C C R G in R G

s

then

[ B , C] g

Proof:

Let G be a Lie group, a ~

5.5.7.

with the aid of L ~

this formula expresses the bracket

. Of course, w e have a similar formula for left t

invariant vectorfields.

We

deduce the following interesting formula:

-88PROPOSITION

a G

5.5.8.

Let G be a Lie ~roup and A, C 6 LG.

If

s G i s the 1-parameter subgroup defined by A, then

[A, Ad : G

where

Proof:

C]

a_

/

dt {Ad( at)}-LL= 0 C

=

> Aut LG denotes the adjoint representation A s in corollary

5.5.7,

we first

obtain

for

of G .

g C G

1

[A, C]g N o w C 6; L G

=

and therefore

t--->0Lim ~-[Cg - ((Rat) ,C)g]

(Latl),c

(R at),C

= C,

i.e.

= Ad(atl)c

This shows

[A, C]g

- dtd{Ad(atl) Cg}/t

-

=0

which can be written

[ -A, C]

= d (Ad(~l) Cg}/ g

But the subgroup

t ------>a

showing thus the desired We have supposed groups.

-I t

dt = a

result. Ad : G

-t

. t=0

corresponds

to the vectorfield

-A6

| > Aut LG to be a homomorphism

This follows from the continuity

of A d

(see section

6. 3).

of L i e

LG,

-89

5. 6.

The homomorphism The knowledge

~: R G

-

> DE

of t h e f o l l o w i n g

for a G - m a n i f o l d

two sections

X.

is not necessary

for the

understanding of the subsequent developments. We shall morphism

show that an operation

v: R G

> DE

T: G

> Aut X defines

of Lie algebras.

a homo-

First w e prove !

LEMMA p :G

5.6. I.

Let X

> G ' a homomorphism

Consider

a C

A*, A'*

defined

Proof:

and

!

s

a = p o a E

by

Let

a,

a '.

Then

~0: X s

by

> X'

a

p-equivariance.

and the Killing vectorfields

A #

and

I

defined

X' a G - m a n i f o l d ,

A'*

are

g~-related.

%D q] be the l - p a r a m e t e r groups of transformations t' t

l

of X , X pr

be a G-manifold,

a,

a

I

: q~t = "rat ' ~ t

!

=

Td t "

It is sufficient to

OF e !

~o

~

=

~

t

o~

t

in v i e w of proposition 5. Z. i. Now

the

p -equivariance of r

signifies the c o m m u t a t i v i t y of the

diagram Gx

X

T

[ G' xX '

As

al =

>X

I w'

> X'

9 o a, the following d i a g r a m is also c o m m u t a t i v e :

-90 -

axl IRxX

x

> G x X

i

I Xf -

IRx

Composing these diagrams,

a Xlx,

~

~ . x~

I

diagram

>X

I

which proves

xX

we obtain the commutative

IRxX

I X'

]Rx

I

>G

~I"

) X'

= x~' t ~ ~0 " |

t

We a r e now in the 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 THEOREM Lie algebra

L e t G be a L i e g r o u p ,

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

of v e c t o r f i e l d s G-manifold

5 . 6 . Z.

on X .

An operation

a~

s G

Let B ~ RG,

DX

9

subgroup defined by

at = B a t '

on X d e f i n e d b y a .

er(B) 6 D X .

u n d e r t h e e f f e c t of a m a p

f o l l o w b y l e m m a 4. 4. Z.

>

the 1-parameter

o-(B) is t h e K i l l i n g v e c t o r f i e l d

p-related

> A u t X d e f i n i n g X a_s_s

induces a homomorphism

If B ~ RG a n d

Proof:

RG the

on G, a n d D X t h e L i e a l g e b r a

v : G

v : RG

then

X a manifold,

p : G

We s h o w t h a t > X.

B and

The theorem

~B)

are

will then

-91-

Choose

x

6 X a n d define

p :G

G

=

> X by

p(g) = V g ( X o ) .

Then

0

the diagram hg

>G

I

pl

X

is commutative. this means

Considering

g

the operation

that p is an equivariance.

Killing vectorfields Killing vectorfield

>

on G a n d

Let

I

X

of G on G b y l e f t t r a n s l a t i o n s ,

Q ~

X are p-related

s

by lemma

on G d e f i n e d b y a i s b y p r o p o s i t i o n

B ~ RG.

But the Killing vectorfield

therefore

B a n d 9 (B) a r e p - r e l a t e d .

KX = 3 r n ~ a n d b y t h e o r e m Example

proposition

5 . 6 . 3.

homomorphi

sm

Example T: G

> Bij

By example of R G '

Let G operate

o-: R G

5.6.4. G'

5.5.4

Let

and the homomorphism

of t h e o r e m

5.6.2.

on G

by left translations.

of G o n

e I

: RG

By

1RG.

defined by

of K i l l i n g v e c t o r f i e l d s

R(p)

X.

is RG and the

> G' b e a homomorphism

the Lie algebra

~(B) a n d

KX i s a L i e a l g e b r a .

> RG the identity

the operation

the element

on the G-manifold

of K i l l i n g v e c t o r f i e l d s

p: G

5.5.3

The

I

5.6.2

5.5. 3 the Lie algebra

5.6.1.

on X d e f i n e d b y a i s j u s t

D e n o t e b y 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 Then

The corresponding

> RG'

and

Tg - L p ( g } . is a subalgebra

the homomorphism

-92-

Example consider and

the natural

Let

V be a finite-dimensional

4. 3. 8 t h a t t h e n

RG = (LG) ~

Now we see from in this c a s e

A ~

5.5.5,

that the homomorphism map

( s (V))

is seen directly

A*

O

> Dr,

----

.~V

*

d A * )(v)A* = (iv 2 v Iv

*

by the same

formula

9

as follows.

T h e n their bracket in (s (V))~ is A z A 1 - A R A I.

[ A 1, A2 ]v

and RG

We know from

But by corollary

s (V) t h e v e c t o r f i e l d

the bracket,

We identify

4.6.8,

R G = ( s (V)) ~ a f t e r i d e n t i f i c a t i o n

is just the canonical

endomorphism conserves

example

isornorphisms.

LG = s

Therefore

]R-vectorspace

of G = G L ( V ) i n V.

representation

IX] w i t h G e b y t h e c a n o n i c a l

proposition have

5.6.5.

we also with G e

~: R G

> KV

assigning

to any

That this map Let

A I, A 2 6 s

O n the other hand

_ (d * )(v) * dvAl v A2 v

as in example

4. 3. 8.

But

d , , ( d-vA2v)lV)Al v

* * = A 2v A i v

= A2AIV

and therefore

[A~, A 2 ]

= (A2A 1

AIA2)v

,

q.e.d.

v

Consider

more

The homomorphism

generally 7 : G

a representation > GL(V) 0

R(v)

> R(GL(V)) = (s (V))

: RG

9 : RG one (s

> DV 0

induces

of t h e L i e g r o u p a homomorphism

and the induced h o m o r n o r p h i s m

is just the composition of this h o m o m o r p h i s m > DV described

G i n V.

before.

with the

-93-

We observe sense

similar

induces

that the general

to that of example

by composition

isomorphism)

CX i s very

of t h i s

map.

RG

> DX

CX,

exactly

dimensional

in CX

a commutative

Lie algebra.

to our question

in general.

Let a commutative

the Lie algebra induced

ve homomorphism,

of o p e r a t o r s

by the representation

on a G-manifold

vectorfields

G on X.

Lie group

The trouble

we can view the induced

G operate

[ 13 ]

the particular for a positive

on X.

the Lie algebra

to ask if any finite X can be

corresponding

We discuss

Palais

is by definition

on a manifold

of Killing vectorfields

See R.

even an

of t h e d i f f e r e n t i a b i l i t y

It is natural

of complete

Lie group

> Aut X

5.6.5.

vectorfields.

as the Lie algebra

(it's

of G i n C X .

speak

of K i l l i n g v e c t o r f i e l d s

Lie algebra

is a surjecti

just

X is in some

T: G

homomorphism

> OX into

of R G

of c o m p l e t e

of s o m e

The operation

about this,

as in example

The Lie algebra a Lie algebra

~-

for a G-manifold

> Aut CX

big and one cannot

on X as the representation

operation

G

But if we don't care

homomorphism

interpreted

5.6.5.

with the natural

a representation

is that

of G i n

situation

KX

As

T : RG

to the case

of

answer

> KX

of K i l l i n g v e c t o r f i e l d s

is c o m m u t a t i v e . We

prove a converse

PROPOSITION

5.6.6.

Let K

be a finite dimensional,

Lie alge_bra of c o m p l e t e vectorfields on X .

Then

there

commutative

is an operation

-94-

of t h e a d d i t i v e g r o u p of K on X , algebra

of K i l l i n g v e c t o r f i e l d s

Proof:

Consider

such thatthe

K.

Then we

set

7A

on K ,

We d e f i n e a n o p e r a t i o n

L e t A ~ K a n d ~0 t t h e 1 - p a r a m e t e r by A.

K is the Lie

of t h i s o p e r a t i o n .

the additive structure

group with Lie algebra

Lie algebra

= ~0 ~ ,

m a k i n g it a L i e of K on X a s f o l l o w s .

g r o u p of t r a n s f o r m a t i o n s

defining thus a map

of X g e n e r a t e d

v: K - - - > A u t

X.

We

show 7 to be a homomorphism. L e t A, B ~ K g e n e r a t e

5. 3. 6, X t = ~ t by A + B.

" ~t

~t'

~t respectively.

is the 1-parameter

Therefore

TA+ B

= X1

Then by proposition

generated

g r o u p of t r a n s f o r m a t i o n s

= ~1

9 ~1

=

vA o 7 B , a s w a s t o

be p r o v e d . We observe corresponding vectorfield

that

7tA

= ~Ot .

to A is t ~ - - - > tA and t h e r e f o r e

A $ on X

A* x

=

d {

d-i

the corresponding

Killing

TtA (x)

}/

d (x) dt ~Pt

=

"t = 0

of t h e o r e m

=

A

=0

9

x

5.6. Z is just the identity

This finishes the proof.

The homomorphism theorem

s u b g r o u p of K

satisfies

This shows that the homomorphism in this case.

Now the 1-parameter

5.6. Z reflects

~: R G particular

> DX d e f i n e d f o r a G - m a n i f o l d properties

of t h e o p e r a t i o n

7 : G

Namely PROPOSITION G - m a n i f o l d and

5.6.7.

~: R G

Let

7: G

> Aut X define X as a

> DX be the i n d u c e d homomorphism.

X

in

> Aut X.

-95-

(i)

If ~" i s i n j e c t i v e

(ii)

If ~

is a free

everywhere Proof:

Let

subgroup.

(i)

operation,

zero

and

(~'B)~t(x)

then ~

r

is injective.

is either

zero.

6 ~G

= ~%(x) ,

the corresponding

where

t~t

=

1-parameter

T'0{t "

D

G'B = O .

q" b e i n g i n j e c t i v e ,

operation,

then a Killing vectorfield

or nowhere

B ~ RG

Then

Suppose

(i. e . a n e f f e c t i v e

Then

T'~t =

1

X

~t(x)

= O and

implies

r t = e

~t(x) and

= x

for every

Be =

~

~o(X)

= 0

O

x and t.

= O,

i.e.

B=O. (ii)

Suppose

for every qt(x)

(~B)

= 0 for some

X

t (x fixed)

= (s

for some

because with

x implies

~

that a free

operation

equivalent

statements.

t

= x .

is injective,

of %" : R G

Example

5.6.8.

A homomorphism

9" is j u s t

G I

R{~) 9 RG

being injective. with injective

(see example

> RG'.

It is sufficient

~,

: Ge

R(~)

>

IR/Z

= lr

"~t

0" B = O

(X)

----" X

(Remember

~" B = O .

are

> DX does not imply the

~ : G 5.6.4)

> G

!

can be injective

> G'e " T h e c a n o n i c a l

is suchacase.

induces

and the induced without

to exhibit a non-injective

e

IR

= x

)

> Aut X.

of G on

B = O and

~t(x)

of

~t(x)=

s o b y (i) B = O a n d

T : G

o ~

and

of t h e s o l u t i o n

'l" b e i n g f r e e ,

= e and as before

of

"r = L

Then

of t h e u n i q u e n e s s

~o(X)

Note that the injectivity injectivity

x ~ X.

an operation homomorphism T = L

~ : G

homomorphism

o

> G

-96-

Remark.

L e t us d i s c u s s

p o i n t of v i e w a l r e a d y Lie group.

the results

mentioned

several

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

times,

This is an effective operation

considering

Aut X as a

on X a n d d e f i n e s t h e r e f o r e

by

~w

5.6. 7 an injection

R ( A u t X)

~>

DX.

i s 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 a Lie algebra,

which destroys

this algebra to be R(Aut X). problem,

every vectorfield

It i s n a t u r a l t o t h i n k t h a t t h e i m a g e

X is c o m p a c t ,

being complete.

c a n be t h o u g h t t o d e f i n e a h o m o m o r p h i s m the homomorphism =

~: R G

one would decrete

t h e r e is h o w e v e r no

Now any operation

R ( v ) : RG

> DX of t h e o r e m

T: G

> R(Aut X).

> Aut X

Then

5.6. Z is just the composition

r 1 7 6R(T). E x e r c i s e 5.6.9.

manifold X

and K X

Let G

be a c o m m u t a t i v e

group operating on the

be the Lie algebra of Killing vectorfields on X .

that every element of K X

5. 7.

Otherwise

many hopes. In case

But this s e t is not n e c e s s a r i l y

on X .

is invariant under the action of G

on

Show

KX.

Killin~ vectorfields and equivariant m a p s . We

shall study the compatibility with equivariant m a p s

homomorphism

~: R G

> DX

defined in section 5.6 for a left operation.

It is clear that considering a right operation of G similarly a h o m o m o r p h i s m

of the

0-: L G

on X ,

one obtains

> DX.

W e prove first the f

L E M I V ~ A 5. 7. I.

Let X, X' be manifolds,

{P: X

> X

a map

A, A I' , A, A z' pairs of X' a__ p-equivariant m a p . I

: LG If either unique

X ' a G ,~ - m a n i f o l d

> Aut X ,

7': G ' ~

> KX

e...->DX,

G

operates

map

~/: K X

9

effectively > KX'

L(P)

!

commutative

> KX

I "y[ ~t I

and this m a p

is a h o m o m o r p h i s m

to a

and

I

~ D X

on X o1" q~ i s s u r j e c t i v e ,

I

LG

with respect

C o n s i d e r the induced h o m o m o r p h i s m s > KX

(5"

to a

> G' a h o m o m o r p h i s m

p : G

I

: LG

making

LG

with respect

I

> KX

of Lie algebras.

of t h e o r e m there

the diagram

5 . 6 . Z.

is a

-98-

Proof: A*

=

~(A)

Let A G LG. and

oJ(L(p)A) are

Suppose first that G 7: G ~

> Aut X

injective.

For

9 (A) = A * .

We

By proposition

operates effettively on X ,

6 KX

define

~'(L(p)A)

defined by A ~ .

are

there is therefore a unique A C L G y(A*)

=

and

COROLLARY l_f ~ : X

> X' is a

=

~(A)

of ~ : L G

> KX

L e t t h e s i t u a t i o n be a s i n p r o p o s i t i o n

5 . 7 . Z.

by lemma

5. 7. 3.

