Machine-translated from Chinese.
Introduction to smooth manifolds by John M. Lee
定义
李群就是指能够成为光滑流形的群, 且乘法和求逆都是光滑映射.
它们往往具有非常好的对称结构. 回忆对于群结构的封闭性,
只需验证$ab^{-1}$的封闭性即可. 类似地,
我们只需说明$(a,b)\mapsto ab^{-1}$的光滑性,
来代替乘法和求逆光滑性的说明.
我们定义左平移, 右平移算子$L_g,R_g:G\rightarrow G,$
$$
L_g(h)=gh,\quad R_g(h)=hg.
$$
它们都是光滑映射, 且为微分同胚.
例子
下面列举一些常见李群.
$GL(n,\mathbb{R})$称为一般线性群,
是矩阵空间$M(n,\mathbb{R})$中的开子流形, 为全体可逆矩阵.
它在矩阵乘法的意义下成为群, 容易验证它是一个李群.
记$GL^+(n,\mathbb{R})$为其中行列式取正的全体矩阵, 它也是一个李群.
一般的, 任意李群的开子群(是开流形的子群)都还是一个李群.
复一般线性群$GL(n,\mathbb{C}),$
实/复线性空间$V\rightarrow V$的全体可逆线性映射$GL(V)$也都是李群.
一些更为常见的空间,
如$\mathbb{R}^n,\mathbb{C}^n,\mathbb{R}^\ast ,\mathbb{R}^+,\mathbb{C}^\ast ,S_1,T^n$等都是李群.
李群的直积还是李群. 离散群也可以是李群.
李群间的同态
李群$G,H$间的同态指一个光滑映射$F:G\rightarrow H,$ 满足群同态的要求.
它是李群间的同构, 若其是一个微分同胚.
举一些李群同态的例子, 一些包含映射是显然的同态,
$\exp:\mathbb{R}\rightarrow \mathbb{R}^+$与其逆映射$\log,$
$\det:GL(n,\mathbb{R})\rightarrow \mathbb{R}^\ast $都是李群同态.
特别地共轭算子$C_g:h\mapsto ghg^{-1}$是李群$G\rightarrow G$上的同构.
定理 1. 每个李群间的同态秩为常数.
我们首先说明对于同态$F,$ 它和平移算子可交换:
$$
F(L_{g}(h))=F(gh)=F(g)F(h)=L_{F(g)}(F(h)).
$$
因此在$e$点考虑微分,
$dF_{g}d(L_g)_e=d(L_{F(g)})_{\widetilde{e} }dF_e.$
由于平移算子是微分同胚, 切映射满秩. 因此$r(dF_g)=r(dF_e).$
这就说明了同态秩恒为常数.
结合秩定理, 我们立即得到:
推论 2. 李群同态是李群间的同构当且仅当它是双射.
万有覆盖群
考虑$G$的万有覆盖空间, 我们可以在其上定义群结构, 从而有如下定理成立.
定理 3. 令$G$为连通李群, 存在单连通李群$\widetilde{G}$与光滑覆盖映射$\pi:\widetilde{G}\rightarrow G,$ 且为群同态. 称$\widetilde{G}$为$G$的万有覆盖群. 它也在群同构的意义下是唯一的, 保持覆盖映射.
李子群
李子群即是赋予了拓扑与光滑结构的子群, 它是一个子流形.
容易说明子群作为嵌入子流形自动是李子群.
引理 4. 设$H\subset G$为开子群, 那么$H$是嵌入李子群, 它也是闭的, 为$G$中的若干连通分支之并.
开流形自动是嵌入子流形. 注意到陪集$gH$总是开的,
因为它是微分同胚$L_g$下开集的像. 从而$G\setminus H$是若干开陪集的并,
故$H$也是闭的, 进而得到结论.
命题 5. 设$G$是李群, $W\subset G$为幺元处邻域, 则$W$生成了$G$中的开子群. 当$W$连通时, 生成子群也连通; 当$G$连通时, $W$生成了$G.$
记$W$生成了$H,$ 那么$\,\forall\,h \in H,$ $hW\subset H,$
因此$h$周围包含了平移过来的幺元处的邻域, 从而$H$是开的. 当$W$连通时,
只需证$\,\forall\,h\in H,$ $e,h$道路连通即可.
