実特殊線形リー群の基本群

平成

22

実特殊線形リー群の基本群

広島大学理学部数学科 B070930 飯田隆太 指導教員 田丸博士 准教授

平成 23 年 2 月 10 日

目次

1

2

2

3

2.1

. . . . 3 2.2

. . . . 4

3

10

3.1

. . . . 10 3.2

. . . . 11 3.3

. . . . 14

4

24

1 はじめに

私は数学を学ぶ中で関手に興味もったのでその一例である基本群を取り上げる

.

.

,

,

.

.

第

2

,

,

.

,

,

.

,

,

.

,

.

第

3

,

.

,

,

.

第

4

,

,

.

2 準備

2.1

この節では

,

,

.

,

,

.

定義

2.1.1 R

: M (n, R ) = { (a _ij ) _{i,j=1, ··· ,n} | a _ij ∈ R} .

2.1.2 M (n, R )

:

GL(n, R ) = { A ∈ M (n, R ) | det A 6 = 0 } .

R

.

,

SL(n, R ) = { A ∈ GL(n, R ) | det A = 1 }

,

.

定義

2.1.3 R ⁿ

:

O(n) = { A ∈ GL(n, R ) | ^t AA = I n } .

.

,

SO(n) = O(n) ∩ SL(n, R )

.

GL(n, R ), SL(n, R ), O(n), SO(n)

,

.

,

.

定義

2.1.4

X = (x _ij )

,

:

|| X || :=

v u u t X ⁿ

i,j=1

( || x ij || ² ).

M (n, R )

2

A, B

d(A, B) = || A − B ||

, M (n, R )

.

,

.

, M (n, R )

GL(n, R ), SL(n, R ), O(n), SO(n)

,

.

GL(n, R ), SL(n, R ), O(n), SO(n)

,

,

,

:

群 弧状連結性 弧状連結成分の個数 コンパクト性

GL(n, R )

2

SL(n, R )

1

O(n)

2

SO(n)

1

2.2

この節では

,

,

.

,

.

X, Y

.

定義

2.2.1

1.

I = [0, 1]

X

u : I → X

X

.

, u(0)

u

, u(1)

.

u

u(0)

u(1)

. 2. x 0 , x 1 ∈ X

,

x 0

x 1

u : I → X

Ω(X, x 0 , x 1 )

で表す

.

3.

u ∈ Ω(X, x ₀ , x ₁ )

, u ⁻¹ (t) = u(1 − t)

u

. 4.

u, v

X

u(1) = v(0)

,

u ∗ v

u ∗ v =

½ u(2t) (0 ≤ t ≤ ¹ ₂ ) v(2t − 1) ( ¹ ₂ ≤ t ≤ 1)

u

v

.

写像

φ : X → Y

,

u ∈ Ω(X, x 0 , x 1 )

, φu ∈ Ω(Y, φ(x 0 ), φ(x 1 ))

.

定義

2.2.2 x ₀ , x ₁ ∈ X

.

Ω(X, x ₀ , x ₁ )

2

u, v

½ F (t, 0) = u(t), F (0, s) = x 0

F (t, 1) = v(t), F (1, s) = x 1

をみたす連続写像

F : I × I → X

,

u

v

, u ∼ v

u ∼ v(F )

.

F

u

v

.

2.2.3 u, v ∈ Ω(X, x ₀ , x ₁ )

u ∼ v ⇔ u

v

∼

.

(

) (i)

, (ii)

, (iii)

. u, v, w ∈ Ω(X, x 0 , x 1 )

.

(i)

.

, u ∼ u

.

F : I × I → X

F (t, s) = u(t)

u ∼ u(F )

.

(ii)

.

, u ∼ v

v ∼ u

. u ∼ v(F )

, F (t, 0) = u(t)

F (t, 1) = v(t)

, F ⁰ : I × I → X

F ⁰ (t, s) = F (t, 1 − s)

F ⁰

, F ⁰ (t, 0) = F (t, 1) = v(t)

F ⁰ (t, 1) = F (t, 0) = u(t)

, v ∼ u(F ⁰ )

.