A*

as before.

of y f o l l o w s f r o m t h e s u r j e c t i • i t 7

Y is a h o m o m o r p h i s m

As

6 IX3.

5.7. I, o"(L(p)A) is uniquely

can therefore define N

The uniqueness

with

~'(L(p)A).

(P-related, by l e m m a

We

i.e. that

B y proposition 5.6. 7, o" is then also

Suppose n o w that (~ is surjective and let A and

the vectorfields

~0-related.

is injective.

A*

5.6.1,

p -equivariant

4 . 4 . Z, q. e. d.

diffeomorphism,

then the followin~

d i a g r a m c o m m u t e s.

Proof:

>

KX g

> DX

LG'

>

KX'~

> DX' are

O-related.

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

7g_ I : X

~,/KX

=

If C

LG

~ DX,

then C

and

r

Therefore

Y , q.e.d.

by a right-operation

T: G ~

> Aut X .

First w e r e m a r k

> X defined

-99 -

LEMMA

5. 7 . 4 .

Proof:

The diagram

Tg_l : X---> X

i_~s g - equivariant. g

T -I g L) X

X

I

I iT T

X

commutes

for

Y6

By corollary

G

operation

homomorphism

Tg

g(Y)

5.7.5.

v : G~

Let

vg

yg-1

OT

g-1

X be a G~

5 . 6 . Z.

LG

=

~

if: IX]

Then

> KX

the following

>KX

e"

> DX the induced diagram

(

remember

a g defines

commutes.

r

LG

=

to a

> DX

f

(T g -1),

.rg_l o "ry

=

with respect

Adg

We just

"r.yg-1

=

we have

> Aut X and

of t h e o r e m

X

>

~ T g -I

5. 7. 3 t h e r e f o r e

PROPOSITION right

:

-1

g

that DX

~

> KX

Adg

= L(gg).

as G-Lie

algebra,


DX

Further

it is clear

v being

a right

that

operation.

We o b t a i n t h e r e f o r e THEOREM

5. 7 . 6 .

X be a G~

homomorphi

sm

--of G

and the operation of G __~ D X

__in L G

T :G ~

Let

> Aut X .

T h e n the induced h o m o m o r p h i s m

with respect

to a

C o n s i d e r the adjoint representation_

~: L G

defined by > DX

ag = ( Tg_l), 9

is an equivariance.

-I00

For

A 6

-

LG and the corresponding

A$ =

~(A) w e h a v e t h e r e f o r e

the formula

(V g _ l ) , A * This shows in particular

that

T

g

~(Ad g A)

=

transforms

.

Killing vectorfields

in

Killing vectorfields. Example commutativity

5. 7. 7.

Let G operate

on G

by r i g h t t r a n s l a t i o n s .

The

of t h e d i a g r a m

> DG

LG

I

expressed

Adg I

~

LG

> DG

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

For effective operations, The commutative

(Rg - 1 ) ,

4. 7. 3,

9 is an injection by proposition

5.6.7.

diagram LG

0"

> DX

P Adg I LG shows that

a: G

> DX

> A u t DX c a n b e i n t e r p r e t e d

adjoint representation L E M M A 5. 7. 8.

0"

as an extension

of t h e

of G . The homomorphism

if a n d o n l y if t h e h o m o m o r p h i s m

a: G

7:

G ~

> A u t X is i n j e c t i v e

> Aut DX defined by

-I01-

(1

g

(Vg_l) * is injective. Proof:

a is the composition G

I

> G

g ~........~>

T

#

> Aut X

g-1 ~

r

g

> A u t DX

-1 ~ ' ~ ~

(V-l), g

I is b i j e c t i v e . * is i n j e c t i v e , and

= 1x .

because

Therefore

For

a commutative

= 1DX i m p l i e s

~0,x = 1Tx(X }

a is i n j e c t i v e if a n d only if * o v i s i n j e c t i v e if

a n d o n l y if "r i s i n j e c t i v e , Remark.

r C Aut X with ~,

q. e. d.

a left-operation

T: G

> Aut X we have similarly

diagram

RG

v

f

> DX

f cr

> DX

RG where

g = (T), g g

(left) operations

and ~ is t h e r e f o r e of G on RG a n d D X .

an equivariance

(see remark

defining a homomorphism

a t t h e e n d of s e c t i o n 5 . 6 ) .

we have a commutative

diagram

to the

L e t u s 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 a t A u t X a s a L i e g r o u p . f r o m the left on X ,

with respect

Aut X operates : R ( A u t X)

As just observed,

naturally > DX

f o r ~0 C A u t X

-102-

O"

R(Aut X)

> DX

r

t

R%tt

. ~

O"

R ( A u t X) being an injection,

we see that

> DX

(p~ : DX - - ~

R(%) , which is the adjoint representation argument sending

i n t h e p r o o f of l e m m a ~ to

~,,

is injective.

r being an injection.

5.7.8

DX is an extension

of A u t X i n R ( A u t X ) .

we see that Aut X

Thin is in accordance

of By the

> Rut DX,

with lemma

5. 7. 8,

-103-

Chapter

6.

THE EXPONENTIAL

MAP OF A LIE GROUP

The relation between 1-parameter and invariant vectorfields a map

6. l,

G

G

s t u d i e d i n s e c t i o n 5 . 4 i s u s e d to d e f i n e

e x p : 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 .

D e f i n i t i o n a n d n a t u r a l i t ~ r of e x p .

we identify e

on

s u b g r o u p s of a L i e g r o u p

LG

with

G

e

In t h e f o l l o w i n g , f o r c o n v e n i e n c e ,

and write A

E

for a tangent vector

Ge

at

. D E F I N I T I O N 6.1. 1,

exp:G e-*G

T h e exponential m a p

is the

map defined by e x p A -- a 1 where

a

is the 1-parameter

to theorem

5.4.6.

Let clearly

and define for a

and moreover

LEMMA 6.1.2. Proof:

s u b g r o u p of

With our convention

a E s

~ ~ s

for

For

A

t

~ Ge

d e f i n e d by

A

according

"

Then

= h0 ,

~s = a s t

we have

~0 = th0

f ~ CG

G

A

"

we h a v e

d f(l~s) [ ~0f = " ~ s=0

= ~sf(Ctst) [ s=0

= th0f

'

This s h o w s that exp(tA) = a t .

PROPOSITION

Proof:

a

6. I. 3.

exp((tl+tz)A ) -

is a h o m o m o r p h i s m .

|

exp(tlA) 9 exp(tzA)

q, e. d.

-104-

The proposition [A,B]

= 0

5.4.10

shows that for

a n d in p a r t i c u l a r

e

with

= exp(tA)"

exp(tB)

for t = 1

exp(A+B) PROPOSITION commutative,

then

vectorgroup

Ge

= exp A.

6 . 1 . 4.

e x p : G e -~ G into

G

Example

6 . 1 . 5.

and

G - GL(V) .

LG

with

exp B

,

showing

If t h e L i e a l g e b r a

LG

is a homomorphism

of

G

i.~s

of the additive

.

To justify the notation

~s

~ G

we h a v e t h e f o r m u l a exp(t(A+B))

a

A, B

Let

exp , let us consider

V

be a finite dimensional

In proposition

G e , we h a v e

LG

the corresponding

the

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

= s

Now let

1-parameter

this case

~-vectorspace

A

~ s

subgroup,

and

We s h o w t h a t in

(2O

exp(tA)

= a t = eta

= Z

n"~', (tA)l

n

nffi0

To prove this,

consider

~t = XOO n (tA)n-lA n=0 n-~. and

a 0 = iV ,

~t = e t a

Therefore

ential equation with the same C onsider multiplicative algebra

of

and

= ~t A

in p a r t i c u l a r

a

is

~t

~0

= 1V "

Then

But also

ht = ata0=at A

= ~ , as both satisfy the same

differ-

i n i t i a l c o n d i t i o n , q. e. d. V -" ~

group of real numbers ~*

= Zoo n--0 ~1 ( tA )n

.

Then

different

GL(V)

from

with the (only possible)

zero,

trivial

" R*

, the

The Lie Lie algebra

-105-

structure.

The map

exp:~

-~ l~*

is just the ordinary

exponential

map.

We now s h o w the n a t u r a l i t y

P R O P O S I T I O N 6 . 1 . 6. Then the following diagram

of

Let

exp,

i.e.

p: G -~ G '

be a local homomorphism.

( t a k e n in the 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

Proof:

Let

A

P

~ Ge

.

If

> G'

at

= exp(tA)

is the corresponding

!

1-parameter

s u b g r o u p of

G

, then

(p 9 a)(t)" It = 0 = p*

e

~0 = p~'

e

A

"

Therefore e x p ( t p , eA ) "- p ( e x p ( t A ) ) and for

t = 1 exp(p, A) e As an application, COROLLARY

= p(expA)

of

LG

with

Adg Ge

q.e.d.

we o b t a i n

6.1.7.

For

exp(Adg A)

Proof:

,

= L(~g)

g

E G

,

= g exp Ag

by definition.

the adjoint representation

A -1

we have

E LG .

Note that after identification operates

in

Ge

.

-106A 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 Consider a finite-dimensional of

GL(V)

Ft-vectorspace

i s by p r o p o s i t i o n 4 . 3 . 8

morphism

det: G L( V ) -~ l~*

algebra homomorphism

e q u a l to

is th e f o l l o w i n g . V .

s

.

The Lie algebra Now th e h o m o -

i n d u c e s by p r o p o s i t i o n 4 . 5 . 1 1 t h e L i e

t r : ~ ( V ) -~ ~t .

T h e n a t u r a l i t y of the e x p o n e n t i a l

mapping proves C O R O L L A R Y 6.1. 8.

For any

det e x p A

T h e i m a g e of t h e m a p

A

-- e x p t r A

e x p : G e -~ G

c o n n e c t e d c o m p o n e n t of t h e i d e n t i t y of shows that

exp

quadratic matrices

Let

SL (2 , •)

with d e t e r m i n a n t

We s h o w t h a t t h e r e i s a n e l e m e n t in This will imply that

g

.

algebra, obtain

exp

Now g

2

t r g2

The following example

det g = 1

(tr

2

-

1 .

G

.

2-rowed

It i s a c o n n e c t e d L i e g r o u p .

SL(2, ~t) w h i c h is n o t a s q u a r e 9

2

where

and therefore,

g+;Id g)

be th e g r o u p of

and c o n s i d e r its c h a r a c t e r i s t i c

X t r g + det g

- tr g. =

.

G o , th e

is not s u r j e c t i v e .

g E SL(2, R )

d e t ( X 3 " d - g) = X2 of

G

is c o n t a i n e d in

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

E x a m p l e 6.1. 9.

Let

~ s

-- 0 9 >

-

2

Z

tr g

polynomial

d e n o t e s th e

trace

by a t h e o r e m of l i n e a r

A p p l y i n g t h e t r a c e f u n c t i o n , we

D

C o n s i d e r the e l e m e n t L

=

of

SL(2, i t )

.

-107As

tr s < -2 , the equation

Remark.

Consider

transformations C onsidering

r

of

g2

~-

a manifold

X

generated

as a 1-parameter

(Pt

has no solution. X

and a 1-parameter

by a vectorfield

subgroup

of

A*

group of on

X

.

A u t X , it i s s u g g e s t i v e

to write as for Lie group r View now, as before,

= exp tA*

an operation G

and let

a

~ s

and

of

T

G

on

X

as a homomorphism

>Aut X

A'be the Killingvectorfield

defined by

a

.

Then by definition ~- = e x p t A * at o r if

A ~ Ge

with

at -

exptA

T e x p tA This expresses

= exp tA*

just the commutativity

of the diagram

G

:> KX

o-

e

ex~

I

G

where

6: G e -*KXa-.DX

G -~ A u t X the complete exp

.

is therefore,

> Aut X

is the homomorphism

In t h e u p p e r vectorfields

r

right,

exp

induced by the operation

we c a n n o t w r i t e DX , b e c a u s e

are sent into

even in this ease,

Aut X

a natural

only

by the exponential transformation

map.

(of suitably

-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

P ROPOSITION 6.2.1.

induced by

exp: G e -. G

Proof: For

exp~ 0 A

A

at t h e i d e n t i t y .

We show now

The tangent linear map

exp,, 0: G e -~ G e

.is the identity map. ~ G

we have

e

= exP, tAA It=O =

{exP*tn d (tA)} t-0

=-gtd

exp(tA) i

-" A

,

q.e,d,

t ;0

B y the inverse function theorem we therefore have THEOREM in

Ge

6.2.2.

There is an openneighborhood

and an open ..neighborhood

exp: N O -.N e

defines a chart of

G

log:N e-.N O at

e

D E F I N I T I O N 6. Z. 3.

morphism exp/N 0 .

of

e

in

G

such that

.is an analytic diffeomorphism.

We denote by

(N e, log)

Ne

N O of

map,

The map log

. A c a n o n i c a l c h a r t of

of an open neighborhood log: N e - . l o g ( N e )

the inverse

-- N O ~

Ne Ge

of

e

in

G G

is a pair and a diffeo-

which is an inverse of

O

-109-

A n 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 f o l l o w i n g

P R O P O S I T I O N 6. Z. 4,

Let

G

c o n n e c t e d c o m p o n e n t of t h e identity,. GO

If

LG

GO

the

is c o m m u t a t i v e , t h e n

is c o m m u t a t i v e . Proof:

of

be a L i e ~ r o u p a n d

e

in

T h e i m a g e of

G .

exp

c o n t a i n s an o p e n n e i g h b o r h o o d

A n y two e l e m e n t s in

U

commute, as

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 . so that a n y two e l e m e n t s of

GO

But

U

U

exp:G e-.G

generates

GO ,

commute,

T o g e t h e r with c o r o l l a r y 4 . 6 , 9 we h a v e t h e r e f o r e

THEOREM 6.2.5. G

Let

G

is c o m m u t a t i v e if and o n l y if

LEMMA 6 . 2 . 6 , L(p): LG -~ LG'

LG

Then

ks c o m m u t a t i v e .

p : G -~ G'

be a h o m o m o r p h i s m .

is i n j e c t i v e ( s u r j e c t i v e ) if a n d o n l y if

( s u r j e c t i v e ) f o r every, Proof:

Let

be a c o n n e c t e d L i e ~ r o u p .

p•g

is i n j e c t i v e

g ~ G ,

p(g~/) = p(g)p(~/)

implies s

~/--e

p,

o

(Lg),

lg (Lp(g))*e'

P~'e

and

psg = ( L p ( g ) ) . e , ~ p - e ~

PROPOSITION 6.2.7, Lie g r o u p .

Then

Proof: exp: LG -. G

Let

exp: L G - . G

G

(Lg~e

e ,

be a c o m m u t a t i v e c o n n e c t e d

is s u r j e e t i v e .

We h a v e s e e n i n 6.1. 4 that f o r c o m m u t a t i v e is a h o m o m o r p h i s m .

q.e.d.

Let

G'

be the i m a g e ,

LG Now

-II0exp, 0

is the identity map and therefore

isomorphism

for every

(and closed) subgroup Consider

(N e, log} exp

A of

E LG

G

.

e x p , A 9 b y 6 , Z.6, an

Therefore

, i.e.

G'

a local homomorphism

be a canonical chart

, we have for

g @ N

(*)

.

= G

.

p:G

-.

is an open

G'

.

Let

By the naturality

6 , 1 . 6 of

e

p(g)

This necessary

of G

G'

= e x p ( L ( p ) i o g g)

condition determines

p/N e

by

L(p)

and has the

following applications,

PROPOSITION homomorphisms,

Proof: of

G .