事实上$h$为$W$中元素及其逆的有限乘, 我们只需证明$h$与$hg$道路连通即可,
$\,\forall\,h\in H,$ $g\in W.$ 这样归纳地即可得出结论.
然而$h,hg\in hW,$ $hW$是$L_h$下连通集的像, 也连通, 命题得证.
最后若$G$连通, 由前面引理, $H$也是闭的, 因此$H$就是全空间$G.$
可见李群$G$中含幺元的连通分支$G_0$很关键,
我们也称它为幺元分支(identity component).
可以证明它是$G$中的正则子群, 且是唯一一个连通开子群.
每个$G$的连通分支都与$G_0$微分同胚.
由前面关于同态秩为常数的结论, 以及秩定理, 我们有:
命题 6. 令$F:G\rightarrow H$为李群间的同态, 那么$\ker F$是逆紧嵌入的李子群, 余维数恰为$r(F).$
命题 7. 若$F:G\rightarrow H$为李群单同态, 那么$F$的像具唯一光滑结构, 使得$F(G)$为李子群, 且$F:G\approx F(G)$为李群同构.
$SL(n,\mathbb{R})$为行列式为$1$的$n$阶实矩阵全体,
称为$n$阶特殊线性群.
它是$\det GL(n,\mathbb{R})\rightarrow \mathbb{R}^\ast $下的核,
因此是逆紧嵌入的李子群. 由于$\det$为满射,
分析秩得到$SL(n,\mathbb{R})$为$n^2-1$维子流形. 同理,
也有复特殊线性群与类似的结论.
我们知道对于光滑子流形, 闭性与嵌入性并没有什么显然的联系,
然而对李子群是有的. 利用拓扑语言分析有:
定理 8. 设$G$为李群, $H\subset G$为李子群, 那么$H$在$G$中是闭的当且仅当它是嵌入的.
由该定理, 每个嵌入李子群也自动是逆紧嵌入的.
因为嵌入子流形逆紧当且仅当其闭.
群作用与等变映射
群作用
李群在$M$上的群作用是李群最关键的应用之一.
李群$G$在黎曼流形$M$上的左作用指映射$G\times M\rightarrow M,$ 满足:
$$
g_1(g_2p)=(g_1g_2)p,\quad ep=p.
$$
右作用类似定义. 当映射光滑时,
称作用为光滑作用. 一般的, 我们会将作用如左作用记为$\theta_g(p)L=gp,$
从而条件变为:
$$
\theta_{g_1}\theta_{g_2}=\theta_{g_1g_2},\quad \theta_e=\mathrm{id}_M.
$$
右作用类似. 对于光滑作用, $\theta_g:M\approx M$为微分同胚,
以$\theta_{g^{-1} }$为逆.
参考经典群论, 我们有轨道与稳定子的概念,
以及作用的可迁与自由性.
一些经典的群作用的例子是矩阵群在$\mathbb{R}^n$上的自然作用,
李群在自己本身上的左作用, 共轭作用等, 以及作用在覆盖空间上的deck变化.
命题 9. 对于覆盖映射$\pi:E\rightarrow M,$ 取离散拓扑的自同构群$\operatorname{Aut}_\pi(E)$为零维李群, 光滑自由地作用在$E$上.
等变映射
称映射$F:M\rightarrow N$关于给定$G$-群作用是等变的, 若对于左作用有
$$
F(gp)=gF(p)
$$
右作用类似.
定理 10. 有光滑流形$M,N$与李群$G.$ 若$F:M\rightarrow N$为光滑映射, 关于$M$上可迁光滑群作用与$N$上光滑群作用是等变的, 那么$F$秩为常数.
对$\varphi_gF=F\theta_g$做微分求切映射, 说明$\,\forall\,p,q\in M,$
$r(dF_p)=r(dF_q)$即可.