(iii)

.

, u ∼ v, v ∼ w

u ∼ w

. u ∼ v(F ₁ ), v ∼ w(F ₂ )

, F : I × I → X

F (t, s) =

½ F ₁ (t, 2s) (0 ≤ s ≤ ¹ ₂ ) F 2 (t, 2s − 1) ( ¹ ₂ ≤ s ≤ 1)

と定義すると

F

, F (t, 0) = F ₁ (t, 0) = u(t)

F (t, 1) = F ₂ (t, 1) = w(t)

F

u

w

.

, u ∼ w(F )

. (

)

X

1

x

, u : I → X

u(t) = x

, u

X

1

,

u

0 _x

0

,

.

定義

2.2.4 1. X

u

x ₀

, u

x ₀

(

)

.

2. x ∈ X

Ω(X, x, x)/ ∼

π 1 (X, x)

. π 1 (X, x)

x

X

.

3.

x ∈ X

π ₁ (X, x) = 0

X

.

基本群

π ₁ (X, x)

[u], [v] ∈ π ₁ (X, x)

, [u][v] = [u ∗ v]

,

[u] ^{− 1} = [u ^{− 1} ]

,

[0 x ]

.

補題

2.2.5 x 0 , x 1 ∈ X, w ∈ Ω(X, x 0 , x 1 )

.

w _∗ : π ₁ (X, x ₁ ) → π ₁ (X, x ₀ ) ; w _∗ ([u]) = [w][u][w ^{− 1} ] = [w ∗ u ∗ w ^{− 1} ]

.

(

) (i) w _∗

, (ii) w _∗

. (i) w _∗

. [u], [v] ∈ π 1 (X, x 1 )

.

w _∗ ([u])w _∗ ([v]) = [w][u][w ^{− 1} ][w][v][w ^{− 1} ]

= [w][u][0 x

][v][w ^{− 1} ]

= [w][u][v][w ^{− 1} ]

= [w][u ∗ v][w ^{− 1} ]

= w _∗ ([u][v])

.

(ii) w _∗

. [u] ∈ π ₁ (X, x ₀ )

. w ^{− 1} ∈ Ω(X, x ₀ , x ₁ )

w ⁻ _∗ ¹ : π 1 (X, x 0 ) → π 1 (X, x 1 ) ; w _∗ ^{− 1} ([u]) = [w ^{− 1} ∗ u ∗ w]

を誘導する

.

,

w ⁻ _∗ ¹ w _∗ ([u]) = w _∗ ^{− 1} ([w ∗ u ∗ w ^{− 1} ])

= [w ^{− 1} ∗ w ∗ u ∗ w ^{− 1} ∗ w]

= [0 x

∗ u ∗ 0 x

]

= [u]

であるから

, w _∗ ⁻¹ w _∗ = 1

.

w _∗ w ⁻¹ _∗ = 1

.

, w _∗

. (

) X

, X

2

x 0 , x 1

, x 0 , x 1

X

: π 1 (X, x 0 ) ∼ = π 1 (X, x 1 ).

x

X

π ₁ (X)

.

補題

2.2.6

φ : X → Y

.

φ _∗ : π 1 (X, x) → π 1 (Y, φ(x)) ; φ _∗ [u] = [φu]

は群準同型写像である

.

(

) (i) well-defined

, (ii)

.

(i) well-defined

. u ∼ v

u, v ∈ Ω(X, x, x)

.

φu, φv ∈ Ω(Y, φ(x), φ(x))

.

, u ∼ v

½ F (t, 0) = u(t), F (0, s) = x F (t, 1) = v(t), F (1, s) = x

をみたす連続写像

F : I × I → X

.

,

φF : I × I → Y

½ φF (t, 0) = φu(t), φF (0, s) = φ(x) φF (t, 1) = φv(t), φF (1, s) = φ(x)

であり

, φu

φv

.