(i : 1, 2) U

of

Take

Then(*)

Pi: G - G' (i = I, 2)

L: s

coincide, in

Pi: G -~ G' (i : 1, 3)

then there exists an open Pl

and

P2

coincide.

a s t h e d o m a i n of a c a n o n i c a l c h a r t

pl(g)

= p2(g)

Let

G

of 6 . 2 . 8

of

for

g E U

If e

L(Pl) : L(p2) { t h e n

generates

is expressed

Pl : PZ

o.{

by saying that the

i s f a i t h f u l on t h e s u b c a t e g o r y

groups and global homomorphisms.

.

be connected and

homomorphisms,

Any neighborhood

s

be local

homomorphisms

G , on w h i c h

U = Ne

6, 2, 9.

This corollary functor

e

shows

COROLLARY

Proof:

Let

If t h e i n d u c e d a l g e b r a

L ( P i ) : L G -. L G ' neighborhood

6.2.8.

of c o n n e c t e d

Lie

-lll-

ker Ad

L(~g)

An application

of this last result

PROPOSITION

6.2.10.

-

ZG,

where

Let

ZG

Proof:

By definition

--- 1LG

implies

~g

is the

G

be connected.

is the center

of

Ad

,

=

= 1G

L o ~ .

G

Then

.

But as seen before,

Therefore

ker Ad

-- k e r ~" -- Z G ,

q.e.d. Now consider morphism

G --G'

LG --LG'

.

the problem

of constructing

a local homo-

inducing a given Lie algebra

homomorphism

Remember

that the isomorphism

not induced by any homomorphism the existence

h: L G -* L G '

6.2.11,

a Lie algebra

a canonical

chart at

e

local homomorphism

Let

with the desired

G, G'

G

G - - G'

.

Let

(N e , l o g )

The restriction

inducing

property.

be L i e g r o u p s . . a n d

homomorphism.

in

is

~ -- R , s o w e o n l y c a n e x p e c t

of a local homomorphism

PROPOSITION

L qr -~ L ~

h

to

N

e

be of a

is necessaril~r

of the

form p = exp~ h olog:N e If

LG

and

this formula

homomorphism

that

are commutative,

is a local homomorphism

Proof:

6.1~ 4

LG'

We have already G --G'

for commutative log

then

Ne

LG

exp

is necessarily

G --G e

of a local

of this form,

is a homomorphism

L(p ): L G -- L G '

defined by

h.

seen that the restriction

to

Let

G'

9: N e -~G'

inducing

is also a local homomorphism

is a homomorphism,

->

.

By

G e -- G , s o Therefore

p

be the induced homomorphism.

-112-

It i s c l e a r

that

P*e"Ge

p:N e -~G'

The rnap h: L G -~ L G '

h

.

of this requires

LG

and

a deeper

6.4,

LG'

method,

given Lie algebra

6, 4 . 2 ,

L(p)

h

even

See also the comments

a local homomorphism

in C h a p t e r

G -~ G'

inducing a

(see 7.2.3).

defined in proposition

p : N e -~ G'

,I

But a direct proof

We shall construct

h: L G -~ L G '

- h

homomorphism

inducing

G -~ G '

of the situation,

homomorphism

this will prove that the

shows

commutative,

analysis

after proposition

7, b y a d i f f e r e n t

This

defined by a Lie algebra

is a local homomorphism

without supposing

in s e c t i o n

is just

-~ G e

By unicity, 6 . 2 , 11 i s a

local homomorphism, Exercise effectively let that

KX

Suppose that a Lie group

on the manifold

X

be the Lie algebra

g ~

o n l y if

6.3,

6.2.12.

G

g

satisfies

with respect

to

1": G -* A u t X

of Killing vectorfields

(Vg).A

i s in t h e c e n t r a l i z e r

= A

operates

G

for every

on A

X

,

and

Show

~ KX

of the identity component

Unicit~ of Lie group structure,

,

if a n d GO

G .

in

We begin by proving the following

important PROPOSITION a homomorphism there

exists

analytic,

Proof: neighborhood

Let

in the algebraic

an

i.e.

6 . 3 , l,

A

E LG,

a 1-parameter Let of

e

(U, l o g ) in

G

G

be a Lie group and

sense,

such that subgroup

which is continuous. a t = e x p tA

of

G

be a canonical with

VV c

a: R -~ G

U

a

is

. chart

,

and hence

Then

of

G

and

V

a

-113-

Let

g

are

defined.

For

t = 1

subgroup as

g

2

~

.

g2

U

. that

g

E > 0

is uniquely

by the

V,

al/z By

= exp(

V

.

log g),

iteration

shows

E V

exp(~

or with

log al

0
0 such that Itl
0

It I < cO

f(gexptA)

= ~

tn --6!-.

n=0

[(+)~ t] f(g exp tA

t -'0

(3O

-tn

.q.e.d.

n=0

We a p p l y t h i s to p r o v e t h e P R O P O S I T I O N 6 . 5 . Z. such that for an Then for (i)

A,B

Let

E > 0 t-~ O(t 3) E LG

O(t 3)

d e n o t e a v e c t o r in

is bounded and analytic for

and s u f f i c m n t l y s m a l l

LG

Itl
0 and f o r (1)

m

is t h e r e f o r e

[AnBmf](e)

t = s f(exp

tA

tn+m exp tB) =

/. n,m >0

[AnBmf](e)

is {[Af](e)+ [Bf](e)}

T h e 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

t2

,

is [l-~.-[A2f](e)+ [ABf](e) +

~[BZf](e)}

On the o t h e r h a n d , by t h e o r e m 6 . 2 . 2 f o r s u f f i c i e n t l y s m a l l e x p t A exp tB = exp Z(t) with

Z: I -~G e , I

a n a l y t i c at

O,

Z(O) Z(t)

for fixed

at

=

= 0 .

E

containing 0,

Z

Then

tZ 1+ t 2 Z z + O(t 3)

Z 2 @ Ge

.

Take any function

f

e o

Z l,

a n open i n t e r v a l of

w h i c h i s l i n e a r in a c a n o n i c a l c h a r t

T h e n it i s a n a l y t i c at

e

and

t

-123-

(z)

f( e x p tA e x p t B ) = f ( e x p { t Z 1 + t Z Z z + O(t3)}) = f ( e x p [ t Z 1 + t Z Z z }) + O' (t 3) (30

= ~= i [(tZI+ tZzz)nf](e) + O'(t 3) 0-~. O' (t)

being a real number

such that for an

is bounded and analytic for T h e c o e f f i c i e n t of is

i

E > 0

t-7

O'

(t)

It[ < E . t

is

t h e c o e f f i c i e n t of

[Zlf](e )

tz

[[Z2f](e) + ~-[Z~f](e)} . Comparing

t h i s w i t h t h e c o e f f i c i e n t of

t

and

tz

in

(1) ,

we obtain

[Zlf](e ) = [(A+B)f}(e) 1 [Zzf](e) = [ ~[A, B]f}(e) This being true for any function canonical

c h a r t at

e

,

f

which is linear in a

we h a v e t h e r e f o r e

Z 1 = A+B 1 Z z = TEA, B] This shows e x p tA e x p tB

t2

= eXp Z(t) = exp [t(A+B) +-,2- [A,B] + O(t3)}

i.e. (i) (ii)

i s o b t a i n e d b y (i) a s f o l l o w s

-124e x p tA exp tB e x p ( - t A )

t [ A , B ] + O(tZ)}l ) e x p ( - t A ) = exp(t[(A + B ) + ~-tz

= exp(t([ }l-A)+--2-[[ ]I' -A]+ O(t3)) t2 t2 = exp ((tB + - T [ A , B]) + -~--[A, B] + O(t3)) exp (tB + t2[A, B ] + O(t3)) (iii) is s h o w n s i m i l a r l y by e x p ( - t A ) e x p ( - t B ) e x p tA exp tB = exp (t{-(A+B) +

,B] +O(t 2

,

exp(t[(A+B) + ~[A, B] + O(t2)]l ) t2 -

exp(t([

]2 + [ } 1 ) + 2 --[[ }2"[}1 ] + O ( t 3 :

= exp (t2[A, B] + O(t3)) Remark.

Let

be a n o p e n n e i g h b o r h o o d of

NO

e x p / N 0 : N O -. N

such that the r e s t r i c t i o n

e

O

,

in

G

q.e.d.

e

is a d i f f e o m o r p h i s m .

Then

one c a n d e f i n e a c o m p o s i t i o n A o B = log(expA if NO

e x p A 9 exp B for which

composition,

O exp

N

e

9 e x p B) 9

is an identity.

and sufficiently small

~ NO

In f a c t , by the v e r y d e f i n i t i o n of t h i s

is an i s o m o r p h i s m

of p r o p o s i t i o n 6 . 5 . 2 .

A,B

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

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

tA o tB

f or

of

G .

NO

with

equipped

Now look at the f o r m u l a (i)

It c a n be r e w r i t t e n (for a r b i t r a r y t ) as

= (tA + tB)+ 1-~-[tA, tB] + O(t 3) I-

Ne

.

A, B g

LG

-125-

The fundamental fact can be proved that (for sufficiently small t) the term

LG

on

A, B .

This means that the composition law in the neighborhood Ne

of

e

G

O(t 3) also is expressable by operations in

is completely determined by the Lie algebra

of 6.5.2

just gives the first two terms Moreover

h: L G -~ L G ' NO .

This incidentally

by

h: L G -~ L G '

homomorphism

according G -~ G'

with respect

to proposition

We a p p l y p r o p o s i t i o n

homomorphism

to the composition

shows that the map

inducing

The formula (i)

of this development.

one can show that a Lie algebra

is a homomorphism

in

LG.

6.2.9

h: L G -~ L G '

p : N e -~G'

defined

determined

i s in f a c t a l o c a l ,

6, 5 . 2 t o p r o v e

PROPOSITION 6.5.3.

Let

A, B

~

LG .

Then the following

conditions are equivalent

(i)

[A,B]

= 0

(hi) e x p sA e x p t B

=

( i i i ) e x p tA e x p t B

= e x p t B e x p tA

P roof:

exptB

e x p sA

for every for every

(i) --->(ii) by proposition 5.4. II.

s

and t

t

.

(ii) ---->(iii) is trivial.

To see (iii) --->(i) observe that (iii) implies by proposition 6.5, 2 exp [t(A+B)+

t2 -~[A,B]+

for sufficiently small [A,B]

=

0

O(t3)]

t .

t2 = exp {t(B+A)+"z-[B'A]+

This implies

O(t3)]

[A, B] = [B,A] and

.

The condition (i) and (iii) of proposition 6.5, Z also imply the following convenient formulas.

in

-IZ6COROLLARY 6.5.4. 6.5.Z,

U n d e r the c o n d i t i o n s of p r o p o s i t i o n

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)

}

;exptA

e x p t B e x p O ( t z)

(it) exp [ t Z ~ , B ] ] - exp (-tA) exp ( - t B ) exp tA exp tB exp O(t 3)

.

P r o o f : (i) f o l l o w s f r o m exp (-tA) exp ( - t B ) exp t(A + B) = exp ( t [ - ( A + B) + ~ [ ' A , B ] + O(t2)]) exp t(A + B) t2 = exp (t([ } + ( A + B ) ) + -.2-[{ } , A + B ] + = exp

(ii)

t 2(T[A, B] +

O(t3))

O(t3))

.

follows from

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) . T h e f o r m u l a (i) s h o w s t h a t the c u r v e the s a m e t a n g e n t v e c t o r at t ~-~-~>exp t(A + B) . O(t 3)

is v a n i s h i n g f o r

e

F r o m p r o p o s i t i o n 5 . 4 . 1 0 it f o l l o w s t h a t the t e r m A,B

with

[A,B] = 0 o

t ,-,~> exp (-~/'tA) exp ( - ~ B )

a s the t a n g e n t v e c t o r at 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 f o r l a t e r a p p l i c a t i o n is the f o l l o w i n g

COROLLARY 6.5.5. t E ~t

we h a v e

has

t h a n the 1 - p a r a m e t e r s u b g r o u p

T h e f o r m u l a (ii) d e s c r i b e s [A, BJ of the c u r v e

t --,-~>exp tA exp tB

Let

A, B ~ LG .

Then for any

e

-127(i) e x p t ( A + B )

=

l exp Ht

lim

n

A exp "H B

n~(x)

(ii) exp [t2[A, B] } -

P r o o f : Let 6 . 5 . 2 f o r fixed

t ~ ~

lim

n~co

and

and t h e r e f o r e

t}n

exp ~ A exp ~ B

t h u s s h o w i n g (i) . -

exp

exp(--~tnA) exp(-~B) exp~A exp~B} n2 t t t sufficiently great.

n

By p r o p o s i t i o n

t

exp ~t A exp t B = exp

exp

t

-~

t (A+ B ) +

2nZ[A, B] + O

- exp t(A+ B ) +'2"~ [A, B] + O

To s e e (ii) it s u f f i c e s to o b s e r v e that by 6 . 5 . 2 2 exp~A

lnl

exp~B

t

=

exp n-2 - [ A , B ] +

-

exp [ t 2 [ A , B ] +

2 ~

O(1)} .

-iZ8-

CHAPTER

7.1.

7.

SUBGROUPS AND SUBALGEBRAS

Lie subgroups.

B e f o r e d e f i n i n g t h e n o t i o n of L i e s u b g r o u p s ,

L E M M A 7.1. 1.

Let

we

prove

H,G

in)ective homomorphism.

L:H-~ G

be Lie ~roups and

Then the induced homomorphism

an L(~): LH -~ LG

is injective. Proof: with

Let

s

ai

= 1, Z)

being injective,

Laa 1 = L oa 2 .

the map

s163

be 1 - p a r a m e t e r

-~ s

i n d u c e d by

s h o w s t h e i n j e c t i v i t y of

L(r

,

r

a 1

=

s u b g r o u p s of

a 2

is injective.

H

Therefore

.

B y 5 . 4 . lZ, t h i s

q.e.d.

N o t e t h a t by l e m m a 6 . 2 . 6

every tangent linear map

C * h : H h ~ Gc(h)

is injective. D E F I N I T I O N 7.1, Z. G

isa

Lie subgroup

of

(i)

H

(ii)

the injection

Let

H

Let G

A subgroup

H

of

if

I,: He,.-, G

b e a L i e s u b g r o u p of (H, L)

D E F I N I T I O N 7.1. 3.

and

be a Lie group.

is a Lie group

f o l l o w i n g it, t h e p a i r

G

G

i s a s u b m a n i f o l d of (i)

H

(ii)

the injection

is analytic. G

B y l e m m a 7, 1. 1 a n d t h e r e m a r k

i s a s u b m a n i f o l d of Let

G

G /according

be a manifold.

A subset

to the H

of

G if

is a manifold

L . h : H h -" G~(h)

L:H ~--~ G

is an immersion,

injective for any

h

~

H

i.e. .

t

differentiable

-129-

Let

H

be a subgroup

locally the analytic

structure

that the group operations subgroup

of

G

in

of

G .

of

H

H

are analytic.

Let

H

: He--. G identify

LEMMA exp:H e-*H Proof:

a

be a Lie subgroup

a:lR

-~

G,

G,

is therefore

G

of

map

G .

By

so a Lie

is a Lie subgroup

7 . 1 . 1, t h e i n j e c t i o n

Lit,): L H -~ L G .

We c a n t h e r e f o r e

LG

L(~): LH~--. L G .

of

7.1.4.

be a Lie subgroup

Let

H

is the restriction

of

After the canonical

There

an open neighborhood

and write

exp:G e-~G

G

.

