Google 翻译生成的粗略草稿，质量较低。术语依照术语表，但句子可能有误；之后会由仔细的译文取代。
§ 7。 HAUSDOFF 的收敛性级数（真实或复杂的情况）
在本段中，我们假设 K K K R \mathbf{R} R C \mathbf{C} C K K K K K K T \mathscr{T} T T \mathscr{T} T A × A A \times A A × A A A A ( x , y ) ↦ x y (x, y) \mapsto xy ( x , y ) ↦ x y K K K A A A K K K ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ \|xy\| \leq \|x\|\ \|y\| ∥ x y ∥ ≤ ∥ x ∥ ∥ y ∥ A A A x , y x, y x , y g g g K K K g g g M > 0 M > 0 M > 0
∥ [ x , y ] ∥ ≤ M ∥ x ∥ ∥ y ∥ for x , y in g .
\|[x, y]\| \leq M \|x\|\ \|y\| \quad \text{for } x, y \text{ in } g.
∥ [ x , y ] ∥ ≤ M ∥ x ∥ ∥ y ∥ for x , y in g . 1. g g g
设 I I I P ( g I ; g ) P(g^I; g) P ( g I ; g ) P ^ ( g I ; g ) \hat{P}(g^I; g) P ^ ( g I ; g ) g I g^I g I g g g P ( g I ; g ) P(g^I; g) P ( g I ; g ) N I \mathbf{N}^I N I P ^ ( g I ; g ) \hat{P}(g^I; g) P ^ ( g I ; g ) P ( g I ; g ) P(g^I; g) P ( g I ; g ) P ( g I ; g ) P(g^I; g) P ( g I ; g ) P ( g I ; g ) P(g^I; g) P ( g I ; g ) [ f , g ] ( x ) = [ f ( x ) , g ( x ) ] [f, g](x) = [f(x), g(x)] [ f , g ] ( x ) = [ f ( x ) , g ( x )] P ^ ( g I ; g ) \hat{P}(g^I; g) P ^ ( g I ; g )
到§ 6, no. 3的命题 2，L ^ ( I ) \hat{L}(I) L ^ ( I ) ϕ I : u ↦ u ~ \phi_I : u \mapsto \tilde{u} ϕ I : u ↦ u ~ P ^ ( g I ; g ) \hat{P}(g^I; g) P ^ ( g I ; g ) p r i \mathrm{pr}_i pr i p r i ∈ P ( g I ; g ) \mathrm{pr}_i \in P(g^I; g) pr i ∈ P ( g I ; g ) i ∈ I i \in I i ∈ I u ~ ∈ P ( g I ; g ) \tilde{u} \in P(g^I; g) u ~ ∈ P ( g I ; g ) u ∈ L ( I ) u \in L(I) u ∈ L ( I ) u ∈ L ( I ) u \in L(I) u ∈ L ( I ) u ~ \tilde{u} u ~ ( t i ) ↦ u ( ( t i ) ) (t_i) \mapsto u((t_i)) ( t i ) ↦ u (( t i )) ϕ I \phi_I ϕ I L ( I ) L(I) L ( I ) P ( g I ; g ) P(g^I; g) P ( g I ; g ) u = ∑ v ∈ N I u v u = \sum_{v \in \mathbf{N}^I} u_v u = ∑ v ∈ N I u v u v ∈ L v ( I ) u_v \in L^v(I) u v ∈ L v ( I ) v ∈ N I v \in \mathbf{N}^I v ∈ N I
u ~ = ∑ v ∈ N I u ~ v , where u ~ v ∈ P v ( g I ; g ) .
\tilde{u} = \sum_{v \in \mathbf{N}^I} \tilde{u}_v, \quad \text{where } \tilde{u}_v \in P_v(g^I; g).
u ~ = v ∈ N I ∑ u ~ v , where u ~ v ∈ P v ( g I ; g ) . 令u = ( u j ) j ∈ J u = (u_j)_{j \in J} u = ( u j ) j ∈ J L ^ ( I ) \hat{L}(I) L ^ ( I ) v ∈ L ^ ( J ) v \in \hat{L}(J) v ∈ L ^ ( J ) w = v ∘ u w = v \circ u w = v ∘ u u ~ = ( u ~ j ) j ∈ J ∈ g \tilde{u} = (\tilde{u}_j)_{j \in J} \in \mathfrak{g} u ~ = ( u ~ j ) j ∈ J ∈ g
(2)
v ~ ∘ u ~ = ( v ∘ u ) ∼ .