, φu ∼ φv

. (ii)

. ∀ [u], [v] ∈ π 1 (X, x)

.

φ _∗ ([u][v]) = φ _∗ ([u ∗ v])

= [φ(u ∗ v)]

= [(φu) ∗ (φv)]

= [φu][φv]

= φ _∗ [u]φ _∗ [v]

となる

. (

)

これを連続写像

φ : X → Y

.

補題

2.2.7

f : X → Y , g : Y → Z

, (f g) _∗ = f _∗ g _∗

.

(

) ∀ [u] ∈ π ₁ (X, x)

.

(f g) _∗ [u] = [f gu]

= f _∗ [gu]

= f _∗ g _∗ [u]

が成り立つ

. (

)

ここで

,

.

定理

2.2.8

x 0 ∈ X , y 0 ∈ Y

.

,

2

:

π ₁ (X × Y, (x ₀ , y ₀ )) ∼ = π ₁ (X, x ₀ ) × π ₁ (Y, y ₀ ).

(

) (i)

, (ii)

.

まず

,

. X, Y

p : X × Y → X, q : X × Y → Y

.

,

i : X → X × Y , j : Y → X × Y

,

i(x) = (x, y 0 ), j (y) = (x 0 , y)

.

pi = 1, pj (y) = x 0 , qj = 1, qi(x) = y 0

であるから

,

2.2.7

,

p _∗ i _∗ = 1 _∗ , p _∗ j _∗ = [0 _x

], q _∗ j _∗ = 1 _∗ , q _∗ i _∗ = [0 _y

]

.

(i)

.

φ

φ : π 1 (X × Y, (x 0 , y 0 )) → π 1 (X, x 0 ) × π(Y, y 0 ) ; φ(γ ) = (p _∗ (γ), q _∗ (γ))

, p _∗

q _∗

, φ

.

(ii) φ

.

まず

,

. ∀ (α, β) ∈ π ₁ (X, x ₀ ) × π ₁ (Y, y ₀ )

.

γ = i _∗ (α)j _∗ (β) ∈ φ 1 (X × Y, (x 0 , y 0 ))

.

,

φ(γ) = (p _∗ (γ), q _∗ (γ))

= (p _∗ (i _∗ (α)j _∗ (β)), q _∗ (i _∗ (α)j _∗ (β)))

= ((p _∗ i _∗ (α))(p _∗ j _∗ (β)), (q _∗ i _∗ (α))(q _∗ j _∗ (β)))

= ((p _∗ i _∗ (α))[0 x

], [0 y

](q _∗ j _∗ (β)))

= (p _∗ i _∗ (α), q _∗ j _∗ (β))

= (α, β).

したがって

, φ

.

次に

,

.

Ker(φ) = { [0 (x

,y

) ] }

.

γ = [u] ∈ Ker(φ)

.

φ(γ) = ([0 _x

], [0 _y

])

.

,

([0 _x

], [0 _y

]) = φ(γ)

= φ([u])

= (p _∗ [u], q _∗ [u])

= ([pu], [qu]).

よって

, pu ∼ 0 x

, qu ∼ 0 y

,

½ F 1 (t, 0) = pu(t), F 1 (0, s) = x 0 , F ₁ (t, 1) = x ₀ , F ₁ (1, s) = x ₀ ,

½ F 2 (t, 0) = qu(t), F 2 (0, s) = y 0 , F ₂ (t, 1) = y ₀ , F ₂ (1, s) = y ₀

をみたす

,

F 1 : I × I → X , F 2 : I × I → Y

.

, F : I × I → X × Y

F = F ₁ × F ₂

,

½ F (t, 0) = (pu(t), qu(t)), F (0, s) = (x 0 , y 0 ), F (t, 1) = (x ₀ , y ₀ ), F (1, s) = (x ₀ , y ₀ )

が成り立つ

.

F

(pu, qu)

0 _(x

_,y

₎

.