The map

.

identifications,

Let

2 , then

(i H 1

--

1,2)

= H 2

this is just the naturality

e

in

be connected

Lie subgroups

9

is an open neighborhood of

of

map,

7, 1. 5.

LH 1 Proof:

H

with a subalgebra

PROPOSITION

G.

of

is injective.

6.1. 6 of the exponential

o_f

is a submanifold

is induced from that of

subgroup

induces an injective LH

H

.

Note that a 1-parameter if a n d o n l y if

If

of

e

in

H Z (take a canonical

H1 chart

which is also at

e

and

use 7.1.4.). We use without proof the following

LEMMA and

~a-X -* Y

6 : X --, S

7 . 1 . 6.

Let

X,Y

a differentiable

is continuous,

be m a n i f o l d s r

map with

it i s d i f f e r e n t i a b l e .

~p(X) c

S

a submanifold

S

.

of

Y

If t h e i n d u c e d m a p

-130-

L E M M A 7.1. 7. Then

LH

= [A ~

Proof:

A

map

]R-~ H

Let

LG/t~ LH

.

H1

and

e x p tA

implies that

-~ H

AE

LH

PROPOSITION If

be a L i e g r o u p a n d

H2

,

H

a Lie subgroup.

is a continuous map ~-~H} t ~

Suppose conversely

a continuous map and therefore

G

exp t A

A ~

LG

.

is a differentiable

with

t,-,--~> e x p tA

T h i s m a p is d i f f e r e n t i a b l e

by lemma

7, 1.6

.

7.1, 8.

Le__t

H1 , H2

be two Lie subgroups

coincide as topological groups,

of

G ,

they coincide as Lie

groups. Proof: structure

7. l, 7 c h a r a c t e r i z e s

al o n e .

is therefore

By

7.1.5,

H10= H20

"

by a i d of t h e t o p o l o g i c a l

The identity map

H 1--, H z

an isomorphism,

T h i s is of c o u r s e a simple,

the Lie algebra

also a consequence

of 6 . 3 , 5, but we h a v e p r e f e r r e d

direct proof,

We state now THEOREM g r o u p of

G

,

7.1, 9.

of

subgroup

,

Proof: of

LG

G

G

be a Lie grou~.

then the Lie algebra

Each subalgebra of

Let

There

LG

of

H

is the Lie algebra

only remains

there exists a connected

to s h o w ,

If

H

is a subalgebra

is a L i e s u b of

LG

of a unique connected

.

Lie

that for a given subatgebra

Lie subgroup

H

of

G

with

~-~

LH = ~,

-131-

Suppose there exists Moreover

exp ~f

therefore

generates

exp ~

.

H

contains an open neighborhood

This permits by

such a Lie subgroup

. of

conversely

The problem

to d e f i n e

is to make

a submanifold

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

W

on

G

versely W. W

H

be a L i e s u b g r o u p of are the maximal

integral

~c

LG

is t h e n i n v o l u t i v e , e

Let

.

t h a t t h e f i e l d of v e c t o r s p a c e s fore the maximal

N o w if

Lg_IH

= H

gE

t h e f i e l d of v e c t o r s p a c e s ~

g r o u p of

G .

Then

Let X

is a

X

be a

L

1h = e

h" then

,

g~

G-manifold

H-manifold.

G,

so

first observe There-

Lh_l H = H0 Therefore

H ,

and

Suppose

W

among themselves

= H} a n d H is a s u b g r o u p of G , g ~l ~ m a n i f o l d of G, we s e e t h a t H is a L i e s u b g r o u p of 7.1, 10.

m a n i f o l d of

by left translations.

= {g/L

Exercise

being a subalgebra,

integral

are just permuted

G,

G

Now given con-

is a s u b g r o u p of

h C H, t h e n for

.

{Ag/A ~}

is invariant

integral manifolds

by left t r a n s l a t i o n s .

If conversely

W

We

o f t h e f i e l d of v e c t o r s p a c e s

of t h e c o s e t s

H

,

for integral

T h e n t h e l e f t c o s e t s of

be t h e m a x i m a l

To see that

G

(see Chevalley theorem

one can reconstruct

H

of

on a m a n i f o l d .

manifolds

is n a m e l y t h e v e c t o r s p a c e

passing through

a

G .

d e f i n e d by t h e t a n g e n t s p a c e s

a subalgebra

Wg

and

H

1), m a k i n g u s e of t h e e x i s t e n c e

modulo

H

in

as the subgroup generated

[ 3 ] , p. 109, t h e o r e m

H

e

H

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

Let

exp(~) ~ H

H .

do not s h o w t h i s h e r e ,

manifolds

Then

H G

being a sub, H

a Lie sub-

v : G -~ A u t X

-132-

to be an effective

operation

of the Lie algebra connected H

KX

of

G

on

X

and let

of Killingvectorfields.

Lie subgroup

H

defines an operation

on

of X

G

S

be a subalgebra

Then there

is a unique

such that the restriction

with

S

as Lie algebra

of

T

to

of Killing-

vectorfields.

7, 2.

Existence LEMMA

groups.

If

of l o c a l h o m o m o r p h i s m s , 7 . 2 , 1.

Let

p: G -~ G'

L ( p ) : L G -. L G '

We begin with

be a l o c a l h o m o m o r p h i s m

i_s a n i s o m o r p h i s m ,

then

p

of Lie

is a local

isomorphism.

Proof: neighborhood p and

If of

L(p) e'

is an isomorphism, in

G

therefore

Example homomorphism

If

b y 6 . 1 . 4,

moreover

surjective

structure

of commutative

by

We h a v e a l r e a d y p: G -. G' groups.

~

G

#

Now 6.2.7,

L(exp)

of

p:G-~G'

e x p : L G -. G

= 1LG: L G -, L G

is a exp

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

the

Lie groups

(see section

in 6 . 2 . 1 1 t h e e x i s t e n c e

it n o w i n t h e g e n e r a l

.

a local homomorphism

and

connected

proved

on an open

|

is commutative,

inducing a given homomorphism We p r o v e

map

is necessarily

a local isomorphism.

7, 2 . 2 ,

exists

a local inverse

being a local homomorphism, p

there

7.3) ,

of a local homomorphism

h: L G -. L G ' case.

is

for commutative

-133-

THEOREM 7.2.3.

Let

G,G'

a homomorphism

of L i e a l g e b r a s .

p : G -* G '

L(p)

with

= h

N o t e t h a t by 6 . 2 . 1 1 , necessarily

of

LG x LG'

Let If

Let

K

on t he d o m a i n of a c a n o n i c a l c h a r t

morphism

L ( k ) : k - * LG

with local i n v e r s e

A E LG .

G X G' -, G'

k

must

Moreover

D : G -0 K

gives a local homomorphism

A r L G , i.e.

L(k)(A,h(A))

k

k .

L(/z)A = ( A , h ( A ) )

with the projection

L(p) = h ,

p r o p e r t y of

L

functor

= A

is a local i s o -

p: G -. G' .

T o g e t h e r with t h e u n i c i t y is a c o m p l e t e l y f a i t h f u l

with L i e a l g e b r a

consider the homomorphism

By lemma 7.2.1, D: G -~ K ,

is a s u b a l g e b r a of d e f i n i t i o n 4 . 6 . 2 .

is t h e m a p g i v e n by

T h e c o m p o s i t i o n of

L(p)(A) = h ( A ) for

morphisms

Then

G x G'

is the n a t u r a l p r o j e c t i o n ,

and t h e r e f o r e an i s o m o r p h i s m ,

that

.

e q u i p p e d with t h e L i e a l g e b r a s t r u c t u r e

.

p

exp o h o log.

be t he c o n n e c t e d L i e s u b g r o u p of

X - p/K:K-.G

for

Then there exists a local homomorphism

k = { ( A , h ( A ) ) / A ~ LG}

p: G x G' -* G

h: LG -. LG'

.

c o i n c i d e with

Proof:

be L i e g r o u p s a n d

By c o n s t r u c t i o n

q.e,d.

6. Z. 8, t h e t h e o r e m e x p r e s s e s

on L i e g r o u p s a n d l o c a l h o m o -

to L i e 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 .

T o be a b l e 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 local homomorphisms, on

a

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

coinciding

n e i g h b o r h o o d of t h e i d e n t i t y . We h a v e a l r e a d y s e e n in 4 . 5 . 6

that a local isomorphism

of L i e g r o u p s

-134-

i n d u c e s an i s o m o r p h i s m of L i e a l g e b r a s , t h e f u n c t o r i a l i t y of

L,

T h i s i s a t r i v i a l c o n s e q u e n c e of

We a r e now a b l e to show

THEOREM 7.2.4,

Two Lie~roups

G

and

G'

i s o m o r p h i c if a n d o n l y if the L i e a l g e b r a s

LG

and

LG'

Proof:

If

h: LG -. LG'

is a n i s o m o r p h i s m ,

a local homomorphism

p : G - - , G'

i s o m o r p h i s m by 7 . 2 . 1 ,

q . e . d,

inducing

h ,

a r e locall~r are isomoprhic.

t h e r e e x i s t s by 7, 2 . 3

and

p

[s a l o c a l

T h i s t h e o r e m i s t h e m o s t i m p o r t a n t fact we h a v e p r o v e d up to now. It t e l l s e x a c t l y w h i c h type of i n f o r m a t i o n o n e c a n h o p e to o b t a i n by t h e L i e a l g e b r a of a L i e g r o u p . s e q u e n c e of 7, 2 . 4 ,

Note t h a t

t h e o r e m 6 . 2 , 5 is a n e a s y c o n -

T o c o m p l e t e the s t u d y o n e w o u l d like to know if e v e r y

finite-dimensional Lie algebra over some Lie group.

e.g.

R

is o c c u r i n g a s the Lie a l g e b r a of

T h i s is in fact so, but we s h a l l not p r o v e t h i s h e r e .

A

p r o o f i s o b t a i n e d by t h e f o l l o w i n g t h e o r e m due to Ado: E v e r y f i n i t e d i m e n s i o n a l JR-Lie a l g e b r a ~(n,

I~)

GL(n, ~ )

of

~

is i s o m o r p h i c to a s u b a l g e b r a of t h e Lie a l g e b r a

G L ( n , IR)

for s o m e

n .

T h e c o n n e c t e d s u b g r o u p of

c o r r e s p o n d i n g to t h i s s u b a l g e b r a i s a L i e g r o u p with Lie a l g e b r a

i s o m o r p h i c to

~.

T h i s s h o w s by t h e way, that a n y L i e g r o u p is l o c a l l y i s o m o r p h i c to a L i e s u b g r o u p of a g r o u p

GL(n, ~ )

for some

n .

A n o t h e r p o i n t 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 l o c a l h o m o m o r p h i s m s and (global) homomorphisms. If

G

Let

p: G -~ G'

be a l o c a l h o m o m o r p h i s m .

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 o n e e x t e n s i o n

-135-

to a g l o b a l h o m o m o r p h i s m lemma

G -* G'

.

on t o p o l o g i c a l g r o u p s . LEM1VLk 7 . 2 . 5 .

Let

G

be a connected,

simply connected topological group, and

p: G ~ G'

G'

a local homomorphism

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 Proof:

Uniqueness

a t o p o l o g y on e

on which

GXG' p

where

W

.

Let

w -,~-,-~> ( w g , p ( w ) g ' )

(G, p / G )

p'(v)

with

g' ~ G'

Therefore

of

e

-~ G

is a covering

q : G ~ G' --. G'

= p(v) ,

9 : G -. G'

.

we define

, a fundamental

and

p

G

with

p'l(wg)

of

system

9

.

N ( g , g ' , W)

/~

G

of

and

The map

If

W

p/G

be t h e i n v e r s e

. B y d e f i n i t i o n of p , vi ~

for V

is a

are open connected subsets

(e, e' )

p

We

.

is locally connected and

i s a n e x t e n s i o n of

For

G

.

p / N ( g , g' ,W)

in

.

GxG'

a covering. Then

a homeomorphism,

G

and define If

V

p = qo/~ ,

v E V,

It r e m a i n s vE

p

,

is the canonical projection.

= p(v)'~(g) = " ~ ( v ) ' ~ ( g ) .

of

V

w},

is t h e d i s j o i n t u n i o n of

g

s p a c e of

W c

W -~ N ( g , g', V~ 9

is c o n n e c t e d a n d

G ~ G'

Let

in

is a covering

denote the connected component

ks a h o m c m o r p h i z m 7(vg)

p:GNG'

N ( g , g' ,W)

being simply connected. where

I~, g. ) ~ G ~ c G '

N ( g , g' ,W) -~ W

N(g,g',W)

"~

Then

be a connected neighborhood

is a h o m e o m o r p h i s m

therefore

homeomorphism:

Let

to a h o m o m o r p h i s m

G

is a n o p e n n e i g h b o r h o o d

of t h i s u n i o n .

topological group

i s d e f i n e d by N ( g , g ' , W) = [ (x, x' )Jx = wg, x' = p ( w ) g ' , w

show that the projection

connected,

an arbitrary

To prove the existence,

V ~ If

locall~r c o n n e c t e d a n d

(of t o p o l o g i c a l g r o u p s ) .

p

is c l e a r .

is defined 9

of n e i g h b o r h o o d s

the

W e 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

and

by i n d u c t i o n

then

to s h o w t h a t g~

G

p

we h a v e

-136-

i vi)g) = (H i'p(vi))~(g )

Therefore "p

and in particular

"~((l'[ i vi)g ) = y(IIi vi)~(g )

.

Y(ni v i) V

As

:

n i Y(v i)

generates

G

is therefore a h o m o m o r p h i s m . T o g e t h e r with l e m m a 6 . 3 . 3 follows P R O P O S I T I O N 7. Z. 6.

Let

G

be 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 group,

G'

morphism.

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

.

an a r b i t r a r y

Li e g r o u p a n d

.

p: G -~ G'

a local homo-

p

to a h o m o m o r p h i s m

--p:G-~ G I

N o t e t h a 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

this proposition. COROLLARY 7.2.7. a homomorphism

Let

G, G'

of L i e a l g e b r a s .

If

be Lie groups and G

is connected and simply connected,

then there exists a unique homomorphism If, m o r e o v e r . isomorphism, Proof:

then

G' p

h: L G -- LG'

p:G-~G'

with

L(p) -- h

is 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 , a n d

h

an

is a n i s o m o r p h i s m .

To a homomorphism

7.2.3 a local homomorphism

h: LG -~ LG'

p : G - * 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 ,

p

inducing

t h e r e e x i s t s by t h e o r e m h

.

If

G

is

c a n , b y 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 . S u p p o s e now a l s o is an i s o m o r p h i s m ,

k: G' -~ G .

Now

G'

connected and simply connected.

its inverse

L(kop)

= ILG

k

If

h

is i n d u c e d by a h o m o m o r p h i s m

and by unicity

Z.ap = iG

.

Similarly

-137p o X = 1G,

and

p

is an isomorphism,

As an application, 6.1. 4 the map

e x p : L G -. G

isomorphism

simpler connected

let

G

7.2.8.

~ : G -. G

such that

H

G

r

Now let

G

p: G -. G'

covering

connected

group of

.

and

(G', r

)

exists a unique homomorphism

connected

in p a r t i c u l a r

Lie groups

More precisely and simply

and local isomorphism of

G

.

