Concrete introduction to classical Lie groups via the exponential map

382 112 899KB

English Pages 74 Year 2002

Report DMCA / Copyright

DOWNLOAD PDF FILE

Recommend Papers

Unitary Representation Theory of Exponential Lie Groups 9783110874235, 3110139383, 9783110139389

193 37 7MB Read more

Lie Groups and Subsemigroups With Surjective Exponential Function 0821806416, 9780821806418

115 101 2MB Read more

Lie Groups Beyond: An Introduction [140.0] 9781475724530

103 45 Read more

Lie groups and compact groups 9780521213400, 0521213401

The theory of Lie groups is a very active part of mathematics and it is the twofold aim of these notes to provide a self

428 59 4MB Read more

Lie groups, Lie algebras, and representations: an elementary introduction [2nd edition, corrected version] 9783319134666, 9783319134673, 3319134663

Table of contents : Part I: General Theory.-Matrix Lie Groups.- The Matrix Exponential.- Lie Algebras.- Basic Representa

101 68 7MB Read more

Lie groups, Lie algebras, and representations : an elementary introduction [2ed.] 3319134663, 9783319134666, 9783319134673, 3319134671

Table of contents : Front Matter....Pages i-xiii Front Matter....Pages 1-1 Matrix Lie Groups....Pages 3-30 The Matrix Ex

117 5 4MB Read more

Lie Groups, Lie Algebras, and Representations: An Elementary Introduction [2 ed.] 9783319134666, 9783319134673

Table of contents : Front Matter....Pages i-xiii Front Matter....Pages 1-1 Matrix Lie Groups....Pages 3-30 The Matrix Ex

120 56 7MB Read more

Lie Groups, Lie Algebras, and Representations: An Elementary Introduction [2 ed.] 9783319134666, 9783319134673

Table of contents : Front Matter....Pages i-xiii Front Matter....Pages 1-1 Matrix Lie Groups....Pages 3-30 The Matrix Ex

115 5 7MB Read more

Isometric actions of Lie groups and invariants [Transformation groups]

100 20 589KB Read more

Introduction to Groups and Geometries

103 75 5MB Read more

Concrete introduction to classical Lie groups via the exponential map

Author / Uploaded
Gallier J.

Similar Topics
Technique
Construction

0 0 0
Like this paper and download? You can publish your own PDF file online for free in a few minutes! Sign Up

File loading please wait...

Citation preview

! "# $ $$

½

!" # $ % # % Ê ¾ Ê ¿ " & ' ( ( % ) ) % *

'

'

#

+

Ê " Ê " " " Æ " Ê " " Ê " " " " % " # # , - % % % $ " % " # . */0" 1 % 23 + %4 Ê ¾ Ê ¿ # , " " ,

% # Ê 5 / 6" Ê " 2 4 (

" " # , " 7 !" #

"

-

8 9℄ & 8(℄ ; % 8

' >

Ê ½

)

@ " ' > ' > A Ê

½ " ½ "" ' ½ ½ ? ¾ ¾ ? ?

& A & = =

" ' " *

' B A = =

" ' " ,

¾

½

¾

½

B A =

Æ

'

Æ

& = ' ' Æ ' Æ '

+, % +, = " = % % + " +

+, % +, + & %

C

=

D

C '

-

" " ' "

*

"

*

"

Æ ' Æ " ' Æ " " Æ " " ' " " '

,

" " ' > " " ' >

" " ' "

" ' >

" ' > " ' >

"

" ' >

5 C ' C

= ( " # " ' - ( " " Æ

- & % &

B

½ ¾ " ? ½ ¾ " ' ½ ? ½ ¾ ? ¾ "

? "

' ?

" ' ? "

7 7

> " ' > " 9

> " "' ? ' , B ' ? 7 ½ " ½ > " > "" # B = =

? "' " ? " 6

.

&

= +

" "

" > ' >

= ½ ? ¾ ½ ¾

"' "? "

½ ? ¾ " ' ½ " ? ¾ "

" ' " "

" ' "

. "

"' ' > " >

=

- "

# @= ·

@ "' "

> > " ' >

" ' > ' >
" " ' > =

E

' >

! =

'

" * (

" ' > " "

"

A &

. " A 7 = ' ? = ' ? Ê

? " ' " ? "

? " ' ? " ? " ' ? ? " " ' " ' ?