, u(t) = (pu(t), qu(t))

, F

u

0 _(x

_,y

₎

.

, γ = [u] = [0 _(x

_,y

₎ ]

, φ

. (

)

2.2.9 R

.

(

) R

, π ₁ ( R , 0) = 0

. R

0

u : I → R

.

F : I × I → R ; F (t, s) = u(t)(1 − s)

u

0

.

, u ∼ 0

, [u] = 0

.

, R

は単連結である

. (

)

また

,

2.2.8

,

R ⁿ

.

3 実特殊線形群の極分解

この章では

,

SL(n, R )

.

,

,

.

3.1

この節では

,

,

.

定義

3.1.1

X ∈ M (n, R )

^t X = X

, X

n

R -Hermite

,

. n

R -Hermite

:

= (n, R ) = { X ∈ M (n, R ) | X = ^t X } .

定義

3.1.2

P ∈ = (n, R )

,

.

∀ x ∈ ( R ⁿ ) ^×

h x, P x i > 0. ( h , i

R ⁿ

.)

, n

R -Hermite

= + (n, R )

.

3.1.3

P ∈ = (n, R )

,

3

.

(1)

P

R -Hermite

. (2)

P

.

(3) P = Q ²

Q ∈ = + (n, R )

.

(

) (1)

(2)

.

λ k

x k

.

, k = 1, 2, . . . , n. ∀ k

,

P x _k = λ _k x _k

.

^t x k

t x _k P x _k = λ _k || x _k || ² , h x k , P x k i = λ k || x k || ²

.

|| x k || ² > 0

,

λ k > 0.

したがって

,

R -Hermite

P

.

(2)

(3)

. ∀ P ∈ = (n, R )

,

A ∈ O(n)

,

.

P = ADA ^{− 1} , D =



  λ ₁

. . . λ _n



  , λ k ∈ R

P

, λ k > 0

,

Q = A



 

√ λ ₁

. . . √ λ n



  A ^{− 1}

とおくと

, Q ∈ = (n, R )

, P = Q ²

. (3)

(1)

. ∀ x ∈ ( R ⁿ ) ^×

.

h x, P x i = ^t xP x

= ^t xQQx

= ^t x ^t QQx (.. . Q ∈ = (n, R ))

= h Qx, Qx i ≥ 0

h x, P x i = 0

, x = 0

. x ∈ ( R ⁿ ) ^×

, h x, P x i > 0

る

. (

)

3.2

この節では

,

,

.

,

2.1.4

,

.

補題

3.2.1 ∀ X, Y ∈ M (n, R )

,

: (1) ∀ (i, j), | a ij | ≤ || X || .

(2) || XY || ≤ || X || || Y || . (

)

(1)

.

(2) ∀ X, Y ∈ M (n, R )

. || X || ≥ 0, || Y || ≥ 0

|| XY || ² ≤ || X || ² || Y || ²

を示す

. A = XY

.

,

|| XY || ² = || A || ²

= X n i,j=1

| a _ij | ²

= X n i,j=1

¯¯ ¯¯

¯ X n k=1

( | x ik || y kj | )

¯¯ ¯¯

¯

2

≤ X n i,j=1

à _n X

k=1

| x ik | ² X n l=1

| y lj | ²

!

(.. .

)

= X n i,k=1

| x _ik | ² X n l,j=1

| y _lj | ²

= || X || ² || Y || ²

.

(

)

3.2.2 X ∈ M _n ( R )

,

exp

:

exp X = X ∞ n=0

µ X ⁿ n!

¶ .

指数写像

exp

,

,

.

命題

3.2.3 X ∈ M (n, R )

exp X

. (

) ∀ (i, j)

.

exp X _ij = X ∞ n=0

µ (X ⁿ ) ij

n!

¶

なので

,

X ∞ n=0

¯¯ ¯¯ (X ⁿ ) ij

n!

¯¯ ¯¯ = X ∞ n=0

1

n! | (X ⁿ ) ij |

≤ X ∞ n=0

1

n! || X ⁿ || (.. .