\tilde{v} \circ \tilde{u} = (v \circ u)^{\sim}.
v ~ ∘ u ~ = ( v ∘ u ) ∼ . 接下来是通过 § 6, no. 3 的连续性公式 (7) 以及可微分和解析流形、R、附录、no. 6 进行扩展。
2. 群由完整规范定义的细菌李代数
令 H = ∑ r , s ≥ 0 H r , s ∈ L ^ ( U , V ) H = \sum_{r,s \geq 0} H_{r,s} \in \hat{L}(U, V) H = ∑ r , s ≥ 0 H r , s ∈ L ^ ( U , V )
(3)
H ~ = ∑ r , s ≥ 0 H ~ r , s ∈ P ^ ( g × g , g )
\tilde{H} = \sum_{r,s \geq 0} \tilde{H}_{r,s} \in \hat{P}(g \times g, g)
H ~ = r , s ≥ 0 ∑ H ~ r , s ∈ P ^ ( g × g , g ) 是收敛的（可微且解析流形，R，3.1.1）。
我们介绍以下正式的幂级数η ∈ Q [ [ U , V ] ] \eta \in \mathbf{Q}[[U, V]] η ∈ Q [[ U , V ]]
(4)
η ( U , V ) = − log ( 2 − exp ( U + V ) )
\eta(U, V) = -\log(2 - \exp(U + V))
η ( U , V ) = − log ( 2 − exp ( U + V )) = ∑ m ≥ 1 1 m ( exp ( U + V ) − 1 ) m
= \sum_{m \geq 1} \frac{1}{m} (\exp(U + V) - 1)^m
= m ≥ 1 ∑ m 1 ( exp ( U + V ) − 1 ) m = ∑ m ≥ 1 1 m ∑ r 1 , … , r m s 1 , … , s m U r 1 V s 1 r 1 ! U r 2 V s 2 r 2 ! ⋯ U r m V s m s m ! .
= \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1, \ldots, r_m \\ s_1, \ldots, s_m}} \frac{U^{r_1} V^{s_1}}{r_1!} \frac{U^{r_2} V^{s_2}}{r_2!} \cdots \frac{U^{r_m} V^{s_m}}{s_m!}.
= m ≥ 1 ∑ m 1 r 1 , … , r m s 1 , … , s m ∑ r 1 ! U r 1 V s 1 r 2 ! U r 2 V s 2 ⋯ s m ! U r m V s m . 因此
(7)
η ( U , V ) = ∑ r , s ≥ 0 η r , s U r V s ,
\eta(U, V) = \sum_{r,s \geq 0} \eta_{r,s} U^r V^s,
η ( U , V ) = r , s ≥ 0 ∑ η r , s U r V s , 哪里
η r , s = ∑ m ≥ 1 1 m ∑ r 1 + ⋯ + r m = r s 1 + ⋯ + s m = s r i + s i ≥ 1 1 r 1 ! … r m ! s 1 ! … s m ! .
\eta_{r,s} = \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_m = r \\ s_1 + \cdots + s_m = s \\ r_i + s_i \geq 1}} \frac{1}{r_1! \ldots r_m! s_1! \ldots s_m!}.
η r , s = m ≥ 1 ∑ m 1 r 1 + ⋯ + r m = r s 1 + ⋯ + s m = s r i + s i ≥ 1 ∑ r 1 ! … r m ! s 1 ! … s m ! 1 . u u u v v v u + v < log 2 u + v < \log 2 u + v < log 2 0 ≤ exp ( u + v ) − 1 < 1 0 \leq \exp(u + v) - 1 < 1 0 ≤ exp ( u + v ) − 1 < 1 U U U u u u V V V v v v ∑ r , s ≥ 0 η r , s u r v s = − log ( 2 − exp ( u + v ) ) < + ∞ .
\sum_{r, s \geq 0} \eta_{r,s} u^r v^s = -\log(2 - \exp(u + v)) < +\infty.
r , s ≥ 0 ∑ η r , s u r v s = − log ( 2 − exp ( u + v )) < + ∞. r , s ≥ 0 r, s \geq 0 r , s ≥ 0 ∥ H ~ r , s ∥ \| \tilde{H}_{r,s} \| ∥ H ~ r , s ∥ H ~ r , s \tilde{H}_{r,s} H ~ r , s
∥ H ~ r , s ∥ ≤ M r + s − 1 η r , s .
\| \tilde{H}_{r,s} \| \leq M^{r+s-1} \eta_{r,s}.