" ' " " ' '

H

! = ? ' ? ' ' >

'

Ê ' >

!

!

" '

" ' # " $

I

% ' >

%

! ' >

! ' > ' >

"

"

' > ' > - , ( > . ' > ? ' (E ? ? ' ? ( ' > ' > ' ' > ' >

* ' ?

*

" ' ? ? (

" ' ' > ' > 7 ' > ' >

" " 6 ' > 1 "'

'

( > = " ( 7 1 . #

! = ½ " $

$

>

½

¾

$

$

'

>

#

7 '

& ( - " = ' ? " ' ? ? " #

( >

7 ' > ' ' > ' 7

" " (I " ;

7 ' > ' > ' " ' " ' # % ½

½

" 7 > ½ ' ½ ' ½ ' ¾ ' 7 > ½ ' ½ ' ½ ' ¾ ' 7

½ ¾" ½ ½ ½ ' ½ ½ ½ ½ ½ > . ( (I " *

& % +, % +, .

% " % . $

! = ½ "

$ $

½

¾

# 7 '

& ( * " = 5 ; " ; ( ' > ' > ' >

6 % ( ' >

% +

) ½

' ½

)

'

& +,

# . B %

+ @= Ê

@

" ' "

7

@ " '

$

@ " @ " ½

@ " ' @

?

;

@ " ' (

" "

+,

'

"

@

½ ' ½ " Ê ¾½ ? ? ¾ '

* ½ @ @ % ½ ; % @

Ê

*

@ " ' @ " ' >

½ @ "

@ " ' (

@ " '

"

(

(

" '

"

6 % +,

! = ½ " $

$

D

Ê > >

¾

'

>

>

%

!

' & ( ' ? Ê # ' > -

> $

½

" ' >

7 '

" ' " ' " ' " ' "

" ' >

& " ' " ' ?

" ' ' > ' " ' ? ' ? > '

*

? " ' >

' > ' > ' >

% ( ' > (

% + > 9

% +, +, ; (G >

-

! = ½ " $

$

½

¾

!

! ! > ! "

' & ( ' ? * ' ' 7 7 "

$

! ! ' !

%

Æ

Æ

" " ' ' " " ' Æ " " '

' ' > ' ' & * ! ! ( > ! > > ! " 7 " ' ? "

+ %

¾ Ê > % ! >

G

# ! $ $

$ '

&

& & &

½ &¾ &'

&

&

%

$

! !

& '

!

!

! ! > ! "

#

B # '

$ "

#

$ ' $

#

' ' $ # B #

&

'

"

!

'

"

'

'

' '

I

' %

# '

' "

7 . ½ " = ½ " , 5 + , . % +, % . 7 . ½ " ½ " = ½ " ½ "

, ½ " ½ "

- " " = " " ½

½

½

½

' ( ( #

"

# ! $ $ ! $ (

$ & ' (&( # & $

&

!

&

!

$ % & '%&( ) *

7 A = 7 $ .

+,

Æ

Æ " " ' " " ' Æ " "

Æ +, Æ Æ +,

Æ Æ $ 7 Æ > 7 Æ " " ' " "

*

Æ " " '

(>

> & Æ Æ > > Æ

Æ Æ

' " " ¾ ½

B = " ' #

¾

)= " '

" ) Æ %" ' " Æ ) % " ) Æ Æ ) ! > Æ Æ " & +, = > 7

# , Æ Æ +, + "

" > - % Æ Æ 3 = 7 " ½

½

7 " &

C " ? 7 " ' " " ? " ' "

!