(G, ~)

and simply

there

has

connected

is a unique homomorphism

G

Lie group,

the pair

Lie group,

G'

Let

an arbitrary

( ~ ~)

G,G'

a universal @:G -.G'

.

covering with

~'o

Lie group

be the universal

to a homomorphism

p: G -~ G'

(LG, exp)

.

Then the local homomorphism

a unique extension

is connected

Suppose

9- H -* G

of a

is a c o n n e c t e d

For any connected

be a connected

G

the existence

manifold

a local homomorphism.

group of

has by 7.2.6 G'

covering

and

.

is a commutative

is the universal

and

property.

7. Z. 7 s h o w s

connected

Lie group.

and a homomorphism

o p = p

Corollary

proof;

is a covering

]By

is an isomorphism.

Then there

and homomorphism

with If'

G (G, cO)

the following universal

p:H-*~

Lie group.

Lie group

Lie group

e x p : L G -. G

.

It i s i n d u c e d b y t h e

Gis a commutative,

group for any connected

be a connected

connected

If

We mention here~without

covering

Lie group G

of Lie algebras.

Lie ~roup,

Remark.

a commutative

is a homomorphism.

1LG: L G -. L G

PROPOSITION

universal

consider

q.e.d.

p o ~: G -. G' @: G -~ G'

g r o u p of ~

G'

there

= e,,.

to be a local isomorphism

The preceding

If

.

shows that

of

~ : ( ~ - . (~'

-138-

is a local isomorphism.

By corollary 7.2.7,

{~ is an isomorphism.

Therefore a local isomorphism of connected Lie groupStnduces an isomorphism 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,

this map is bijective.

By the above mentioned theorem of Ado

The problem of classifying all possible connected

Lie groups is therefore decomposed in two steps, algebras.

First find all R-Lie

Secondfind all discrete normal subgroups of a simply connected

Lie group. Consider the restricted problem of classifying all possible commutative connected Lie groups. dimension.

A commutative Lie algebra is characterized by its

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.

7. 3.

We shall do this in the next section.

Discrete subgroups.

Can

H

Let

H

be a L i e g r o u p a n d

be d e f i n e d a s a L i e s u b g r o u p of

F o r a g i v e n t o p o l o g y on of

G

in

G), s u c h t h a t

H

H

H

a subgroup.

G ?

(not n e c e s s a r i l y

the r e l a t i v e topology

i s a t o p o l o g i c a l g r o u p , t h e r e is by 7. 1. 8 at

-139-

most one Lie group structure Lie subgroup

of

G

The example

of

G

on

H

of t h e r ~ t i o n a l s H , there

Q~

~

a

s h o w s t h a t if we t a k e t h e

does not necessarily

exist a Lie group

inducing this topology and making

H

a Lie subgroup

. We c a n a l w a y s

H a Lie subgroup is

H

.

induced topology on structure

inducing this topology and making

consider

of

G

.

0 , and the subalgebra

subgroup

therefore

from this trivial

0

.

H

as a

0-dimensional

The Lie algebra

of LG corresponding The example

manner

of a

there

0-dimensional to a

Q~,~ ~t

islpossibly~no

manifold,

making

Lie group

0-dimensional

Lie

again shows that apart

way of turning a subgroup

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 .

DEFINITION

7.3.1.

subgroup

H

When

is a Lie group,

as a

G

of

G

Let

0-dimensional

of a L i e g r o u p Hausdorff

is a subgroup

is a closed

group.

which is a discrete

it i s n a t u r a l

Lie subgroup

G

be a topological

of

subspace

to view a discrete G

subgroup

.

A discrete

subgroup

Note that a discrete

(use the fact that

of

G

G. H

subgroup is a

space),

Example subgroup

G

of

7o 3 . 2 . •n

Let

0 _~ p

~ n

.

Then

~P

is a discrete

.

We show now

PROPOSITION p : G -~ G'

7.3.3,

a homomorphism

Let

G, G '

be topological

and local isomorphism:

groups

and ,m

The_n t h e k e r n e l

of

-140-

p

is a discrete

Proof:

G, G'

normal

There

such that

ker p N N

subgroup

exists

and

is discrete,

e

N, N'

of

is an isolated

point of

every point of

e, e'

in

Therefore ker p

ker p

.

The trans-

is isolated and

|

COROLLARY kernel

,

is a homeomorphism.

lations being homeomorphisms, ker p

G

open neighborhoods

p / N : N -, N'

-- {e}

of

7.3.4.

Let

of the homomorphism

additive group of

G

be a commutative Lie group.

e x p : L G -. G

is a discrete

Th__._e

subgroup

of the

LG .

Proof: The homomorphism

e x p :LG -. G

is by example 7.2. Z a

local isomorphism.

This raises

the problem

additive vectorgroup such subgroup precisely

a discrete

dimension linearly

is isomorphic

to a group

•-vectorspace

2E p, w h e r e

7.3.5.

Let

subgroup

of the additive vectorgroup.

of the subspace

independent

Proof: induction.

of a finite dimensional

subgroups

of the V

,

p ~ dim V

Every .

More

we show

LEMMA D

of f i n d i n g a l l d i s c r e t e

V

generated

vectors,

We a s s u m e

be a n-dimensional

b~"

v I .....

known the case

Suppose the lemma

D

Vp

,

L.e t

Then there

_in

p = 1

true for all

E. - v e c t o r s p a c e

V

and

p _< n

be the

exists

P

generating

D

and prove the lemma k < p

and let

D

. by

generate

-141a p - d i m e n s i o n a l subspace subspace

A of

U

U

of

V .

T h e r e is a ( p - l ) - d i m e n s i o n a l

g e n e r a t e d by e l e m e n t s of

be l i n e a r l y independent v e c t o r s in D + A[A ~ D/D N A .

V

D ,

generating

Let

vI , , . . ,

D N A .

Vp_l

Now

That this a l g e b r a i c i s o m o r p h i s m is a topological

i s o m o r p h i s m follows from the fact that t h e s e g r o u p s a r e locally compact and that

D+A

has a countable

of S. Helgason [ 6 ], p. Ill). discrete. group

base (for a proof we r e f e r to c o r o l l a r y 3, 3

Using this, we see that

Being a subgroup of the l - d i m e n s i o n a l v e c t o r s p a c e

D + A/A

is g e n e r a t e d by an e l e m e n t

a r e l i n e a r l y independent and g e n e r a t e

We a r e now a b l e to d e t e r m i n e connected

D + A/A

Vp + A .

is

U/A , the

Then

vI 9 . . . .

Vp

D .

the structure

of the commutative

Lie groups.

THEOREM of d i m e n s i o n

7.3.6.

n

.

Let

G

be a c o m m u t a t i v e

Then there is an integer

p

,

connected Lie group 0 _~ p ~ n ,

such that

--~ n-p~Tp O-- E

Proof: The h o m o m o r p h i s m 6.2.7.

We h a v e t h e r e f o r e

algebraic

sense.

of L i e g r o u p s . general

7.3.5.

an isomorphism

is s u r j e c t i v e by p r o p o s i t i o n LG/ker

exp ~

G

in the

It i s n o t d i f f i c u l t t o s e e d i r e c t l y t h a t it i s a n i s o m o r p h i s m We o m i t t h i s h e r e ,

statement,

by lemma

exp: LG -~ G

Now

ker exp

Therefore

G

a s w e s h a l l p r o v e 9 i n 7, 7 . 6 , Z p En/z

for some P

p

with

which proves

a more 0 ~_ p _~ n

the theorem

-142-

COROLLARY of dimension

n

7, 3 . 7 ,

.

Then

As mentioned problem

connected

PROPOSITION

7, 3 . 8 .

center

.

Proof:

Let

is continuous.

H

one step in the classification

H

G

is greatly

The map

simplified

be a discrete Then

H

G-~H

a subgroup

connectedness.

which is an open subset

a Lie subgroup

H

of

H0 = GO .

of

G

V

necessarily

7.4.1.

G L ( V ) -~ ~ *

connected, of

G

Remember

Example and det:

subgroups by

normal

subgroup

is contained

defined by

of

in t h e

g ~

g h g "1

it m u s t b e a p o i n t a n d t h e r e f o r e

.

Let of

G ,

G H

The Lie algebra

is LG, the injection being a local isomorphism. group

normal

q.e.d.

Open subgroups,

therefore

.

7.2,

The image being connected,

equal toh,

7.4.

Let

Lie group

.

This

group

h ~ H

connected

in f i n d i n g a l l d i s c r e t e

Lie group.

topological

G

be a compact

= T n

consists

the connected of

G

G

a t t h e e n d of s e c t i o n

for Lie groups

of a simply

Let

GL(V)

with the same

can be continuously

contains

GO,

that an open subgroup Let

V

is not connected,

transformed

is a submanifold

Therefore as

and

of an open subgroup an open sub-

LH = LG

is necessarily

be a finite-dimensional

the determinant

orientation

be a Lie group and

homomorphism. On the other hand,

implies closed.

l:t-vectorspace R*

being not

any two bases

(after the choice of an orientation) one into the other by automorphisms

of

-143-

V.

This shows readily that

of t h e i d e n t i t y in Let

H

GL(V),

:

be an o p e n s u b g r o u p of

G

H ,

topology is discrete

G/H

where

is the connected component

~+

l e f t c0sets m o d u l o

manifold.

d e t - l ( ~ +)

{x ~ ~ * / x and

G/H

A l l l e f t ocsels b e i n g o p e n in

and

G/H

If in p a r t i c u l a r

c a n be c o n s i d e r e d

can be considered

H

as a

is a normal

>

0]

the s e t of

G,

the quotient

as a

0-dimensional

s u b g r o u p of

0-dimensional

.

G ,

then

Lie group 9

T h i s a p p l i e s to t h e c o n n e c t e d c o m p o n e n t o f t h e i d e n t i t y a n d ~/-- G / G 0

is a 0 - d i m e n s i o n a l

Example

79 4 . 2 .

Let

of a f i n i t e - d i m e n s i o n a l

Lie group.

G = GL(V)

vectorspace

be the g r o u p of a u t o m o r p h i s m s

V

9

Then

~ =Z 2

by example

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 manifold

G

is d i f f e o m o r p h i c

canonical diffeomorphism. e -~G0 -. G -. ~ - . e

G0xT~

defined by

= ~S(r

many particular

cases.

Example

7 . 4 9 3.

V

.

forms

Any splitting

,

w ith

G0 x~/ .

There

s: ~ - . G

~

for

G = GL(V)

i m a g e of

~ ,

e -. G O -. G -~ ~ -. e

no

with the semi-

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

together

Z 2 -" G / G 0 group

G O , the

of the exact sequence

for an odd-dimensional V

to

is, h o w e v e r ,

of G

T: ~ -~ A u t G O

where

In t h e c a s e o f a c o m m u t a t i v e the exact sequence

.

G Ox

T h e r e f l e c t i o n at the o r i g i n of an isomorphic

is diffeomorphic

g i v e s r i s e to an i s o m o r p h i s m

direct product Tr

to

G

G ,

in

~-vectorspace

with the i d e n t i t y of G

V

9

a splitting

s: ~ -~ G

defines an isomorphism

of

of G

-144-

E x a m p l e 7, 4 . 4 .

Let

V

Consider a vectorsubspaceU u n i o n of group

U G

be a f i n i t e d i m e n s i o n a l

and a vector

and its translates

i s o m o r p h i c to

•

.

G

of a L i e g r o u p c l o s e d in

G

G .

G

.

H

Let

by d e f i n i t i o n of

(i), exp t(A+B)

showing

on

and

G ~ UxZ

g H

.

G

o

H

a subgrou p

Then there exists

s u c h t h a t th e c o r r e s p o n d i n g t o p o l o g y

and such that

H

i s a L i e s u b g r o u p of

~

G

Now let

At

G

.

Let ~c ~ H

LG

be d e f i n e d by

for every

t

i s a s u b a l g e b r a of ~ . iu

~,

S u p p o s e now

for every

a n d by 6 . 5 . 5 , ~,B]

has an

B o t h t y p e s of s u b g r o u p s a r e

be a L i e g r o u p a n d

H

[A ~ L G / e x p t A

We f i r s t p r o v e t h a t

~ ~,

-~ G

-~ 0

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,

~=

A+B

G

H

be a c l o s e d s u b g r o u p of

~

is a L i e

i s th e c o n n e c t e d c o m -

0 -~ U -*G - ~

is a c l o s e d s u b s e t of

i s t h e i n d u c e d t o p o l o g F on

tA

s:Z

are Lie subgroups.

a u n i q u e L i e Group s t r u c t u r e

H

U

a

T h e n th e

We s t a t e now m o r e g e n e r a l l y

Suppose

Proof:

U .

We h a v e s e e n t h a t d i s c r e t e a n d o p e n s u b g r o u p s

T H E O R E M 7 . 5 , 1. o.f

V .

T h e e x a c t sequemce

Closed subgroups.

a

a n d 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

evident splitting homomorphism

7.5.

~ V ,

by i n t e g e r m u l t i p l e s of

in t h e r e l a t i v e t o p o l o g y of

p o n e n t of t h e i d e n t i t y of

a

1R-vectorspace.

t ~lR,

(ii), e x p t 2 [ A , B ]

as

A ~

implies

A,B ~ ~ . Then by 6.5.5, H

is closed. Therefore

H

for any

is t h e r e f o r e a s u b a l g e b r a of

IR,

t LG

.

- 145

Consider LH'~

=~.

now t h e c o n n e c t e d

By construction

H* c H , H* Let

being generated

H

e

in

H

(using that

that

H$

H$

= H0

H

Lie subgroup ~

we h a v e

by

V

of

e

This will prove that H~'r

H

ks open in

H$

of

exp~

H

in

t o p o l o g y of H*

H$

H0

It is then c l e a r that the multiplication This is in fact sufficient to see that

in

H .

G

c 1,

...

c k,

be a complementary

...

subspaee

We s h a l l of

H$

in

H) ,

Then

H ~ H -~ H H

,~ill be differentiable.

is a Lie subgroup of

Suppose

V

in

H - V

of

~

V

LG .

open, connected neighborhoods

of

G ,

e

in

H$

is not a neighborhood of

with

in

G.

with the aid of translations,

We show that this leads to a contradictLon.

sequence

bounded,

H .

.

V = H~' j m o r e o v e r .

T h e r e r e m a i n s to show, that a neighborhood in

G

is t h e r e f o r e a Lie subgroup of

can now be turned into a submanifold of

e

and therefore

is a topological subgroup of

being an inner point of

as topological groups.

a neighborhood of

with

is a n e i g h b o r h o o d

is continuous), and taking

H ( e

G

exp~z.

be e q u i p p e d w i t h t h e r e l a t i v e

prove that a neighborhood H

of

-

U1,

is e

T h e r e exists a

K-.co'lim c k = e

.

By 6.3.2, U2

of

exp B

for

Let

M

there exist

O

in

M

and t/

respectively,

such that

is a diffeomorphism G

9

4:(A,B)~

of

U I•U

We can, therefore,

expA

2 onto

assume

A ~M,

an open neighborhood

that

of

c k = exp A k exp B k

B ~ ~,~ e

in

with

\

Ak

U 1, B k E U 2 Since

rkA k ~ U 1 assume,

Ak ~ 0 , and

and

expB k E V

.