∥ H ~ r , s ∥ ≤ M r + s − 1 η r , s . 令
r i , s i r_i, s_i r i , s i 在
N \mathbf{N} N 中，对于
1 ≤ i ≤ m 1 \leq i \leq m 1 ≤ i ≤ m ，有
s m = 1 s_m = 1 s m = 1 ；我们写
r = ∑ i r i , s = ∑ i s i r = \sum_i r_i, s = \sum_i s_i r = ∑ i r i , s = ∑ i s i 并考虑
L ( { U , V } ) L(\{U, V\}) L ({ U , V }) 的以下元素：
Z = ( ( ∑ i = 1 m − 1 ( a d U ) r i ( a d V ) s i ) ( a d U ) r m ) ( V ) .
Z = \left( \left( \sum_{i=1}^{m-1} (\mathrm{ad}\ U)^{r_i} (\mathrm{ad}\ V)^{s_i} \right) (\mathrm{ad}\ U)^{r_m} \right)(V).
Z = ( ( i = 1 ∑ m − 1 ( ad U ) r i ( ad V ) s i ) ( ad U ) r m ) ( V ) . 然后是
Z ~ = f ∘ p \tilde{Z} = f \circ p Z ~ = f ∘ p ，其中
f f f 是
g r + s g^{r+s} g r + s 到
g g g 的以下
( r + s ) (r+s) ( r + s ) 线性映射：
( x 1 , … , x r , y 1 , … , y s ) ↦ ( a d ( x 1 ) ∘ ⋯ ∘ a d ( x r ) ∘ a d ( y 1 ) ∘ ⋯ ∘ a d ( y s ) ) ( y s )
(x_1, \ldots, x_r, y_1, \ldots, y_s) \mapsto (\mathrm{ad}(x_1) \circ \cdots \circ \mathrm{ad}(x_r) \circ \mathrm{ad}(y_1) \circ \cdots \circ \mathrm{ad}(y_s))(y_s)
( x 1 , … , x r , y 1 , … , y s ) ↦ ( ad ( x 1 ) ∘ ⋯ ∘ ad ( x r ) ∘ ad ( y 1 ) ∘ ⋯ ∘ ad ( y s )) ( y s ) 其中
p p p 是
g 2 g^2 g 2 的映射到
g r + s g^{r+s} g r + s 的结果：
( x , y ) ↦ ( x , … , x ⏟ r , y , … , y ⏟ s ) ;
(x, y) \mapsto (\underbrace{x, \ldots, x}_{r}, \underbrace{y, \ldots, y}_{s});
( x , y ) ↦ ( r x , … , x , s y , … , y ) ; 因此
∥ Z ~ ∥ ≤ ∥ f ∥ ≤ M r + s − 1 \| \tilde{Z} \| \leq \| f \| \leq M^{r+s-1} ∥ Z ~ ∥ ≤ ∥ f ∥ ≤ M r + s − 1 （可微分和解析流形，R，附录）。将应用这些不等式与§ 6, no. 4的公式(9)右侧的各个项，我们得到：
∥ ( H r , s ′ ) ∼ ∥ ≤ M r + s − 1 r + s ∑ m ≥ 1 1 m ∑ r 1 + ⋯ + r m = r s 1 + ⋯ + s m = s r 1 + s 1 ≥ 1 , … , r m − 1 + s m − 1 ≥ 1 1 r 1 ! … r m ! s 1 ! … s m ! .
\|(H'_{r,s})\sim\|
\leq \frac{M^{r+s-1}}{r+s} \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_m = r \\ s_1 + \cdots + s_m = s \\ r_1 + s_1 \geq 1, \ldots, r_{m-1} + s_{m-1} \geq 1}} \frac{1}{r_1! \ldots r_m! s_1! \ldots s_m!}.
∥ ( H r , s ′ ) ∼ ∥ ≤ r + s M r + s − 1 m ≥ 1 ∑ m 1 r 1 + ⋯ + r m = r s 1 + ⋯ + s m = s r 1 + s 1 ≥ 1 , … , r m − 1 + s m − 1 ≥ 1 ∑ r 1 ! … r m ! s 1 ! … s m ! 1 . 类似的论证给出
∥ ( H r , s ′ ′ ) ∼ ∥ ≤ M r + s − 1 r + s ∑ m ≥ 1 1 m ∑ r 1 + ⋯ + r m − 1 = r − 1 s 1 + ⋯ + s m − 1 = s r 1 + s 1 ≥ 1 , … , r m − 1 + s m − 1 ≥ 1 1 r 1 ! … r m − 1 ! s 1 ! … s m − 1 !
