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8/14/2019 Klein .
1/21
1 Klein
- Klein
, Klein (~1870)
, . , , Hilbert , .
, , :
1. .2. (
. ) R n.
. ( . . ) Minkowski .
KLEIN
() } 1"-"1f XX:f /f { .
, ()
.
:
8/14/2019 Klein .
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2 Klein
f () f -1 , 1-1
f f -1=id x idx(x)=x Xx 1-1
.
,
f , g 1-1 , f g 1-1
f g . idx =f Xf f idf x = () , (G, ).
Klein:
(G, ) , Klein
, (G, ) (() , ).
Klein;
Klein .
(:=
)
G .
G;
G , f( )
f G.
:
8/14/2019 Klein .
3/21
3 Klein
J (:= )
( P( X) )
(J)={A P( X) : J(A) }
J f : X X
f(A) (J).
: J G
, f G.
G .
,
:
:
,
.
, ( u ,
f ,
u =: f(u) )
.
8/14/2019 Klein .
4/21
8/14/2019 Klein .
5/21
8/14/2019 Klein .
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6 Klein
. .. .
, ,
.
. .
,
,
.
. :
, .
, :
,
. ,
.
,
, .
, dE
, ,
8/14/2019 Klein .
7/21
7 Klein
d
2 , )( E d ,
:
2 2( ), )d
:
2
2
1 ,2 f :
. f(1)=2
.
8/14/2019 Klein .
8/21
8 Klein
, 1 2
, g 1
2
. g : g( 1)= 2
~ 1~ 2
.
: ~
:
:
1 ()= ,
1 : .
1 G G . . : S(X) .
1~ 2 . 1 2: ( ) g g = . g
1-1 , , 1 1 1
1 2
11 2
11 2
2 1
( ) ( )
1 ( ) ( )
( )
~
g g g g
g
g
=
=
=
8/14/2019 Klein .
9/21
9 Klein
g-1 G , G
( S(X) )
~ ,
1 2 2 3
1 2 1 1 2
2 2 3 2 1 1
2 1 1 2 2 3
1 3
~ ~
, : ( )
( ) . , ( )
( ( )) ( )
~
g g G g
g T g g
g g g
=
= = = =
2 1 g g G g1 g2
G
.
,
G , .
;
,
g
.
R n n >3 .
R n n
.
8/14/2019 Klein .
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10 Klein
R n .
;
,
,
:
f: V V ,
[ , ] V , ( [ , ]=[f(x),f( )] , V)
:
:F(x+ )=f( )+f( ) f( )= f( )
,
: [f( ) f(x) , f( )- f( )]=0 . (1)
,
(1) f( )- f( )=0 f( )= f( ).
, :
[f( ) f(x) , f( )- f( )]=
[ f( ) , f( )- f( )]-[ f(x) , f( )- f( )]=
[f( ) , f( )]-[ f( ) , f( )]-[ f(x) , f( )]+[ f(x) , f( )]=
[f( ) , f( )]- [ f( ) , f( )]- [ f(x) , f( )]+ 2[ f(x) , f( )] =
[ , ]- [ f( ) , f( )]- [f( ) , f(x)]+ 2[ f(x) , f( )] =
8/14/2019 Klein .
11/21
8/14/2019 Klein .
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12 Klein
[ , ]+[ ,]+[, ]+[,]-[ , ]-[, ]-[ ,]-[,]-[,]-[ , ]+
[x ,x]+[ x, ] +[ ,x]+[ ,]=0
.---
[,] f f:X X ,
:
( ) f
( ) f f(0)=0
:
( ) ( ) > &
:
f(0+0)=f(0)+f(0)
f(0)=f(0)+f(0)
f(0)=0
( ) ( ) > f ,
.
8/14/2019 Klein .
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13 Klein
,
, !
, , f
.
:
( ) ( ) , f x f y x y x y X = (1)
=0 (1) :
( ) (0) 0 ,0 f x f x x X = ( f(0)=0 )
( ) 0 0 ,0 f x x x X =
( ) , f x x x X =
f .
, y X :
8/14/2019 Klein .
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14 Klein
2
2
2 2
2 2
2 2
[ ( ) ( ), ( ) ( )] ( ) ( )
[ , ][ , ] 2[ , ] [ , ]
2[ , ] ( )
( ) 2[ , ] ( )
:
[ ( ) ( ), ( ) ( )] ( ) 2[ , ] ( )
f x f y f x f y f x f y
x y
x y x y x x x y y y
x x y y f
f x x y f y
f x f y f x f y f x x y f y
= =
=
= + =
+ =
+
= +(*)
:
2 2
[ ( ) ( ), ( ) ( )] [ ( ), ( )] 2[ ( ), ( )] [ ( ), ( )]
( ) 2[ ( ), ( )] ( ) (**)
f x f y f x f y f x f x f x f y f y f y
f x f x f y f x
= + =
+
(*) (**) :
[f(x),f(y)]=[x,y] f ,
.---
8/14/2019 Klein .
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8/14/2019 Klein .
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16 Klein
( ) ( ) ( ( ) (0)) ( ( ) (0))( ) ( )
g x g y f x f f y f f x f y x y
= + + = =
g .
, g .
. f : = f(0).
g
f , :
( f g)(x) = f (g(x))= g(x)+f(0)= f(x) x V
g
f
:
, g1 f b
:
g1 f b =f
:f(x)=f(x)
(g1 f b)( ) =( f g)(x)
g1( )+b =g(x)+a (1)
8/14/2019 Klein .
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17 Klein
g g1 ,
g1(0)=0= g(0) (1) :
0+=0+b a=b. g1( )= g(x) V.
.
,
, nxn
g
V. dimV=n :
1 11 1 1 11 1 1
1
1 1
n n n
n m mn n m mn n
x a a x a x a x
g x
x a a x a a x
= =
: f: V V
, ( ) x f x x V =
: f , :
0 ( ) (0) x f x f x V = f , f(0)=0
8/14/2019 Klein .
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18 Klein
0 ( ) 0 x f x x V =
( ) x f x x V =
: :
( ) x f x x V = , x-y
:
( ) ( )
( ) ( ) ( )
f x y x y f
f x f y x y f
=
=
g : R 2 R 2
(2 2) :
( , ) ( , )
x a ax y g
y y
g x y ax y y
+ = = +
= + +
8/14/2019 Klein .
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8/14/2019 Klein .
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20 Klein
2 2 1
x a b ax y b f
y y c y c
x a b ax y b f
y y c y c
a
+ = + = + + + = + = + + + = = = b=c=0 .
:
x f
y y
=
.
=1 , =0 ,
f(x,y)=(x,y)+(b,c)
=1 , =0 b=c=0
f(x,y)=(x,-y).
,
, .
, f(x ,y i)=g(x ,y i) i=1,2,3 , f=g.
8/14/2019 Klein .
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21 Klein
, , b, c
.
, ,
.
.-