# ! =

Æ Æ

C ' C " C ' C " C ' 7 " C ' 7 " 7 " ' 7 " C " ' C "

Æ

Æ (

" ' "" ' " 7 " ' > Æ " > " ' > C C Æ " ;

" " ' Æ " " " ' > 7 Æ " " ' > C Æ " C C ' C Æ " C ' C Æ " ;

" ' "

C ' 7 " C ' 7 " *

7 " ' C " 7 "" ' 7 " C ' 7 " C " ' 7 ""

7 " ' 7 " C " ' C " -

7 " ' 7 Æ "" ' 7 Æ "" Æ Æ % +, ! = *= ' *¾ ' * Æ * # C ' C *

½

*

¾½ ¾

¾

((

*

+, ) ½ "

½ %

Ê * > ' ¾ > 7 *= ½ "

" '

*

' *¾ * +, ½ " " 7 * ) = +, ¾

' ) ½ " )

½ ) > 6

" ' " '

)

¾

) )

"" '

¾

¾ 7 ½ ) ½ ¾ ( . ,

) "

' ½

' ' ( * > ' ¾ ' ¾ ¾½ ¾ ½ ) ' * * & C ' C * ½ * ¾½ ¾

# + * 8(9 (D℄ + Æ #

%

7 ;% 8I G()" !" * * . # * 1 % 44 . */0 * 1 % + % I>E" #1 % I (" &

I>( I 9" A % K */0 I)" &

" 7

" 7 ' C # ) = " = " ' ) # " ' ) * "" ' ) " ' ) " ' ' " ' " )Æ* " + " ' ) * "" ' ) > " ' > )Æ* " + ? " C ' C * "

' ) Æ * ) " " # ' * Æ ) 6

" ' Æ " " * ' Æ Æ "" " ' Æ* " "" ' * " ' " ' ' + " ' Æ " " ' "" ' > * + ?

" C ' 7 " * * ½

½

½

·½

½

½

½

½

½

·½

½

½

½

½

¾

¾ ¾

¾ ½

¾ ½

¾

¾

¾ ¾

·½

+, *¾¾ 9(

*¾ " ' (9

¾

+

*¾ " ' > + ? ½ " *¾ ½ ·½ ' ' ' >" *½ *¾ ½

&

' " *¾ Æ )" " ' *¾ ) "" ' *¾ " '

*½ *¾ Æ Æ +, Æ ' *¾½ Æ ' * ½ *½ *¾ ½

Æ Æ - *½ *¾

¾ ¾

*

Æ ' *¾½

9( *½ *¾ Æ ' Æ *½ ' *¾

Æ ' *¾½ ) ) ' Æ *½ ½

7

*

7 9D -

' -

& -

"

' (

'

'

'

' %

7 . 9) ) 7 ( . ,

' (,

(

6 > " B

A ½ " % " > % ½ " ½ " 7 " ' > , M 7 " > > ( - 1 1

(E

* " ' > , M M 9D "

$ (

$ & ' - & ( & # !

½

¾ &'

-

½

·½ ' ' ' > '

# " >

" ' " ' ?

(

&

-

( & - "

(

(

-

(

-

' - & ( ()*

7 * 8(9℄ ( - & ( ' .¾ - ' .½ & ' N */0

' .½ & . ¾ .½

.¾ ' .½ & .¾ .¾ " .½ " */0 7 ( - &

' -&(

& ½ ½ ·½ ' ' ' > 7 */0 % 7 ' ½ ½ ¾ & ' ¾ & ¾ ' ½ ¾ & ¾ ' - & (

-

' ½ ¾

( ' ¾ (G

B */0 ' - & ( ' - ( ' (& ( 7

' - ( (& ( ' - & ( ' 6

" ' ?

" >

99 # * 8(9℄" &

9) 9D $ = A 7

= ; ,

= =

" ¾ ' " ½

" "¾ ' Æ " " ½

' " # ' " / ' # " % / / ½ " C " ½ " ""

Æ '

C " = ! % / ' # "

Æ ' C " Æ ' > C 7 # % / ' # " "

' %

#

#

7

'

%

>

#"

> >

' %

%

# # ' " #%

(I

!