\|(H''_{r,s})\sim\|
\leq \frac{M^{r+s-1}}{r+s} \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_{m-1} = r-1 \\ s_1 + \cdots + s_{m-1} = s \\ r_1 + s_1 \geq 1, \ldots, r_{m-1} + s_{m-1} \geq 1}} \frac{1}{r_1! \ldots r_{m-1}! s_1! \ldots s_{m-1}!}
∥ ( H r , s ′′ ) ∼ ∥ ≤ r + s M r + s − 1 m ≥ 1 ∑ m 1 r 1 + ⋯ + r m − 1 = r − 1 s 1 + ⋯ + s m − 1 = s r 1 + s 1 ≥ 1 , … , r m − 1 + s m − 1 ≥ 1 ∑ r 1 ! … r m − 1 ! s 1 ! … s m − 1 ! 1 引用了 1 · 被以下引用 1 引用了 II § 6. 。豪斯多夫级数
被以下引用 命题 1
从而，由 (8)
∥ H ~ r , s ∥ < η r , s M r + s − 1 r + s ≤ η r , s M r + s − 1 ,
\|\tilde{H}_{r,s}\| < \eta_{r,s} \frac{M^{r+s-1}}{r+s} \leq \eta_{r,s} M^{r+s-1},
∥ H ~ r , s ∥ < η r , s r + s M r + s − 1 ≤ η r , s M r + s − 1 , 这就证明了引理。
Ω = { ( x , y ) ∈ g × g ∣ ∥ x ∥ + ∥ y ∥ < 1 M log 2 } .
\Omega = \left\{ (x, y) \in g \times g \mid \|x\| + \|y\| < \frac{1}{M} \log 2 \right\}.
Ω = { ( x , y ) ∈ g × g ∣ ∥ x ∥ + ∥ y ∥ < M 1 log 2 } . 令u , v u, v u , v > 0 > 0 > 0 u + v < 1 M log 2 u + v < \frac{1}{M} \log 2 u + v < M 1 log 2
(12) M ∑ r , s ≥ 0 ∥ H ~ r , s ∥ u r v s ≤ ∑ r , s ≥ 0 η r , s M r + s u r v s = − log ( 2 − exp M ( u + v ) ) < + ∞
\text{(12)} \quad M \sum_{r,s \geq 0} \|\tilde{H}_{r,s}\| u^r v^s \\
\leq \sum_{r,s \geq 0} \eta_{r,s} M^{r+s} u^r v^s = -\log(2 - \exp M(u + v)) < +\infty
(12) M r , s ≥ 0 ∑ ∥ H ~ r , s ∥ u r v s ≤ r , s ≥ 0 ∑ η r , s M r + s u r v s = − log ( 2 − exp M ( u + v )) < + ∞ 由（9）。
令h : Ω → g h : \Omega \to g h : Ω → g H ~ \tilde{H} H ~
(13) h ( x , y ) = ∑ r , s ≥ 0 H ~ r , s ( x , y ) = ∑ r , s ≥ 0 H r , s ( x , y ) for ( x , y ) ∈ Ω .
\text{(13)} \quad h(x, y) = \sum_{r,s \geq 0} \tilde{H}_{r,s}(x, y) = \sum_{r,s \geq 0} H_{r,s}(x, y) \quad \text{for } (x, y) \in \Omega.
(13) h ( x , y ) = r , s ≥ 0 ∑ H ~ r , s ( x , y ) = r , s ≥ 0 ∑ H r , s ( x , y ) for ( x , y ) ∈ Ω. 这个函数被称为 g g g M M M Hausdorff 函数 （或者在不会引起混淆的情况下简称为 g g g H r , s ( U , − U ) = 0 H_{r,s}(U, -U) = 0 H r , s ( U , − U ) = 0 r + s ≥ 2 r + s \geq 2 r + s ≥ 2
(14) h ( x , − x ) = 0 for ∥ x ∥ < 1 2 M log 2.
\text{(14)} \quad h(x, -x) = 0 \quad \text{for } \|x\| < \frac{1}{2M} \log 2.
(14) h ( x , − x ) = 0 for ∥ x ∥ < 2 M 1 log 2. 同样
(15) h ( 0 , x ) = h ( x , 0 ) = x for ∥ x ∥ < 1 M log 2.
\text{(15)} \quad h(0, x) = h(x, 0) = x \quad \text{for } \|x\| < \frac{1}{M} \log 2.
(15) h ( 0 , x ) = h ( x , 0 ) = x for ∥ x ∥ < M 1 log 2.