Then

there exists an integer

(rk+l)A k ~

U1 .

p a s s i n g to a s u b s e q u e n c e ,

Now

U1

A k~r rk > 0

0

lira A k - - 0 .

such that

is b o u n d e d ,

that the sequence

and

s o we c a n

(rkAk)

converges

-146-

toa

limit

A ~ U1 .

t h e b o u n d a r y of Let

Since

(r k+l)A k~

U 1 , in p a r t i c u l a r

p, q

be a n y i n t e g e r s

A 4

tk

But then

A 6~,

.

in c o n t r a d i c t i o n

The previously either a discrete where

~"

discussed

0

Let

LH

A ~

G

= [A ~

U 1CM

cases

The Lie algebra

of

H

and

of 7 . 5 . 1 ,

.

sk

for every

G , correspond

be a L i e g r o u p a n d of

H

defined in 7.5.1. LG/ exptA

0 _~ t k < q

t G

A~

0 . |

where

H

is

to the case

LG .

be t h e L i e a l g e b r a

unique Lie group structure

Proof:

Let

and

e x p tA 6 H

particular

o r e q u a l to

COROLLARY 7 . 5 . Z.

LH

to

is on

A

A k = l~m (expAk)

q

o r a n o p e n s u b g r o u p of

is e i t h e r

subgroup.

Prk

By continuity

A k-~ 0 ,

T h e n we c a n w r i t e

are integers

exp-P--A = itkm e x p q H

and

0 .

(q > 0) .

pr k = qs k+ tk, where s k, tk Then lim--q A k = 0, so

w h i c h b e l o n g s to

U1

~ H

H

a closed

with respect

to t h e

Then for every

t ~

IR}

.

w a s d e f i n e d i n t h e p r o o f of 7 . 5 . 1

by this property, The corollary

Remark. subgroup

H

S. H e l g a s o n

of

Lie

See

which has countably many components.

[ 6 ], p, 108.

An important kernels

G

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

c l a s s of c l o s e d s u b g r o u p s

of h o m o m o r p h i s m s

starting

from

of a L i e g r o u p G.

G

are the

-147-

P R O P O S I T I O N 7 . 5 , 3. L~e g r o u p s .

Then

= k e r L(p),

ker p

where

Le_~_t p : G -. G'

be a h o m o m o r p h i s m

is a L i e s u b g r o u p , of

L(p): LG -~ LG'

G

and

of

L ( k e r p)

is t h e i n d u c e d h o m o m o r p h i s , m

of L i e a l g e b r a s . Proof:

ker p

s u b g r o u p of G. e'

is a c l o s e d s u b g r o u p of

B y 7 . 5 , Z,

d e n o t i n g t he i d e n t i t y of

p ( e x p tA) = e' for every E > 0, signifies

for every

t ~ Ft . such that

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' f o r e v e r y t G' .

By th e n a t u r a l i t y 6, 1. 6 of exp,

t ~ l~

i s e q u i v a l e n t to

e x p ( L ( p ) t A ) - e'

T h i s a g a i n is e q u i v a l e n t to t h e e x i s t e n c e of a n L(p)tA

L(p)A = 0 .

= 0

for every

Therefore

It ] < E .

L(kerp)

The latter property

= kerL(p)

,

q. e, d.

T h i s s h o w s in p a r t i c u l a r t h a t the k e r n e l of t h e h o m o m o r p h i s m L(p): LG -. LG'

is a Lie algebra.

T h i s is of c o u r s e t r u e f o r t h e k e r n e l

of 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 . We a l s o w o u l d l i k e t h e i m a g e of a h o m o m o r p h i s m be a L i e g r o u p .

I n d e e d we h a v e

P R O P O S I T I O N 7, 5, 4, groups. and

Suppose

G

Let

connected,

L(im p) -- im L(p), where

morphism

of L i e a l g e b r a

Proof: LH = i m A ~ LG .

of L i e g r o u p s to

Let

L(p) . Now

H H p(G)

p:G -. G' Then

be a homomorphism of Lie

imp

L(p): LG -. LG'

is a Lie subgroup of is the induced homo-

s,

be t he c o n n e c t e d L i e s u b g r o u p of i s g e n e r a t e d by th e e ~ e m e n t s is g e n e r a t e d by th e e l e m e n t s

G'

with

exp(L(p)A)

with

p(exp A)

with

G'

~ •],

-148-

A ~LG p(G)

.

But

p(expA)

= exp (L(p)A)

= H , as both groups

Remark.

There

is a homomorphism,

Consider

i. e, a n a l y t i c .

now a sequence G'

and the induced sequence

Therefore

are connected.

is the question,

('V)

by 6,1.6.

if t h e i n d u c e d m a p

~ : G -. p ( G )

This is indeed so (see 7.7.6)

of homomorphisms

P' > G

.

of Lie groups

P " >G"

of homomorphisms

of Lie algebras

(a) I..P~' L(p').>LG .L(p"),,LG" PROPOSITION exactness

of

(~)

Proof:

If

= L(ker p ")

implies

i m p'

7.5.6.

TG

, then

by 7.5.3

G

0 -~ G e --, T G -. G -. e

of the first

G'

the exactness

Let

0 -. G e -. L ( T G ) -~ L G -~ 0 Gr

Suppose

= ker p"

= ker L(p")

Example sequence

7, 5 , 5 .

of

(a)

= L(im p')

q.e.d.

be a connected of Lie groups

Then the

.

im L(p')

and 7.5.4,

of Lie algebras.

sequence

connected,

Lie group.

The exact

induces an exact sequence

Note that the natural

defines a splitting

LG-~ L(TG)

splitting of the

second sequence. Observe if all groups

that the converse

7.5.5

is not true,

even

are connected.

Example isomorphism,

of proposition

7.5.7. Then

Let 0

p: G -~ G' :> L G L ( P / >

be a homomorphism LG'

>0

and local

is an exact sequence.

-149-

But

e -* G -~ G' -* e'

i s not n e c e s s a r i l y e x a c t ,

i.e.

G

and

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 , T h e f o l l o w i n g p a r t i a l r e s u l t is s o m e t i m e s u s e f u l . P R O P O S I T I O N 7 . 5 , 8. Lie g r o u p s .

Suppose

if a n d o n l y if Proof:

Let

G

and

G'

L(p): LG -~ LG' If

p

p: G -~ G'

connected.

Then

p

is s u r j e c t i v e

is s u r j e c t i v e .

is s u r j e c t i v e , t h e L i e a l g e b r a

m u s t c o i n c i d e with the L i e a l g e b r a L(p}

be a h o m o m o r p h i s m of

LG'

of

L(p}LG

of

p(G)

G' , w h i c h s h o w s that

is s u r j e c t i v e . Suppose conversely

L(p}

is s u r j e c t i v e by 6 . 2 . 6 f o r e v e r y (and h e n c e c l o s e d } s u b g r o u p of T h e c o n d i t i o n that

G'

surjective. g ~ G. G'

,

i.e.

Then

p(G) p(G)

p , g : Gg -. G' p(g)

is t h e r e f o r e an o p e n :

O'

.

|

is c o n n e c t e d c a n n o t be o m i t t e d , a s s h o w n

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 i d e n t i t y into a n o n - c o n n e c t e d L i e g r o u p , w h i c h i n d u c e s a n i s o m o r p h i s m of L i e algebras. T h e c o r r e s p o n d i n g s t a t e m e n t f o r i n j e c t i o n s is not t r u e , s e e n in 7. I. I t h a t an i n j e c t i o n

L(p ): L G -+ LG' injectivity of R - ~

.

9 p ,

p : G -. G'

But t h e i n j e c t i v i t y of

We h a v e

i n d u c e s an i n j e c t i o n L(p)

d o e s not i m p l y the

a s s h o w n 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 07.6.

C l o s e d s u b g r o u p s of t h e full l i n e a r g r o u p .

dimensional

~-vectorspace

and

GL(V)

Let

~: VX V -~ 1R

be the s u b g r o u p of

H

= {g ~ G L ( V ) / r

d e f i n e d by

GL(V)

GL(V) .

leaving

gw) = r

v, w ~ V

w)

the m a p

r

i s c l o s e d in G L ( V ) .

H

.

gw) .

As

~

is c o n t i n u o u s ,

S(v, w) = [g ~ G L ( V ) / # ( g v , gw) - r

Now =

S(v, w)

N

v,w ~V

is a c l o s e d s u b g r o u p of

We i d e n t i f y t h e L i e a l g e b r a of 4.3.8).

v, w ~ V]

G L ( V ) -* G L ( V ) X G L ( V ) -* VXV-*

T h e r e f o r e the set

H

T h i s s h o w s that

V .

invariant:

for any

g --~-~>(g, g) --~-,~>(gv, gw) ~----->r

this map is continuous,

be a f i n i t e

be a b i l i n e a r a n d n o n - d e g e n e r a t e d f o r m on

H

C o n s i d e r for f i x e d

V

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 .

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 Let

Let

GL(V)

GL(V). with

s

(see proposition

T h e n we h a v e t h e f o l l o w i n g c h a r a c t e r i z a t i o n of

LH.

P R O P O S I T I O N 7 . 6 , 1. LH = {A C s 1 6 2 Proof: r

Let

A ~ LH .

v, exp(tA)w) = r

r e s p e c t to

t

w) + r Then

w)

we o b t a i n f o r

Suppose conversely that

exptA E H

for

t=0

,

by

A*

by

#(Av, w) - r

A* w) = 0.

by

A*

We s h a l l show that

t E ~t,

= -A

.

which means

for any

v, w ~ V.

A~ s

the adjoint linear map of

Aw) = 0 f o r

A

r

v , w ~ V} t ~ l~

and

D i f f e r e n t i a t i n g with

w)+ r

Aw) = 0 .

satisfies this condition. w i t hrespect to

Denote

~ , characterized

The hypothesis can therefore be expressed

exptA ~ H

(exp tA)*

for e v e r y

= (exp tA) -1

t C~ IR .

for e v e r y

This implies

w)}

-15 i-

AE

LH

.

There

remains

to s h o w t h a t

A*

But this follows from the expression

=

implies

-A

( e x p tA)* = ( e x p tA) -1

oo (tA) n = Eh=0 n,--Hl---

e x p tA

given in

6.1.5.

Exercise.

if

r

g r o u p of

is moreover

V

algebra

Deduce also proposition

with respect

symmetric, the group

with respect

consists

r

Example

to

r

Consider

metric

by t h e d e f i n i t i o n

a ~ V Then

s

of

a, v, w .

d e f i n e d on L O ( V , ~) verify that

v~ V

V .

O(V,~)

which are

e l,

Then

.

The Lie

antiselfadjoint

e2 , e3

V

of

defined by

= ~([a,v],

V

be an orthonormal

can be turned into a Lie [ e z , e 3] = e 1 9

GL(V) (see 4.3.8). Av=[a,v]

w)+ ~(v,[a,w])

for = 0 ,

Let v~

V

.

as is seen

([a, v ] , w) a s t h e o r i e n t e d v o l u m e of t h e p a r a l l e l e p i p e d Therefore

of t h e o r t h o g o n a l g r o u p

~IW-~ L O ( V , ~)

for every

.

A ~ ~(V)

by the interpretation

map

is the orthogonal

[ e l , e 2] = e 3 , [ e l , e 3] = - e 2 ,

~ ( A v , w ) + ~(v, A w )

L O ( v , 9)

Let

with the Lie algebra

and consider

defined by

V

H

a 3-dimensiona 1 lR-vectorspace

~ .

base with a positive orientation

We identify

of

from 6.4.5.

.

7 . 6 . Z.

with a Euclidean

algebra

a n d denoter

of t h e o p e r a t o r s

to

7.6.1

d e f i n e d by

implies

a --,---,> A

3: V -~ L O ( V , ~)

in t h e L i e a l g e b r a

with respect is linear.

for this particular

that

3 , so

is contained

O(V, 9)

a=0

This means

have dimension

A

3" is i n j e c t i v e . ~

to

r

But

.

[ a, v] = 0

Lie algebra But both

is a l i n e a r i s o m o r p h i s m .

is an isomorphism

The

of L i e a l g e b r a s .

structure

V

and

We f i n a l l y Let

-152-

A iv = [a i , v ]

for

v~

i = 1, 2 .

V,

[A1, A 2 ] v

= A1Azv-AzAlV

using the

Jacobi

- [a 1,[a 2 . v ] ] -

where

[a 2 , [ a 1 , v ] ] :

[ [ a 1,a 2 ] , v ]

identity.

We h a v e t h e r e f o r e s e e n that the g r o u p

Then

O(V, r

by t h e m a p

Av = [a,v]

V

is i s o m o r p h i c to the L i e a l g e b r a of

~ : V -. LO(V, r

for every

, d e f i n e d by

a ~,--,,~> A ,

vE V .

T o s e e t h e g e o m e t r i c s i g n i f i c a n c e of t h e 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 satisfying

~t = A a t

by

~rt = &tv = A a t v " A v t

5.4.5.

o p e r a t e s in V .

V

O(V,r

o-

a ~ V ,

V

o-

V

vt = a t v

and

.

Then

T h i s show s

d e f i n e d by

with r o t a t i o n a x i s

A,

a

A

is t h e

.

~ : V -*LO(V, r .

If

o-: O(V, r

is d e f i n e d by

O(V, r

defines therefore a

v~

-~ Aut L O ( V , r

(O-gA)(v) = [ga, v]

V .

for

This representation

, because

= [ga, v] = g[a, g ' l v ] =

Let

d e f i n e d by

~zt = [ a , v t] 9

and e v e r y

is j u s t 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

and therefore

v ~ V

O(V, 9)

LO(V, r

then

A = J(a)

(O-gA)(v)

O(V,r

by a u t o m o r p h i s m s of t h e L i e a l g e b r a s t r u c t u r e

in

denotes this representation, O(V,r

of

The i s o m o r p h i s m s

r e p r e s e n t a t i o n of

g~

Let

at

1 - p a r a m e t e r g r o u p of r o t a t i o n s of

d e f i n e d in

of

c a n be w r i t t e n a s

that the 1 - p a r a m e t e r subgroup

O(V, r

a

a ~ A ,

g ( A ( g ' l v ) ) = (gAg-1)(v)

for e v e r y

v ~ V

o-gA = gAg 1 be a g a i n of a r b i t r a r y f i n i t e d i m e n s i o n a n d

b i l i n e a r a n d s y m m e t r i c f o r m on V . the s a m e a r g u m e n t as for

Suppose

~

~

a non-degenerated

positive definite.

GL(V) ( s e e e x a m p l e 7 . 4 . 1 ) ,

By

one s h o w s t h a t t h e

c o n n e c t e d c o m p o n e n t of t h e i d e n t i t y i s the k e r n e l of t h e h o m o m o r p h i s m

-15 3-

det: O(V, 4)-~]R*

.

T h i s g r o u p is d e n o t e d by

PROPOSITION 7.6.3. space,

Let

V

SO(V, 4)

9

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 -

~ a p o s i t i v e d e f i n i t e , s ~ - m m e t r i c b i l i n e a r f o r m on

the orthogonal group 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 with d e t e r m i n a n t

and

1 .

Then

V ,

SO(V, 4) O(V, ~)

9

r

the g r o u p and

SO(V, 47 a r e c o m p a c t . Proof: closed.

SO(V, 4)

i s a n o p e n s u b g r o u p of

H e n c e it i s s u f f i c i e n t to p r o v e

i s a c l o s e d s u b g r o u p of

GL(V) .

O(V, ~b)

i s a l s o c l o s e d in

O(V, ~b7

i s b o u n d e d in

Now let to

s

O(V, ~b7

GL(V) 9

O(V, 4)

and t h e r e f o r e

compact.

Now

O(V, 4)

b e i n g a n o p e n 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 s h o w t h a t

s (V) 9

I I: s

be the n o r m

on

s

d e f i n e d with r e s p e c t

~b b y

=

{AI T h e n any

g~

O(V,r

~(A v,,,, Av) I/z

v~oSUp~ V satisfies

~(v. v)i/2.