# ! = *½ =

*¾ =

! ) =

Æ

Æ

' ) *½ ' *¾ ) # + *½

*¾ ½

# ! ) *½ *¾

*½ ' *¾

Æ

Æ

*¾ =

Æ ' *

¾ ½

Æ

*½ = ' *¾¾ & 9)

; 9( " +,

C ' C *½

+ % ½ " *½ ½ " ½

*½ " ·½ " C ' C *½ ½ " C " ' 7

" '

BF* $ " + " + ' >" &

" 7 "

½

½

½

·½

7 ' C

/ ' # " ) =

" = " ' ) / " ' > ) / ? 6 + / L 9) " ' " )Æ* " + " ' ) * "" ' ) > " ' > )Æ* " + ? " C ' C * "

' ) Æ * ) % "

" ½

½

½

½

·½

½

½

½

½

)>

#

' *¾ Æ ) L 9) *¾¾ " ' ¾

+

*¾¾ " ' Æ " " ' "" ' > + ? # ·½ " C ' 7 " * *¾ +, *¾¾ 9( *¾ " ' + *¾ " ' > + ? # ½ " *¾ ½

·½ ' ' ' >" *½ *¾ ½

&

*¾ Æ )" " ' *¾ ) "" ' *¾ " ' ' " # * *½ *¾ Æ Æ +, Æ ' *¾½ Æ ' *¾¾ ½ *½ *¾ ½ Æ Æ - *½ *¾ Æ ' *¾½ Æ ' *¾½ 9( *½ *¾ ) ) ' Æ *½ ½ 7 * Æ ' Æ *½ ' *¾ 7 9< - #

% #

' # # $ & " ' 7 . 9< % ) % 7 # % # ( . ,

' (,

)

9< ½ " ½ " ½ " *½ ½ " + *¾ - ½ " 7 ·½ " C ½ " 7 ·½ " C H = +

!

# ! = ½ "

½ " + $

# $ &

½

¾ &' > > >

½

¾ &'

> >

> >

> >

> >

>

Æ Æ

' ' > / ' # " # ½ " Æ ½ " Æ

" ' / ' # " A & # $ & 9E

½

·½ '

"

# ! # $ ( "

- # #"

# $ & ' - & ( &

½

¾ &' > > >

½

¾ &'

> )(

> > > >

> >

> >

½

·½ '

' ' > ' " '

(

-

/

#

& ( & - " ' - & ( ()* */0 + * 8(9℄ 9< + - # B # % # ½

' ½ * ¾ &

' ¾ & ¾

' ½¾ & ¾ # ½ ¾ % 7

½ ¾" ½ ¾ " ' ¾ ½½"¾ #

½½ ' %

½ ¾" ½ ¾ " ' % ½ ¾ % # - */0 ' - & ( ( - # # & #

- # - # $ & & # $

- & ' - & # ' - & ( ))

( - # % & & % # - ( &

' - & ( # B # # # % # ½

' ½ * # # ¾ # # &

' ¾ & ¾

' ¾& ¾½ # ¾½ % 7

¾½ " ¾½ ' ½ ¾ ¾"½ ' ½½ ¾ #

½ ' ½

%

>

> >

> ¾ ¾ > > ¾½ % * # ¾½" ' ½¾ % & # % # - ( &

' - & ( # # # - % # ( ( % " # # &

' - & ( 7 - ( ' & %

½ " ½ '

)D

+ , -. 7 2 $4 7 Æ ; " & 8(℄ 5 8I℄ 3 8 E℄ 5 8 >℄ . 8 D℄ *

# 8(>℄

B "

' % ?

$ " %

'

/!

'

/!

' % O

+

'

$ " $ $

' $

%

'

$ "

$ " $ /!

,

!

$

'

/!

$

/

-

&

$ '

/ ' >

' %

" '

$ "

$ $ $ $

)9

$

" '

"

-

" $ "

$ /!

'

" / !

'

0!

7 & % '

! ! >

>

# >

!

!

>

'!

>

>

!

>

!

>

' !

>

>

1'

>

>

' %

?

' ! % ' ! 1 ' ! % ' ! 1

! 1 ! % ! 1 ? ! % ! (! )! D! )
3 > 3 ' $ ? ' " ' 3 % #

3 3 3 3 3 3 ' % ? ! (! % )! ? D! % ? 9! + " ' ( $ ? ' ' > &

+ " ( ( -

$

'

$ > $ '

>

>

$

& ½ -

'

+

(

$

½

' ( (

'

( (

"