Ω ′ = { ( x , y , z ) ∈ g × g × g ∣ ∥ x ∥ + ∥ y ∥ + ∥ z ∥ < 1 M log 3 2 } .
\Omega' = \left\{ (x, y, z) \in g \times g \times g \mid \|x\| + \|y\| + \|z\| < \frac{1}{M} \log \frac{3}{2} \right\}.
Ω ′ = { ( x , y , z ) ∈ g × g × g ∣ ∥ x ∥ + ∥ y ∥ + ∥ z ∥ < M 1 log 2 3 } . 如果 ( x , y , z ) ∈ Ω ′ (x, y, z) \in \Omega' ( x , y , z ) ∈ Ω ′
(16) ( x , y ) ∈ Ω , ( h ( x , y ) , z ) ∈ Ω , ( y , z ) ∈ Ω , ( x , h ( y , z ) ) ∈ Ω
\text{(16)} \quad (x, y) \in \Omega, \quad (h(x, y), z) \in \Omega, \quad (y, z) \in \Omega, \quad (x, h(y, z)) \in \Omega
(16) ( x , y ) ∈ Ω , ( h ( x , y ) , z ) ∈ Ω , ( y , z ) ∈ Ω , ( x , h ( y , z )) ∈ Ω 和 (17) h ( h ( x , y ) , z ) = h ( x , h ( y , z ) ) .
\text{(17)} \quad h(h(x, y), z) = h(x, h(y, z)).
(17) h ( h ( x , y ) , z ) = h ( x , h ( y , z )) . 设 ( x , y , z ) ∈ Ω ′ (x, y, z) \in \Omega' ( x , y , z ) ∈ Ω ′ ( x , y ) ∈ Ω (x, y) \in \Omega ( x , y ) ∈ Ω ( y , z ) ∈ Ω (y, z) \in \Omega ( y , z ) ∈ Ω
∥ h ( x , y ) ∥ ≤ ∑ r , s ∥ H r , s ∥ ∥ x ∣ r ∥ y ∣ s ,
\|h(x, y)\| \leq \sum_{r, s} \|H_{r,s}\| \|x|r\|y|s,
∥ h ( x , y ) ∥ ≤ r , s ∑ ∥ H r , s ∥∥ x ∣ r ∥ y ∣ s , 因此由 (13)
∥ h ( x , y ) ∥ ≤ − 1 M log ( 2 − exp M ( ∥ x ∥ + ∥ y ∥ ) ) .
\|h(x, y)\| \leq -\frac{1}{M} \log(2 - \exp M(\|x\| + \|y\|)).
∥ h ( x , y ) ∥ ≤ − M 1 log ( 2 − exp M ( ∥ x ∥ + ∥ y ∥ )) . 现在M ( ∥ x ∥ + ∥ y ∥ ) < log 3 2 − M ∥ z ∥ M(\|x\| + \|y\|) < \log \frac{3}{2} - M\|z\| M ( ∥ x ∥ + ∥ y ∥ ) < log 2 3 − M ∥ z ∥ u = exp ( M ∥ z ∥ ) u = \exp(M\|z\|) u = exp ( M ∥ z ∥ ) 1 ≤ u ≤ 3 2 1 \leq u \leq \frac{3}{2} 1 ≤ u ≤ 2 3
M ( ∥ h ( x , y ) ∥ + ∥ z ∥ ) < − log ( 2 − exp ( log 3 2 − M ∥ z ∥ ) ) + M ∥ z ∥ = − log ( 2 − 3 2 u ) + log u = log 2 u 2 4 u − 3 = log ( 2 + 2 ( u − 1 ) ( u − 3 ) 4 u − 3 ) ≤ log 2.
\begin{align*}
M(\|h(x, y)\| + \|z\|) &< -\log(2 - \exp(\log \frac{3}{2} - M\|z\|)) + M\|z\| \\
&= -\log\left(2 - \frac{3}{2u}\right) + \log u = \log \frac{2u^2}{4u - 3} \\
&= \log\left(2 + \frac{2(u - 1)(u - 3)}{4u - 3}\right) \leq \log 2.