[g[ = 1

and

O(V,r

is boundedin

s L e t now f o r m on

4: V ~ V - ~

be a s k e w - s y m m e t r i c

V (V of e v e n d i m e n s i o n ) .

i n v a r i a n t i s the s y m p l e c t i c g r o u p of Sp(V, 47 9

consists,

T h e s u b g r o u p of V

V

are isomorphic.

a c c o r d i n g to 7 . 6 . 1 ,

with r e s p e c t to

~ .

GL(V)

with r e s p e c t to

As t h e r e is e s s e n t i a l l y a unique

s y m p l e c t i c g r o u p s of

b i l i n e a r and n o n - d e g e n e r a t e d

~

~ , denoted

of t h a t t y p e , a n y two

T h e L i e a l g e b r a of

of the o p e r a t o r s of

leaving

V

Sp(V, 47

which a r e a n t t s e l f a d j o i n t

-154-

det: GL (V) -~ JR*

C o n s i d e r now t he h o m o m o r p h i s m is d e n o t e d by

SL(V)

Proof: L ( k e r det)

7.7.

SL(V)

V

be a f i n i t e - d i m e n s i o n a l 0

By 4.5.11,

L(det)

group.

Let

G/H

G / H ( s e c t i o n 1. 4). G/H

Let

H

,

.

Now 7 . 5 . 3 s h o w s

q.e.d.

be a L i e g r o u p a n d

H

be t h e o r b i t s p a c e of t h e o p e r a t i o n of Z. Z. 3).

a closed subon

G

by

C o n s i d e r th e n a t u r a l o p e r a t i o n of

G

on

Then there exists a unique structure

H

of anal~rtic m a n i G-manifold.

be e q u i p p e d w i t h t h e s t r u c t u r e of L i e g r o u p of 7 . 5 . 1 . of

LG

such that

t he c a n o n i c a l p r o j e c t i o n .

LG = M @LH .

There exists a neighborhood

e x p / U : U - . e x p (U)

is a h o m e o m o r p h i s m

is a h o m e o m o r p h t s m

Let

D e n o t e by

T h e n t h e o r e m 7 . 7 . 1 is b a s e d on t h e

w h i c h p r o o f we o m i t ( s e e S. H e l g a s o n ,

L E M M A 7 . 7 , 2.

- . p ( e x p U)

It is th e

i n d u c i n g t he q u o t i e n t t o p o l o g y a n d m a k i n g it a

following lemma,

such that

,

Le~t G

be a v e c t o r - s u b s p a c e

p: G -~ G / H

is a L i e a l g e b r a .

factor groups.

right-translations(see

fold on

= tr

= L(SL(V)) = k e r t r

Coset spaces,

lR-vector-

,

THEOREM 7.7.1.

M

Let

T h e s e t of o p e r a t o r s w i t h t r a c e

L i e a l g e b r a of

The kernel

.

P R O P O S I T I O N 7, 6 . 4 . space.

,

U and

onto a n e i g h b o r h o o d of

[ 6 ] , p. 113) .

of

0

in

M

,

p/exp(U):exp(U) p(e)

in

G/H.

-155-

The structure follows.

If

interior at

of

p(e)

NO

of a n a l y t i c m a n i f o l d o n denotes the interior

U, t h e n

E G/H .

( e x p / ~ ) -1 o ( p / e x p Now

G

operates

so that this defines also charts that these G/H,

charts

Then,

are compatible,

by construction,

i.e. G

follows from the last statement PROPOSITION operation

be t h e i s o t r o p y b_~

~(gH)

7.7.3.

Let

group of_ x 0 .

defined above.

Let

Then

(p

U

is a c h a r t

by homeomorphisms G/H

.

the

on

G/H ,

It i s t o s h o w

define an analytic structure by a n a l y t i c m a p s on

G/H

on G/H ,

a s a n n o u n c e d in 7 . 7 . 1

in X

T:G ~Aut X

= Vg(X0) .

o

and

(~))-1 : N o - * Ua c M

operates on

is t h e n d e f i n e d a s

p ( e x p U)

at a n y p o i n t o f

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

a transitive

of

G/H

be a G - m a n i f o l d .

Select

x0 ~ X

Consider, the map G/H

with respect and let

(p: G / H -. X

to H

defined

have the analytic manifold structure

is d i f f e r e n t t a b l e .

If

~

is a homeomorphism,

t h e n it i s a d i f f e o m o r p h i s m . Proof: and write permits : Br Then

We u s e

NO

B - exp(~)

.

defining

B

and

~

Denote by and

of

G

4:G-~X

p/B:B

the map

q~(g)=l"g(X0),

differentiable.

(see remark

if t h e t a n g e n t l i n e a r m a p of

(p b e i n g a n e q u i v a r i a n c e

-- N O

, making the injection

~0 i s t h e r e f o r e

t o be a h o m e o m o r p h i s m

w i l l be a d i f f e o m o r p h i s m

meaning as before

Then the homeomorphism

~0/N 0 = qlo L o (p/B) -I

is an isomorphism,

with the same

as a submanifold

differentiableo

Now suppose

0

U

r

(see 1.4.10),

below), at a n y p o i n t it is s u f f i c i e n t

-156-

to p r o v e t h i s f o r t h e p o i n t

x 0 . Now the d e c o m p o s i t i o n r

s h o w s t h a t it is s u f f i c i e n t to p r o v e shall prove

k e r ~*e = He

"

= dim G - dim H = dim G/H homeomorphism),

~ e: Ge -" Tx0(X)

Then rank

the c o r r e s p o n d i n g

A ~ RG,

5.6.2.

and

A

k e r @*e = He "

HeC k e r @*e "

Consider theKilltngvectorfield

A*

A* i.e. are

which shows

A =A* = 0. x0 ~*e e t ~ ~ . Thus A E H

every

e

Remark.

The map

homeomorphism,

=dim G-

We

dim ker ~e a

is a

a n d t h i s w i l l f i n i s h the p r o o f .

x 0 , clearly

Then

to be s u r j e c t i v e .

= d i m X (the l a s t e q u a l i t y b e c a u s e

T h e r e r e m a i n s to show t h a t g r o u p of

~

0 = @oI. o ( p / B ) "1

if

G

e

r

being the isotropy

H

Let c o n v e r s e l y

on

X

A*

= 0-(a)

A e E k e r @.

. e Ae , respectively

d e f i n e d by

in t h e n o t a t i o n of t h e o r e m

- r e l a t e d ( s e e the p r o o f of 5 . 6 . 2 ) , T h e n by 6 . 3 . 1 ,

exptA e ~ H

for

in view of 7 . 5 , 3, q. e. d. -~ X

d e f i n e d in 7 . 7 . 3

i s in fact a

has countably many components.

Under this

c o n d i t i o n , the a r g u m e n t in the p r o o f a b o v e s h o w s , f o r an a r b i t r a r y G-manifold that

r

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 ) a n d --. X

i s a d i f f e o m o r p h i s m onto t h e o r b i t of

B e f o r e t u r n i n g to t h e c a s e w h e r e

H

x0~

X,

x0 .

is a n o r m a l s u b g r o u p of

G ,

we g i v e a d e f i n i t i o n . DEFINITION 7.7.4. An

i d e a l ~ v of ~ "

for e v e r y If ~

A ~ ,

Let

~

be a L i e a l g e b r a o v e r a r i n g

i s a v e c t o r s u b s p a c e of BE

~

satisfying

,~

.

[A, B] ~: ~ ' ,

~.

is an i d e a l of ~

, the q u o t i e n t v e c t o r s p a c e

c a n o n i c a l l y e q u i p p e d with a L i e a l g e b r a s t r u c t u r e ,

and

~

~/~tl

is

is t h e k e r n e l

-15 7-

of the canonical h o m o m o r p h i s m

~ -. ~ / ~

of a Lie algebra h o m o m o r p h i s m ,

.

Conversely, the kernel

with domain ~t~

~-, and the vectorspace isomorphism of ~ / ~

is an ideal of

with the image is a

Lie algebra isomorphism. We p r o v e now PROPOSITION 7.7.5, theLie

group

G

.

Let

LG/LH

induces

be a c l o s e d n o r m a l s u b g r o u p of

The factorgroup

d e f i n e d in 7 . 7 , 1 i s a L i e {~roup. p : G -* G / H

H

G/H

with the m a n i f o l d s t r u c t u r e

The canonical homomorphism

L(p): LG -~ L ( G / H )

with k e r n e l

LH , s u c h t h a t

"~- L ( G / H )

Proof:

The factorgroup

to t h e q u o t i e n t t o p o l o g y . 7.7.1on

G/H

(g, x H ) ~

is a n a l y t i c .

G/H

Consider Therefore

L(p)

G~g G / H -* G / H

of

given by

There remains to show that the group

are analytic, which is immediate.

of t h e m a n i f o l d s t r u c t u r e on a homomorphism

is a t o p o l o g i c a l g r o u p w i t h r e s p e c t

C o n s i d e r th e u n i q u e m a n i f o l d s t r u c t u r e

such that the map

gx H

o p e r a t i o n s in

G/H

G/H,

p : G -~ G / H

By c o n s t r u c t i o n

is a n a l y t i c , a n d t h e r e f o r e

of L i e g r o u p s .

L(p): L G -~ L(G/H) ,

By

induces an isomorphism

Note that if H

7.5.3

LG/LH

is a normal subgroup of

the factorgroup is not Hausdorff. A s a c o n s e q u e n c e we o b t a i n

ker

G

L(p) = L ( k e r p) = LH .

"= L(G/H) which is not closed,

-15 8-

PROPOSITION 7.7.6. Lie g r o u p s .

Suppose

p : G / k e r p -'p(G)

G

Let

p : G -* G'

p ,

Then

Lie groups, where

G/ker p

of 7 . 7 . 5 and

with t h a t of 7 . 5 . 4 .

the m a p

p(G)

~:G-~p(G)

Proof:

L(p)

p

is a.nal~-tic.

commutative diagram

L ( G / k e r p)

T h e r e is at m o s t one m a p

Moreover

L(p(G)) -~ LG'

is i n j e c t i v e .

f i l l i n g in, a s

i n d u c e d by

Hence

L(p)

C o n s i d e r the c a n o n i c a l

This proves that

in c a n o n i c a l c h a r t s .

L(~ " y

> LG'

r

y: L ( G / k e r p) -, L(p(G))

m a k e s the d i a g r a m c o m m u t a t i v e . y

> p (G) C----------------> G '

L(p )LO

y: L G / L ( k e r p) -~ L(p)LG

e . being just

P

L(p (G))

LG/L(ker p )

isomorphism

,is a n i s . o m o r p h i s . m , o f

T h i s show s in p a r t i c u l a r t h a t

G/ker p

is s u r j e c t i v e and

p

is e q u i p p e d with the L i e g r o u p s t r u c t u r e

induced by

C o n s i d e r the

C o n s i d e r the _canonical m.ap

connected,

i n d u c e d by

be a h o m o m o r p h i s m of

"~

L(p): LG -. LG' ~

.

is a n a l y t i c at

is e v e r y w h e r e a n a l y t i c

is a n i s o m o r p h i s m a n d t h e r e f o r e

~

an isomorphism

-15 9-

of Lie groups.

The map

~: G -*p(G)

is the c o m p o s i t i o n

a n a l y t i c h o m o m o r p h i s m s and hence a n a l y t i c ,

"po p

of

-160-

CHAPTER

8, 1,

The automorphism

dimensional ~I x

8,

GL(!il}

vectorspace, Then

g r o u p of a n a l g e b r a .

~-algebra,

~I -~ ~I .

A u t tl

GROUPS OF AUTOMORPHISMS

i, e. a v e c t o r s p a c e

Let

~I

be a finite

with a bilinear

map

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 the underlying

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 h e a l g e b r a

~I ,

A u t ~I c GL(~I) Example

8, 1, 1.

~I a

LEMMA 8.1.2. Proof:

Let

~-Lie

A u t ~I

A, B

algebra.

i s a c l o s e d s u b g r o u p of

~ !ll a n d c o n s i d e r

GL(~I)

.

the map

GL(Ill) -~ GL(~I) x GL(92)-~ ~I X ~I -~ ~1 defined by

q~---~-~(~,~p)-------->(cpA,~B}--~-~--> ~ A . ~pB

, The multiplication

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 } , The set of

r

Aut(~l)

S(A,B) B)

= {r

= A, BN~ ~I SCA, B)

closed in

and therefore

A u t ill

is the inverse GL(~I) .

c l o s e d in

image

Now GL(tl)

we h a v e t h e r e f o r e

PROPOSITION A u t ~I

= ~0(A,B)}

under this map and therefore

By 7.5.2

Then

~ G L ( ~ I ) / ~ A . (pB

t h i s m a p is c o n t i n u o u s .

8, 1. 3,

Let

92 b e a f i n i t e d i m e n s i o n a l

i s a c l o s e d L i e s u b g r o u p of

is characterized

GL('ll)

.

R-algebra,

Its Lie algebra

by

b(~I) = [ D E s

Aut~I

for every

t ~: IR} .

-161-

Here

s

d e n o t e s the L i e a l g e b r a of e n d o m o r p h i s m s

underlying vectorspace

of

~/ .

D E F I N I T I O N 8.1. 4. D

s

of the

A derivation

D

of

~I is an element

satisfying D(A.B)

= D A . B + A, DB

for e v e r y

The Lie algebra

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 Proof:

A,B

~

is the s e t of

9/ , Let

D E ~(~/) .

exp t D ( A . B )

By

8, 1. 3

= (exp t D . A ) , ( e x p t D . B)

f o r e v e r y A, B E ~/ , t ~ JR,

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 o b t a i n f o r

D(A. B) = DA, B + A . DB Conversely, let

D

be a d e r i v a t i o n of

Dn(A, B) i+~--n (For

n=0

and

this is true,

nl i; j;

~ .

DiA. Dj B

D~

D

t=0 is a d e r i v a t i o n of

~/ .

B y i n d u c t i o n we get i>0,

j >0

b e i n g the i d e n t i t y . )

Now, b y 6 . 1 . 5 we h a v e

O0

exp tD =n=~ (tD)nn ~. Therefore

CO

(tD)n exp tD(A. B) = ~ n--0

(A.B)

(exp tD. A) . (exp tD. B)

-162 -

and

e x p tD

E Aut~l

Remark. algebra

for every

t

~

~ .

B y 8, I. 3 t h i s s h o w s

The fact that the set of derivations

of the L i e a l g e b r a

s

of

~

is a s u b -

follows also directly and is true

without any restriction

on the dimension

suggests,

also,viewing het~ristically the Lie algebra

in t h i s c a s e

of derivations

as the Lie algebra

In p a r t i c u l a r , of f u n c t i o n s

let

X-~ ~

is the Lie algebra

.

X

of

Let now v : G -. A u t X

X .

of

CX

So

DX

be a G - m a n i f o l d

.

the

the homomorphism

of Lie a l g e b r a s

The ad~oint representation for a Lie algebra

Any element

~,

N o w b y 4 . 1 . 3,

(ad A)(B)

L E M M A B. 2.1.

ad A

of

on

X

Aut X

places before.

to an operation

1"* : G -~ A u t C X .

of 5 . 6 . Z .