&

( (

)G

(

½

( ( ½ " ( ( ½ ½ ½ ' /! ' /! ' ( /! ( ' ( ( ¼

¼

*

¼

$

$½ ½ $½ ¾ $½ ¿ $½ ½ > $¾ ¾ $¾ ¿ $¾ ½ > > $¿ ¿ $¿ ½ > >

> >

$½ $¾ $¿

$½ ½ $½

> $

> >

! $ $ $ $ 4 ' 4 ½ $ ' >

#

= ½ " - ' 7 ½ ½ # ¾ ½ "

½ ¾ "

¾ " 7 ½ ¾ "

$½ ½ $½ ¾ $½ > $¾ ¾ $¾ >

$ ¾ $

½ ½

½ ¾ "

/= , / ½ " ' > / " ' (

) = / Æ " " $ "¾ ¾ " ;

¾ " ) ' "½½ .

'

½

)I

" * / , ' )= /

Æ " '

" ' $ ? '

$ ; " ' 4 4

7 . $ $ > > $ > >

> >

> >

" "

>

' 4 4 #

D>

$ $ $

$

! $ $

½ #

; & " *

%

'

'

½

'

!

!

! !

' ½ ' ½ ' '

,

# ' ) % " % % . - ) ) %

> ' ' > $ ' $ >

! ' $¾ ? '¾ ? ¾

¾ $ $' $ ' $' '¾ ' $ ' ¾

. - /012 ' $ = )"

!

! !

' ! %¿ ?

' %¿ ?

!

!

!

!

! ' > ¼¿ ' %¿

?

?

!" !¾

!" !¾

-

¾

' ! ¾ % ? ' ' > D

- =

"

"

, ! L * " '

=

" "

, ! & = "

= "

"

"

,

' $ = "

"

= "

"

-

- %

'

% .

" ' " ' '

" ' ' ' ' % " ' % * " '

% . " ' ?

- , DE - ; DE ( & ' (&( - & "

9

? & ! ? ! ' &

&'

½

¾

7

!

!

'

!

% .

' &

' ( (

'

% . 7 ?

½

' ½

! ?

? ! ' >

% . $

&

' ( (

'

% .

# G .

9(

½

6 " " ) 3 &4 -. 3 . " # . " . " . "

!

!

!

. . & . % 8 ℄ . . "

# ! # ! '

$ $ $ $

7 > .

' (&(

& (

!

!

> = " " , 7 7 P , "

" " " & P > = - ( .

!

' (

!

!

, " " " ; " ¾

, " " ¾ 7 , 7 "

" " "

7 ' ( ( " ' ( + " . "

" > " ' " ' " " , " " " ""

!

9)

!

7 " Æ

# + , (" )"

, 0 - 1 - Æ Ê

Æ

?

"

?

"

9 Ê

9 " ' ? ( "" ( Ê

Æ " A

(

?

?

"

"

>

9 "

'

>

9 " ' ? (

'

Q %

(

? Q >

"

( >

?

"

( Ê "

" "

# = " " , -

%

9D

? "¢ ?

!

" $

(

Q >

'

>

( ¾ Ê

Q ! $

%

Q ' ,

-

&

- ( >

'

Q ( >

Q >

'

' % ?

Q 0 ? "!

0

Q ( >

Q >

'

'

0!

' % ? ' '

% ?

Q >

0!

Q ( >

>

- ( >

½

(

# #

% 7 -

-

' Q

% "

99

-

Q

H % ! - . =

Q - ' % ? ' 0 ? "!

D ,

' $

"

= "

* Q % E( +

= "

"

, % Q ' - ' % ?

Q 0 ? "!

; DD % Q Q ' & & %

& &'

&

&

% & > + & '

> !

! >

! ! > & ! ' 0 (" 0 ! ' 0 (" ' % & + %

$ - Q ' & ' &

- ' % ?

Q 0 ? "! 9
Q > " ' / = Q ( /

3 ; Ê ( ; ( ' ; Ê * = Q ( = ( Q * Q Ê / ; $ = Q ( ; / /" ' " Ê ½

½

½

½ / ½ # ; #" 7 2M 4 - (+ " -

# # ' > ( 7 > ,

7

Ê *

& (+

¾ ' "

Ê ¿

¾

¾ Ê ¿

? ¾ ? ¾ '

(+ $ ¾ = ( ( ¾ ? ¾

½ = " ¾ ? ¾ ? ¾ ? ¾ ? ¾ ? ¾ ?