\end{align*}
M ( ∥ h ( x , y ) ∥ + ∥ z ∥ ) < − log ( 2 − exp ( log 2 3 − M ∥ z ∥ )) + M ∥ z ∥ = − log ( 2 − 2 u 3 ) + log u = log 4 u − 3 2 u 2 = log ( 2 + 4 u − 3 2 ( u − 1 ) ( u − 3 ) ) ≤ log 2. 我们也看到类似的( x , h ( y , z ) ) ∈ Ω (x, h(y, z)) \in \Omega ( x , h ( y , z )) ∈ Ω
现在我们证明(17)。在李代数L ^ ( { U , V , W } ) \hat{L}(\{U, V, W\}) L ^ ({ U , V , W })
H ( H ( U , V ) , W ) = H ( U , H ( V , W ) )
H(H(U, V), W) = H(U, H(V, W))
H ( H ( U , V ) , W ) = H ( U , H ( V , W )) 命题 4 / § 6, no. 5。根据 no. 1、公式 (2)，我们有 P ^ ( g × g × g , g ) \hat{P}(g \times g \times g, g) P ^ ( g × g × g , g )
H ~ ∘ ( H ~ × I d g ) = H ~ ∘ ( I d g × H ~ ) .
\tilde{H} \circ (\tilde{H} \times \mathrm{Id}_g) = \tilde{H} \circ (\mathrm{Id}_g \times \tilde{H}).
H ~ ∘ ( H ~ × Id g ) = H ~ ∘ ( Id g × H ~ ) . 通过*可微分和解析流形*，R，3.1.9，存在数ε > 0 \varepsilon > 0 ε > 0 ∥ x ∥ , ∥ y ∥ \|x\|, \|y\| ∥ x ∥ , ∥ y ∥ ∥ z ∥ \|z\| ∥ z ∥ ≤ ε \leq \varepsilon ≤ ε ( x , y , z ) ↦ h ( h ( x , y ) , z ) (x, y, z) \mapsto h(h(x, y), z) ( x , y , z ) ↦ h ( h ( x , y ) , z ) ( x , y , z ) ↦ h ( x , h ( y , z ) ) (x, y, z) \mapsto h(x, h(y, z)) ( x , y , z ) ↦ h ( x , h ( y , z )) Ω ′ \Omega' Ω ′ g g g Ω ′ \Omega' Ω ′
上述结果意味着：
令α \alpha α 0 < α ≤ 1 3 M log 3 2 0 < \alpha \leq \frac{1}{3M} \log \frac{3}{2} 0 < α ≤ 3 M 1 log 2 3
G = { x ∈ g ∣ ∥ x ∥ < α } ,
G = \{x \in g \mid \|x\| < \alpha\},
G = { x ∈ g ∣ ∥ x ∥ < α } ,
Θ = { ( x , y ) ∈ G × G ∣ h ( x , y ) ∈ G }
\Theta = \{(x, y) \in G \times G \mid h(x, y) \in G\}
Θ = {( x , y ) ∈ G × G ∣ h ( x , y ) ∈ G } 并且
m : Θ → G m : \Theta \to G m : Θ → G 是
h h h 至
Θ \Theta Θ 中的限制。然后： (1)
Θ \Theta Θ 在
G × G G \times G G × G 开，
m m m 解析。 (2)
x ∈ G x \in G x ∈ G 意味着
( 0 , x ) ∈ Θ (0, x) \in \Theta ( 0 , x ) ∈ Θ 、
( x , 0 ) ∈ Θ (x, 0) \in \Theta ( x , 0 ) ∈ Θ 和
m ( 0 , x ) = m ( x , 0 ) = x m(0, x) = m(x, 0) = x m ( 0 , x ) = m ( x , 0 ) = x 。 (3)
x ∈ G x \in G x ∈ G 意味着
− x ∈ G -x \in G − x ∈ G 、
( x , − x ) ∈ Θ (x, -x) \in \Theta ( x , − x ) ∈ Θ 、
( − x , x ) ∈ Θ (-x, x) \in \Theta ( − x , x ) ∈ Θ 和
m ( x , − x ) = m ( − x , x ) = 0.
m(x, -x) = m(-x, x) = 0.