Consider

It c a n be t h o u g h t o f b e i n g

induced by the homomorphism

of a Lie algebra. ~

. a d A: ~ t -~

.

is a derivation

of

~-* ,

We b e g i n with s o m e

r i s e to a l i n e a r m a p

= [A, B ]

~I .

c a n b e t h o u g h t of a s t h e L i e

over a ring

A E ~'gives

8.1. 5

1R-algebra

of v e c t o r f i e l d s

with respect

It i n d u c e s a n o p e r a t i o n 0":RG -. DX

by the definition

DX

CX

, as we have indicated at several

the homomorphism

remarks

Proposition

of the group of automorphisms

The Lie algebra

of d e r i v a t i o n s

Aut X

~l .

be a manifold and

can be identified with Aut CX . algebra

of

~t~

.

-

Proof:

The J a c o b i

16 3 -

identity

can be w r i t t e n in the f o r m

[A, [Bl, ]32]] = [[A, Bl] , B2] + [B V [.4.,B2]] which proves the desired result. DEFINITION 8.2.2. d e r i v a t i o n of ~

Let

defined by

C o n s i d e r the m a p

~

be a Lie a l g e b r a .

A r ~," is the m a p s

ad:

The i n n e r

ad A : ~ - ~ ~t~ .

into the Lie a l g e b r a

of e n d o m o r p h i s m s of ~ - . LEMMA 8 . 2 . 3 .

ad: ~ -~ s

is a h o m o m o r p h i s m of Lie

algebras.

Proof: T h i s is a g a i n a c o n s e q u e n c e of the J a c o b i a n i d e n t i t y , namely (ad [A1, A2])(B ) = [[-4,1, A 2 ] , B ] = [AI,[A2, B]] - [Az,[A1,

B]]

= ( a d A 1 ~ a d A z ) ( B ) - ( a d A 2 o ad A I ) ( B ) = [adA1, We have s e e n b e f o r e that

a d ( 7 ) c 3(~') ,

the Lie a l g e b r a of d e r i v a t i o n s of o ~ , s

.

We s h a l l a l s o w r i t e

induced by

ad: g,

[D, a d A ] B [D, a d A ]

by 8 . 2 , 6 .

i s an i d e a l of

t h e r e i s to show

for every

(Aut LG)0

g E G,

[D, a d A ]

In v i e w of 8 . 2 , 1 1 t h e r e

L(Aut LG) = ~(LG) Let namely ~ ad ~ .

D ~ For

= D [ A , B ] - [A, DB] = i D A , B ]

,

B ~

(Aut LG)0

,

i s o n l y to show

T h i s is t r u e for a n

~),

A ~ ~ ~

.

Then

we h a v e

= (adDA)B,

The group

Ad G

which shows

i s not n e c e s s a r i l y c l o s e d in

T h e a u t o m o r p h i s m g r o u p of a L i e g r o u p . Aut G

Let

6.3.4).

L : A u t G -, Aut LG

into the g r o u p of a u t o m o r p h i s m s

The functor

L

Example 8.3.1.

C o n s i d e r the L i e g r o u p

Aut

" l r - . Aut (L"II') = G L ( ~ )

= ~*

Aut

"11" = {12r, -1,]/,} , w h e r e

-1T

G

be a L i e g r o u p

defines a homomorphism

~s c o n n e c t e d , 6. Z. 9 s h o w s t h a t t h i s h o m o m o r p h i s m

-l~(see

G

Aut L G .

the g r o u p of a u t o m o r p h i s m s (of the L i e g r o u p s t r u c t u r e ;

however, remember

If

Then

= adDA.

Remark.

by

q.e.d.

.

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

Now

and

H

i s a n o r m a l L i e s u b g r o u p of Proof:

8.3.

;Yg(H) c

of

If

G

is i n j e c t i v e .

"It = ~ t / z

is injective.

LG .

.

Then

In f a c t ,

d e n o t e s the m a p i n d u c e d on

-ff

8.3.4). 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

L : A u t G-* Aut LG

i s an i s o m o r p h i s m by 7. Z. 7.

-168Example

8.3,2.

G = ~

More generally,

let

connected Lie group. AutG

-. A u t LG Let

G

morphism

r

.

G

Then

= GL(LG)

G

the homomorphism

A u t ~t = ~{*

be a commutative exp:LG

-, G

connected and simply

is an isomorphism

and connected Lie group,

defines an automorphism e x p : L G -* G

i n v i e w of t h e c o m m u t a t i v e

.

by 7 . 2 . 8 .

is an isomorphism.

be a c o m m u t a t i v e of

Then

.

Then

L(r L(r

of

An autoLG

.

Consider

ker exp c ker exp

diagram

LG

_L(@)

> LG

e x pi

lexp

G

~

PROPOSITION 8.3.3.

Let

>G

W e h a v e p r o v e d h a l f of G

be a commutative

group.

T h e n t h e i m a g e of t h e h o m o _ m o r p h i s m

consists

of the automorphism

Proof: ~ker

~

LG

We h a v e t o s h o w t h a t g i v e n

exp c ker exp, there exists

(exp,~) ker exp of e x p , r

of

= e

through

L: A u t G -~ G L ( L G )

with

~ ker exp c ker exp.

~ 6 GL(LG)

r 6 Aut G

connected Lie

with

with

L(~o) = cp 9

implies that there exists a factorization exp

and clearly

L(cp) = $

.

But

~:G-* G

-169-

Remark. an arbitrary

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

connected Lie group 9

universal covering group

~

Zn

.

and the coverLng h o m o m o r p h i s m

for

A u t G,

~-. G .

as

We s h o w

PROPOSITION 8.3.4.

Let

Proof:

We d e n o t e b y

T..n

Then

T n ~ LG/Z n

This shows

Aut G

O n e h a s o n l y to c o n s i d e r t h e

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 G ~-- L G / k e r e xp

of

Aut"Jr n ~

Aut Z n

G = 'it n

Then

a s u b g r o u p of

Now

Aut

0 q.e.d.

Tn=[~

LG

AutG

~" A u t Z n

i s o m o r p h i c to G L ( L G ) / ~ ( Z n) c z n ~

-170-

Appendix. Cate_~pries and fu.nctors

Definition.

A category

e

c o n s i s t s of

(i)

a c l a s s of o b j e c t s

(ii)

for e a c h p a i r

A, B, C, . . .

(A,B)

of o b j e c t s a set

are called morphisms from range

;

B (we w r i t e

A

a:A-. B

to or

[A, B ] ,

which e l e m e n t s

B

or with d o m a i n

A

a >B

t h e s e s e t s being p a i r w i s e d i s j o i n t : (A,B)

A

and

for

a~[A,B]

(A',B')

implies

[A,B] r] [A',B'] : r (iii) for each triple [A,B]

(A, B, C) x

of o b j e c t s a map

[B,C]

> [A,C]

(a, 8) ~

~a

c a l l e d c o m p o s i t i o n of m o r p h i s m s ; (iv) f o r e a c h o b j e c t

A

1A e [A,A], c a l l e d i d e n t i t y

an e l e m e n t

m o r p h i s m s; t h e s e data being s u b j e c t to the two a x i o m s (I) If

ae

(Z) If

a e[/%,B], then

Remark.

[A,B],

Be

The m o r p h i s m

1A

is u n i q u e l y defined by c o n d i t i o n Z. with thee s a m e p r o p e r t i e s , t h e n Examples.

The c a t e g o r y

[B,C], alA

~ e[C,D],

=a

then

, IBa = a .

whose e x i s t e n c e is r e q u i r e d by (iv) B e c a u s e if

1A' 1A = l A, Ens

l~ =I A

is a s e c o n d m o r p h i s m .

whose o b j e c t s a r e the s e t s and

m o r p h i s m s the m a p s b e t w e e n s e t s with the u s u a l c o m p o s i t i o n s . category ~

The

of g r o u p s is defined by the g r o u p s a s o b j e c t s , group

)1

-171-

homomorphisms

as morphisms

a n d th e 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 t he 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 a n d th e as morphisms

continuous

maps

w i t h t h e u s u a l c o m p o s i t i o n , we o b t a i n t h e 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 b y t a k i n g t h e 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 a n d differentiable maps as morphisms. Let

a:A -B

be a c a t e g o r y a n d

with

va lenc e

a:A -~B, then

isomorphic : ~A

o b j e c t s of

i s c a l l e d an e q u i v a l e n c e o r a n i s o m o r p h i s m ,

6: B ~ A

a: A

A, B

~a = 1A

A ~

B

.

and

a~ = 1B

A

and

B

.

~ .

A morphism

if t h e r e e x i s t s

If t h e r e e x i s t s a n e q u i -

a r e s a i d to be e q u i v a l e n t o r

An a u t o m o r p h i s m

of

A

is a n e q u i v a l e n c e

.

Definition:

F : e - e'

from

Let ~

to

(i)

of a n o b j e c t

(ii)

of a m o r p h i s m

a:A ~ B

of

and ~' FA

~'

be c a t e g o r i e s .

is th e a s s i g n m e n t of

~'

to e a c h o b j e c t

F a : FA - . F B

of

~'

~ ;

s u b j e c t to the t w o c o n d i t i o n s

(i)

F(I A)

(Z) F(~a)

= IFA = F(~)F(a)

If t he c o n d i t i o n (Z) is r e p l a c e d by

(Z~ F(~a)

A covariant functor

= 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

of

~ ;

to e a c h m o r p h i s m

-172-

Examples.

Let

e

be a category

One can define a covariant hA(X)

= ~%,X]

~p:X-~ X ' ,

hA(x)

for any object

a:A-.X

Similarly

,

X'

a:X'

$

(covariant)

-. GX

from

F

to

G

e' ~'

X

of

K-vectorspaces

K-linear

and

CX

~x(X)(X' ) = < x, x'~% formation

K

hA: e - . E n s b y

e

h A ( ~ ) ( a ) = 040

hA(~p): iX'

.

A natural

e,

9

A]

and

.

for

-. iX, A ]

9

F, G:~-~ e'

transformation

of a m o r p h i s m

such that the following diagram

~ GX

, > GY

be a c o m m u t a t i v e

D: K~ -. K ~

map its dual map) . over

-~ [ A , X ' ]

~p:X -, Y

to e a c h v e c t o r s p a c e

vectorsubspace

for

functor

is t h e a s s i g n m e n t

q~y

Let

h A(cp)(a) = Ca

and

~ .

in the following way:

be c a t e g o r i e s

FY

Example.

an object of

h A ( ~ ) : [A, X ]

of

to

to e a c h o b j e c t

for every

X

and

FX

(assigning

~,

Here we have

9

Let

functors from

commutes

of

H e r e we h a v e

-. A

Definition.

r

X

for any object

A

hA: e -. E n s

we c a n d e f i n e a c o n t r a v a r i a n t

= iX, A]

r

functor

and

KI~ t h e c a t e g o r y

of

t h e f u n c t o r d e f i n e d by t h e d u a l i t y X

its dual space

The functor

K

its bidual.

for

x ~ X ,

E:IK~-. D z ,

field,

X'

D2: K~ - . K~

and to each assigns

to e a c h

The evaluation x' ~ X '

defines a natural trans-

-173-

Natural transformattons A natural

are composed in an obvious way.

transformation

~: F -* G

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 @~ = 1F , ~@ = 1G , 1F formations

F-* F

and

and

G -~G

Product and sums, is terminal K -. T

.

Let

Let

families

qj'a

Kj/j

= qj

i s a p r o d u c t of t h e

~ ~ }

for

T -.T

f a m i l y of m o r p h i s m s

(qj)-~(qj')

j

.

K

in

P(Kj)

with a common is an

o b j e c t in

,

pj: P - . Kj

~ [y

a:Q-*Q'

P (Kj)

is an object

for

The product,

like any terminal

9 (Kj)

P

of

j ~ Z , such that any

c a n be w r i t t e n a s

be a c a t e g o r y , K

i n d e x e d by a s e t

of

A terminal

of (Kj)j

In p a r t i c u l a r ,

i s u n i q u e up to a n e q u i v a l e n c e in

each object

iT , and any two

qj: Q -~ Kj

up to a n e q u i v a l e n c e in

R

is

whose objects are indexed

of m o r p h i s m s

A product

together with morphtsms

Let

of

Kj , t h u s

DEFINITION 9

a:Q-~ P

T

there is exactly one morphism

P (Kj)

Q , while a morphism

for which

P

K

An object

b e a f a m i l y of o b j e c t s o f ~

the category

[qj:Q-.

domain

unique

be a category,

~ are equivalent.

(Kj)j ~ Z

Consider

such that

respectively.

, if to e a c h o b j e c t

objects in

~: G -* F

denoting the identical natural trans-

Hence the only morphism

terminal

Z .

1G

is a natural equivalence

qj = p j a

for a

object, is unique

the product-object

~ .

An object

S

of

there is exactly one morphism

R

is initial,

S -~ K

.

if to

Hence the

-174-

S --. S

only m o r p h i s m

in

1

and any two initial objects a r e equi-

va lent.

Let (Kj)j s g g .

be a family of objects of

C o n s i d e r the c a t e g o r y

~9k:K j - ~ R / j a morphism

s

} ofmorphisms (pj)-,(pj')

apj : pjl

for

ofthe

thus

Kj,

j g ;~ .

DEFINITION,

f a m i l y of m o r p h i s m s

in

indexed by a set

w h o s e objects are indexed families

of

~'(Kj)

with c o m m o n is an

a:R--R'

range

R,

while

for which

A n initial object in this category is a s u m

A sum of

t o g e t h e r with m o r p h i s m s

unique

~(Kj)

~

(Kj)j g g

0-j: Kj - S

9j:Kj -R

for

is an object j

~

~Y o

can be w r i t t e n a s

S

of

such that any a~j = pj

for a

a:S-~ R . The sum is unique up to an equivalence 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 equivalence in

~ .

-175

-

B IB LIOGRA PHY

[1] Bruhat, F . ,

A l g ~ b r e s de Lie et g r o u p e s de Lie,

T e x t o s de

M a t h e m a t i c a , Univ, do R e c i f e , Vol. 3 (1961). EZ] B r u h a t , F , ,

L e c t u r e s on Lie g r o u p s and r e p r e s e n t a t i o n s of

l o c a l l y c o m p a c t groups.

Tata Institute of f u n d a m e n t a l r e s e a r c h ,

B o m b a y , 1958, [3] C h e v a l l e y , C . , T h e o r y of Lie Groups, Vol, I, P r i n c e t o n Univ. Press,

P r i n c e t o n , 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 ] G r a e u b , W., L i e s c h e G r u p p e n und affin z u s a m m e n h ~ n g e n d e Mannigfaltigkeiten,

Acta Math, 106 (1961), 65-111.

[61 H e l g a s o n , S., D i f f e r e n t i a l g e o m e t r y and s y m m e t r i c s p a c e s , A c a d e m i c P r e s s (1962). [7] Hof/man, K. H , , Einfi~hrung in die T h e o r i e der L i e g r u p p e n , T e 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. J n s t , Universit'~tt

Tilbingen (1963), ES] Koszul, J. L . , E x p o s e s s u r les s p a c e s h o m o g ~ n e s s y m ~ t r i q u e s . I1. Soc, Math, S~o Paulo ~959).

[91 L i c h n e r o w i c z , A . , G ~ o m e t r i e des g r o u p e s de t r a n s f o r m a t i o n s , Dunod, P a r i s (1958). El0] M a i s s e n , B . , L i e - G r u p p e n m i 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 transformation groups, Memoirs AMS, Vol. 2Z (1957). [14] Pontrjagin, L. S . , Topologische Gruppen, VoL II, Teubner Leipzig (1958).