3.2.1(1))

≤ X ∞ n=0

1

n! || X || ⁿ (.. .

3.2.1(2))

= exp || X ||

< ∞ .

したがって

, exp X

. (

)

,

.

補題

3.2.4

X, Y ∈ M n ( R )

,

: (1) exp(AXA ^{− 1} ) = A(exp X)A ^{− 1} (

, A ∈ GL(n, R )).

(2) X

λ ₁ , λ ₂ , . . . , λ _n

, exp X

e ^λ

, e ^λ

, . . . , e ^λ

.

(3) det(exp X) = e tr (X) . (

)

(1) (AXA ^{− 1} ) ^k = AX ^k A ^{− 1}

, exp(AXA ^{− 1} ) =

X ∞ k=0

(AXA ^{− 1} ) ^k

n! =

X ∞ k=0

AX ^k A ^{− 1} n! = A

à _∞ X

k=0

X ^k n!

!

A ^{− 1} = A(exp X)A ^{− 1}

.

(2) X ∈ M (n, R )

,

B ∈ O(n)

,

.

B ^{− 1} XB =



 

λ ₁ ∗ ∗ . . . ∗ λ _n



  .

よって

,

B ^{− 1} (exp X )B = exp(B ^{− 1} XB) =



 

e ^λ

∗ ∗ . . . ∗ e ^λ



 

となる

.

, exp X

, e ^λ

, e ^λ

, . . . , e ^λ

. (3)

(2)

.

det(exp X) = det(B ^{− 1} (exp X)B)

= det(exp(B ⁻¹ XB))

= e ^λ

e ^λ

· · · e ^λ

= e ^λ

^+λ

^{+ ··· +λ}

= e tr (B

XB)

= e tr (XBB

)

= e tr (X)

となる

. (

)

3.3

この節では

,

.

,

:

D = (n, R ) = { X ∈ = (n, R ) | X

} , D = + (n, R ) = { X ∈ = + (n, R ) | X

} .

3.3.1

exp : D = (n, R ) → D = + (n, R )

.

(

)

D =



  λ 1

. . . λ n



  ∈ D = (n, R )

ならば

,

exp D =



  e ^λ

. . . e ^λ



  ∈ D = + (n, R )

である

.

,

exp : D = (n, R ) → D = + (n, R )

,

.

,

exp

log : D = + (n, R ) → D = (n, R ) ; log



  µ 1

. . . µ n



  =



 

log µ 1

. . .

log µ n



 

であるから

,

. (

)

3.3.2 X ∈ = (n, R ), A ∈ O(n)

, A(exp X ) = (exp X)A

, AX = XA

.

(

)

, X

D =



 

λ

. . . λ



 

,

示す

. D ∈ = (n, R ), A = (a _ij ) ∈ O(n)

, exp D =



 

e

. . .

e



 

から

,

A(exp D) = (exp D)A.

成分でみると

,

(a _ij e ^λ

) = (e ^λ

a _ij ).

すなわち

, i, j = 1, 2, . . . , n

a ij e ^λ

= e ^λ

a ij

が成り立つ

.

,

a _ij (e ^λ

− e ^λ

) = 0.

つまり

,

a _ij = 0

(e ^λ

− e ^λ

) = 0

,

a _ij = 0

(λ _j − λ _i ) = 0.

よって

,

a _ij (λ _j − λ _i ) = 0, a ij λ j = λ i a ij

となるので

,

AD = DA.

次に

,

X ∈ = (n, R )

. X ∈ = (n, R )

,

B ∈ O(n)

, B ^{− 1} XB = D ∈ D = (n, R )

. A(exp X) = (exp X)A

,

A(exp BDB ^{− 1} ) = (exp BDB ^{− 1} )A

となる

.

3.2.4(1)

,

AB exp(D)B ^{− 1} = B exp(D)B ^{− 1} A

,

B ⁻¹ AB exp(D) = exp(D)B ⁻¹ AB

となる

.