m ( x , − x ) = m ( − x , x ) = 0. (4) 设 x, y, z 为 G 的元素，使得 (x, y) ε θ，(m(x, y), z) ε θ，(y, z) ε θ 且 (x, m(y, z)) ε θ。然后 m(m(x, y), z) = m(x, m(y, z))。
在其他字 (章 III, § 1) 中，如果我们写 −x = σ(x)，则四元组 (G, 0, σ, m) 是 K 上的李群胚芽。
3. 完全范数结合律中的指数代数
在这个没有。我们用 A 表示一个完全规范的单位结合词代数（一般拓扑、章 IX, § 3, no. 7）。然后\|x.y\| ≤ \|x\|。 \|y\|对于 A 中的 x、y。
令 I 为有限集合，并令 P ^ ( A I ; A ) \hat{P}(A^I; A) P ^ ( A I ; A ) A I A^I A I
f . g = m ∘ ( f , g ) for f , g in P ^ ( A I ; A ) ,
f.g = m \circ (f, g) \quad \text{for } f, g \text{ in } \hat{P}(A^I; A),
f . g = m ∘ ( f , g ) for f , g in P ^ ( A I ; A ) , 其中m : A × A → A m : A \times A \to A m : A × A → A u ↦ u ~ u \mapsto \tilde{u} u ↦ u ~ A ^ ( I ) \hat{A}(I) A ^ ( I ) P ^ ( A I ; A ) \hat{P}(A^I; A) P ^ ( A I ; A ) p r i pr_i p r i L ^ ( I ) \hat{L}(I) L ^ ( I ) P ^ ( A I ; A ) \hat{P}(A^I; A) P ^ ( A I ; A ) u = ∑ v u v u = \sum_v u_v u = ∑ v u v u v ∈ A v ( I ) u_v \in A^v(I) u v ∈ A v ( I ) v ∈ N I v \in \mathbf{N}^I v ∈ N I u ~ = ∑ v u ~ v \tilde{u} = \sum_v \tilde{u}_v u ~ = ∑ v u ~ v u ~ v \tilde{u}_v u ~ v ( t i ) i ∈ I ↦ u v ( ( t i ) ) (t_i)_{i \in I} \mapsto u_v((t_i)) ( t i ) i ∈ I ↦ u v (( t i ))
令u = ( u j ) j ∈ I u = (u_j)_{j \in I} u = ( u j ) j ∈ I A ^ ( I ) \hat{A}(I) A ^ ( I ) v ∈ A ^ ( J ) v \in \hat{A}(J) v ∈ A ^ ( J ) w = v ∘ u w = v \circ u w = v ∘ u
( v ∘ u ) ∼ = v ~ ∘ u ~ .
(v \circ u)^{\sim} = \tilde{v} \circ \tilde{u}.
( v ∘ u ) ∼ = v ~ ∘ u ~ . 接下来通过 § 5, no. 1 的连续性公式 (2) 以及可微分和解析流形、R、附录、no. 6 进行扩展。
特别是，我们取 I = {U}、认同 A 和 A ( U ) A^{(U)} A ( U ) P ^ ( A ; A ) \hat{P}(A; A) P ^ ( A ; A ) e ( U ) = ∑ n ≥ 1 U n / n ! e(U) = \sum_{n \geq 1} U^n/n! e ( U ) = ∑ n ≥ 1 U n / n ! l ( U ) = ∑ n ≥ 1 ( − 1 ) n − 1 U n / n l(U) = \sum_{n \geq 1} (-1)^{n-1} U^n/n l ( U ) = ∑ n ≥ 1 ( − 1 ) n − 1 U n / n e ~ \tilde{e} e ~ l ~ \tilde{l} l ~ ∥ U ~ n ∥ ≤ 1 \|\tilde{U}^n\| \leq 1 ∥ U ~ n ∥ ≤ 1 ∥ x 1 … x n ∥ ≤ ∥ x 1 ∥ … ∥ x n ∥ \|x_1 \ldots x_n\| \leq \|x_1\| \ldots \|x_n\| ∥ x 1 … x n ∥ ≤ ∥ x 1 ∥ … ∥ x n ∥ x 1 , … , x n x_1, \ldots, x_n x 1 , … , x n e ~ \tilde{e} e ~ l ~ \tilde{l} l ~ ≥ 1 \geq 1 ≥ 1
我们将用e A e_A e A l A l_A l A e ~ \tilde{e} e ~ l ~ \tilde{l} l ~ exp A ( x ) = 1 + e A ( x ) \exp_A(x) = 1 + e_A(x) exp A ( x ) = 1 + e A ( x ) x ∈ A x \in A x ∈ A
log A ( x ) = l A ( x − 1 )
\log_A(x) = l_A(x - 1)
log A ( x ) = l A ( x − 1 ) （对于 x ∈ A , ∥ x − 1 ∥ < 1 x \in A, \|x - 1\| < 1 x ∈ A , ∥ x − 1∥ < 1
exp A x = ∑ n ≥ 0 x n n ! ( x ∈ A )
\exp_A x = \sum_{n \geq 0} \frac{x^n}{n!} \quad (x \in A)
exp A x = n ≥ 0 ∑ n ! x n ( x ∈ A ) (20) log A x = ∑ n ≥ 1 ( − 1 ) n − 1 ( x − 1 ) n n ( x ∈ A , ∥ x − 1 ∥ < 1 ) . \log_A x = \sum_{n \geq 1} (-1)^{n-1} \frac{(x-1)^n}{n} \quad (x \in A, \|x-1\| < 1). log A x = ∑ n ≥ 1 ( − 1 ) n − 1 n ( x − 1 ) n ( x ∈ A , ∥ x − 1∥ < 1 ) .