½

¾ = "

( ( ¾ ¾ ¾ ? ¾ ? ¾ ? ¾ ? ¾ ? ¾ ?

½ , 6 5 ' > > " ' > ¾

, * ' > > " ' > -

5

<

½ "

" '>

¾ "

- = 7 ,

# % ½ $ ¾ 5 ¾ $ ¾

" H

½ $ % <

' > ½ 7 % ½

2 4 - % % ! #

A 7 ¾ $ 7 "

" " # "

" '

Q

!

> "" ' ;

(( 7 %

$

# ;

/ ; ! ( ( '

!

= Q (

= Q ( ; /

Æ = ( ( " ( ( " "

3

Æ = ( () - (

( " ( ( "

Ê * = Æ =

= Ê " ¼ > " , ¼ > " Ê ,

& $

%

8

= %

;

8 1

1

8 1

;

%

=

/

+ Ê - ¼ ;

4 ;

/

Ê A

;

Ê >" = ' >" #

/

=

/

=

4 ;

4 ;

;

' "

, ' ( , ' >

(G : Q " > Q "

% = Q

&

&

Q " '

" ( 7 ? " " ? " "

(

>

>

'

- ? ( >

>

½

½

(

& " " ;½ #½ ;¾ #¾

;½ ;¾ #½ ? #¾ " " " ? "

A " " + " " " 7 " "

B % (G :

+

& + 8 ℄= + $ ' = 8$ $℄ ' >

+ 4 ! = 8$ 8' ℄℄ ? 8 8$ '℄℄ ? 8' 8 $℄℄ ' >

" ' % # "" ' ℄ > > 8 H

' >

""

>"" ' >

7

" ' + " + >"" ' > ' >" & Ê "

" '

" ËÇ " >" ' %

Ê * " '

' + " ' > 5 Ê "

* "

" " ' %

> > 8 % ' >

℄

>" >" ? >" >" ' >

>" ' % ' >"

>" ? >" ' >

' >" % * %

! ' '¾ ? ¾ ? ¾ '

¼

¿

!¾

?

!" !¾

¾

> ' > ' '

¾

? ! ¾ %¿

>

¼¿ ' %¿

H " " & "

" & " " " * I 7

>

# - ; ; # + # + Ê & = ; ; / ; $ = Q ( ; / ? = Q ( ; /" (" (

? Æ Æ = Q Ê ¾

H () 0 ( > $ = Q ( ?= Q ( & " ,

# ; ; # + # + Ê - = ; ; / ; ¼ = 4; 4 ;

/ / " / = 4; = = % ; = >" ' / ¼ = >" ' ¼ " ' = "¼ >"

Æ

¼ 4 0

= ' " = " = ' = "

Æ

Æ Æ

Æ

Æ

Æ

"

() ¼

= 7 ¼

- E>

B :

½ :¾ + = :½ :¾ :½ :¾ " B ½ ¾ + = ½ ¾ ½ ¾ %

½

℄" ' 8

8

"

"℄

& + , ½ + ,

½ 7 = :½ :¾ ¼ = ½ 7 ¾ :½ :¾ +

"

¼

& 7 :

4 : % ' 4 : % # M + ,

+

$!

B : $ : 7 = : : 7 '" ' $' ' : = : :

'" ' '$ ' : 7 ½ 7

½ 7 " 6

Æ

½ 7 '" ' $'$ ½

½ 7 "¼ =

½ 7 % = : "

"

:

$

" ,

7

$" " ' " ' $$ ½ E

=

$ :

= ¼

% $=

"

"

" "

7

$

"

" ' 8 ℄

% L

$ % $ =

$ 8 ℄" ' 8$

"

$ "℄

% " % $ % %

, " # % 8

℄ ' $

"

"

= : - +

3 Æ " 5 8 >℄ *

# 8(>℄ 3 8 E℄ 5 # 8(E℄ 5 8I℄

* * 8( ℄ * 8 I℄ - & 8 ℄ # 8( A = B H * / 8D℄ ; ; B 6 6 "6 7 6 6 5 S P II( A = 0

B 6 9 * / 89℄ LA ; ,8 6 6 6 IG 8