, B ⁻¹ AB ∈ O(n)

,

, B ^{− 1} ABD = DB ^{− 1} AB.

したがって

,

ABDB ^{− 1} = BDB ^{− 1} A

,

AX = XA.

(

)

3.3.3 (1) = (n, R )

M (n, R )

.

(2) D = + (n, R )

GL(n, R )

. (

)

(1)

f : M (n, R ) → M (n, R ), f(X ) = ^t X − X

,

= (n, R )

1

{ 0 } ⊂ M (n, R )

f

, = (n, R )

M (n, R )

.

(2)

D = ⁰ (n, R ) = { D ∈ D = (n, R ) | D

λ _k ≥ 0, k = 1, 2, . . . }

M (n, R )

.

, D = + (n, R ) = D = ⁰ (n, R ) ∩ GL(n, R )

D = + (n, R )

M (n, R )

. (

)

3.3.4 D m ∈ D = (n, R ), m = 1, 2, . . .

, exp D 1 , exp D 2 , . . . , exp D m , . . .

, D 1 , D 2 , . . . , D m , . . .

.

, lim

m →∞ exp D m = P, lim

m →∞ D _m = D

P ∈ D = + (n, R ), D ∈ D = (n, R )

, P = exp D

.

(

)

3.3.1

,

exp : D = (n, R ) → D = + (n, R )

,

log : D = + (n, R ) → D = (n, R )

.

, D m ∈

D = (n, R ), m = 1, 2, . . .

, exp D ₁ , exp D ₂ , . . . , exp D _m , . . .

,

D 1 , D 2 , . . . , D m , . . .

.

P ∈ D = + (n, R )

. exp D m ∈ D = + (n, R )

,

3.3.3 (2)

, D = + (n, R )

GL(n, R )

.

, D = (n, R )

GL(n, R )

. lim

m→∞ exp D _m = P

P ∈ D = + (n, R )

. D ∈ D = (n, R )

P = exp D

. lim

m →∞ exp D m = P

log

,

m lim →∞ D _m = log P

.

, lim

m →∞ D _m = D

, P = exp D

,

D ∈ D = (n, R )

. (

)

写像

exp : M (n, R ) → GL(n, R )

.

3.3.5 X, X 1 , X 2 , . . . , X m , . . . ∈ M (n, R )

|| X || ≤ a, || X m || ≤ a (m = 1, 2, · · · )

a ∈ R

,

.

|| X _m ⁿ − X ⁿ || = na ⁿ⁻¹ || X _m − X || . (

)

|| X m n − X ⁿ || = ¯¯¯¯ X m (X m n − 1 − X ^{n − 1} + (X m − X)X ^{n − 1} ) ¯¯¯¯

≤ ( || X _m |||| X _m ^{n − 1} − X ^{n − 1} || + || X _m − X |||| X ^{n − 1} || )

≤ (a || X m n − 1 − X ^{n − 1} || + || X m − X || a ^{n − 1} )

≤ · · ·

≤ (a(n − 1)a ^{n − 2} || X m − X || + || X m − X || a ^{n − 1} )

= ((n − 1)a ^{n − 1} + a ^{n − 1} ) || X m − X || )

= na ^{n − 1} || X m − X ||

(

)

3.3.6

exp : M (n, R ) → GL(n, R )

.

(

) lim

m →∞ X m = X

, lim

m →∞ exp X m = exp X

.

{ X _m } ^∞ m=1 ⊂ M (n, R )

. X ∈ M (n, R )

, lim

m →∞ X _m = X

, X ₁ , X ₂ , . . . , X _m , . . .

,

.

, || X || ≤ a, || X _m || ≤ a

(m = 1, 2, . . .)

a ∈ R

.

,

¯¯ ¯¯

¯

¯¯ ¯¯

¯ X N n=0

X m n

n! − X N n=0

X ⁿ n!