作为 ( e ∘ l ) ( U ) = ( l ∘ e ) ( U ) = U (e \circ l)(U) = (l \circ e)(U) = U ( e ∘ l ) ( U ) = ( l ∘ e ) ( U ) = U e ~ ∘ l ~ = l ~ ∘ e ~ = I d A \tilde{e} \circ \tilde{l} = \tilde{l} \circ \tilde{e} = \mathrm{Id}_A e ~ ∘ l ~ = l ~ ∘ e ~ = Id A exp A ( log A ( x ) ) = x ( x ∈ A , ∥ x − 1 ∥ ≤ 1 ) \exp_A(\log_A(x)) = x \quad (x \in A, \|x-1\| \leq 1) exp A ( log A ( x )) = x ( x ∈ A , ∥ x − 1∥ ≤ 1 ) log A ( exp A ( x ) ) = x ( x ∈ A , ∥ x ∥ < log 2 ) \log_A(\exp_A(x)) = x \quad (x \in A, \|x\| < \log 2) log A ( exp A ( x )) = x ( x ∈ A , ∥ x ∥ < log 2 ) ∥ x ∥ < log 2 \|x\| < \log 2 ∥ x ∥ < log 2 ∥ exp A ( x ) − 1 ∥ ≤ exp ∥ x ∥ − 1 < 1 \|\exp_A(x) - 1\| \leq \exp \|x\| - 1 < 1 ∥ exp A ( x ) − 1∥ ≤ exp ∥ x ∥ − 1 < 1
最后我们将 A 视为完全范数李代数。然后
∥ [ x , y ] ∥ = ∥ x y − y x ∥ ≤ 2 ∥ x ∥ ⋅ ∥ y ∥ .
\|[x, y]\| = \|xy - yx\| \leq 2\|x\|\cdot\|y\|.
∥ [ x , y ] ∥ = ∥ x y − y x ∥ ≤ 2∥ x ∥ ⋅ ∥ y ∥. H ~ \tilde{H} H ~ Ω = { x , y ) ∈ A × A ∣ ∥ x ∥ + ∥ y ∥ < 1 2 log 2 } .
\Omega = \{x, y) \in A \times A \mid \|x\| + \|y\| < \frac{1}{2} \log 2\}.
Ω = { x , y ) ∈ A × A ∣ ∥ x ∥ + ∥ y ∥ < 2 1 log 2 } . H ~ \tilde{H} H ~ h : Ω → A h : \Omega \to A h : Ω → A h ( x , y ) = ∑ r , s H r , s ( x , y ) h(x, y) = \sum_{r,s} H_{r,s}(x, y) h ( x , y ) = ∑ r , s H r , s ( x , y )
对于∥ x ∥ + ∥ y ∥ < 1 2 log 2 \|x\| + \|y\| < \frac{1}{2} \log 2 ∥ x ∥ + ∥ y ∥ < 2 1 log 2 exp A x ⋅ exp A y = exp A h ( x , y ) . \exp_A x \cdot \exp_A y = \exp_A h(x, y). exp A x ⋅ exp A y = exp A h ( x , y ) . e u e v = e u + v e^u e^v = e^{u+v} e u e v = e u + v
m ∘ ( 1 + e ~ , 1 + e ~ ) = ( 1 + e ~ ) ∘ H ~
m \circ (1 + \tilde{e}, 1 + \tilde{e}) = (1 + \tilde{e}) \circ \tilde{H}
m ∘ ( 1 + e ~ , 1 + e ~ ) = ( 1 + e ~ ) ∘ H ~ 在
P ^ ( A × A ; A ) \hat{P}(A \times A; A) P ^ ( A × A ; A ) 。因此，我们从
可微分和解析流形 , R, 3.1.9 中推断出 (23) 对于足够接近
( 0 , 0 ) (0, 0) ( 0 , 0 ) 的
( x , y ) (x, y) ( x , y ) 成立，从而命题遵循解析延拓 (
可微分和解析流形 , R, 3.2.5）。
习题
参见习题为 § 7 。