¯¯ ¯¯

¯

¯¯ ¯¯

¯ =

¯¯ ¯¯

¯

¯¯ ¯¯

¯ X N n=0

1

n! (X _m ⁿ − X ⁿ )

¯¯ ¯¯

¯

¯¯ ¯¯

¯

≤ X N n=0

1

n! || (X m n − X ⁿ ) ||

= X N n=0

1

n! na ^{n − 1} || X m − X || (.. .

3.3.5)

= X N n=0

a ^{n − 1}

(n − 1)! || X _m − X ||

N → ∞

, || exp X _m − exp X || ≤ e ^a || X _m − X ||

.

,

m → ∞

|| exp X m − exp X || → 0

, lim

m →∞ exp X m = exp X

せた

. (

)

これらの補題をを用いて写像

exp : = (n, R ) → = + (n, R )

.

3.3.7

exp : = (n, R ) → = + (n, R )

.

(

)

3.3.6

,

, (i)

, (ii)

,

(iii)

.

(i)

. ∀ P ∈ = + (n, R )

. P

,

A ∈ O(n)

, A ⁻¹ P A ∈ D = + (n, R )

.

, exp : D = (n, R ) → D = + (n, R )

, A ^{− 1} P A ∈ D = + (n, R )

, D ∈ D = (n, R )

exp D = A ^{− 1} P A

. X = ADA ^{− 1}

, X ∈ = (n, R )

,

exp X = exp(ADA ^{− 1} )

= A exp(D)A ^{− 1}

= AA ^{− 1} P AA ^{− 1}

= P

.

(ii)

. X, Y ∈ = (n, R )

, exp X = exp Y (= P )

. X, Y

,

A, B ∈ O(n)

, X = AD X A ^{− 1} , Y = BD Y B ^{− 1}

.

exp D _X = A ⁻¹ P A , exp D _Y = B ⁻¹ P B

, exp D _X

exp D _Y

.

, D _X

, D _Y

.

ち

, C ∈ O(n)

, CD X C ^{− 1} = D Y

.

, exp X = exp Y

= exp(BD Y B ^{− 1} )

= exp(BCD _X C ^{− 1} B ^{− 1} )

= exp(BCA ⁻¹ XAC ⁻¹ B ⁻¹ )

= exp((BCA ^{− 1} )X(BCA ^{− 1} ) ^{− 1} )

= (BCA ^{− 1} ) exp(X)(BCA ^{− 1} ) ^{− 1}

となり

, BCA ^{− 1} ∈ O(n)

3.3.2

, X = (BCA ^{− 1} )X (BCA ^{− 1} ) ^{− 1}

.

,

X = (BCA ^{− 1} )X(BCA ^{− 1} ) ^{− 1}

= BCA ⁻¹ XAC ⁻¹ B ⁻¹

= BCD _X C ⁻¹ B ⁻¹

= BD _Y B ^{− 1}

= Y

.

(iii) exp

.

, exp

を示す。つまり

, F

= (n, R )

, exp F

= + (n, R )

.

,

X 1 , X 2 , . . . , X m , . . . ∈ F

, lim

m →∞ exp X m = P, P ∈

= + (n, R )

,

X ∈ F

P = exp X

.

X m

,

A m ∈ O(n)

,

A m − 1

X m A m = D m ∈ D = (n, R )

とできる

. O(n)

A 1 , A 2 , . . . , A m , . . .

A _k1 , A _k2 , . . . , A _{k m} , . . .

. lim

m →∞ A _{k m} = A

.

,

X 1 , X 2 , . . . , X m , . . .

X k1 , X k2 , . . . , X _{k m} , . . .

,

A _k ⁻ _m ¹ X _{k m} A _{k m} = D _{k m} ∈ D = (n, R )

.

,

m→∞ lim exp D _{k m} = lim

m→∞ exp(A _k ⁻ _m ¹ X _{k m} A _{k m} )

= lim

m →∞ A k − 1

m (exp X _{k m} )A _{k m}

= A ^{− 1} P A