等变秩定理有一个重要的应用. 对左作用$\theta,$
定义$\theta^{(p)}:G\rightarrow M,$
$$
\theta^{(p)}(g)=\theta_g(p).
$$
称为轨道映射, 它的像是轨道$Gp.$
命题 11. 轨道映射$\theta^{(p)}:G\rightarrow M$光滑且秩为常数, 故稳定子群$G_p$是逆紧嵌入的李子群. 若作用自由, 轨道$Gp$是$M$中的子流形.
轨道映射是光滑的, 因为有如下分解:
$$
G\approx G\times\{p\}\hookrightarrow G\times M\xrightarrow{\theta}M
$$
事实上它是一个等变映射, $\theta^{(p)}(gh)=g\theta^{p}(h).$
因此由上一命题即可得到结论. 若作用自由, $\theta^{(p)}$是单射,
由整体秩定理它是浸入, 从而轨道都是子流形.
我们可以构造等变映射来计算李群的维数.
文章最后更新于 2022-09-28 22:31:58
Introduction to smooth manifolds by John M. Lee
definition
Li Qun It refers to a group that can become a smooth manifold, and multiplication and inversion are both smooth mappings.
They tend to have very good symmetry. Recall the closedness of group structures,
Just verify that $ab^{-1}$ is closed. Similarly,
We only need to explain the smoothness of $(a,b)\mapsto ab^{-1}$,
to replace multiplication and inverse smoothness instructions.
we define pan left, pan right Operator$L_g,R_g:G\rightarrow G,$
$$
L_g(h)=gh,\quad R_g(h)=hg.
$$
They are both smooth maps and diffeomorphisms.
Example
Here are some common Lie groups.
$GL(n,\mathbb{R})$ is called general linear group,
is an open submanifold in matrix space $M(n,\mathbb{R})$ and is a globally invertible matrix.
It becomes a group in the sense of matrix multiplication, and it is easy to verify that it is a Lie group.
Let $GL^+(n,\mathbb{R})$ be the overall matrix in which the determinant is positive, which is also a Lie group.
In general, any open subgroup of a Lie group (a subgroup of an open manifold) is still a Lie group.
Complex general linear group$GL(n,\mathbb{C}),$
All invertible linear maps $GL(V)$ of the real/complex linear space $V\rightarrow V$ are also Lie groups.
Some of the more common spaces,
For example, $\mathbb{R}^n,\mathbb{C}^n,\mathbb{R}^\ast ,\mathbb{R}^+,\mathbb{C}^\ast ,S_1,T^n$ and so on are all Lie groups.
The direct product of a Lie group is still a Lie group. A discrete group can also be a Lie group.
Homomorphisms among Lie groups
Between Lie groups $G,H$ homomorphism Refers to a smooth map $F:G\rightarrow H,$ that satisfies the requirements of group homomorphism.
It is among Lie groups isomorphism, if it is a diffeomorphism.
To give some examples of Lie group homomorphisms, some inclusion maps are obviously homomorphisms,
$\exp:\mathbb{R}\rightarrow \mathbb{R}^+$ and its inverse mapping $\log,$
$\det:GL(n,\mathbb{R})\rightarrow \mathbb{R}^\ast $ are all Lie group homomorphisms.
In particular, the conjugate operator $C_g:h\mapsto ghg^{-1}$ is an isomorphism on the Lie group $G\rightarrow G$.
Theorem 1. The homomorphism rank between each Lie group is constant.
We first show that for the homomorphism $F,$ it is commutative with the translation operator:
$$
F(L_{g}(h))=F(gh)=F(g)F(h)=L_{F(g)}(F(h)).
$$
Therefore, considering the differential at point $e$,
$dF_{g}d(L_g)_e=d(L_{F(g)})_{\widetilde{e} }dF_e.$
Since the translation operator is diffeomorphism, the tangent map has full rank. Therefore $r(dF_g)=r(dF_e).$
This shows that the homomorphic rank is constant.
Combined with the rank theorem, we immediately get:
Corollary 2. A Lie group isomorphism is an isomorphism between Lie groups if and only if it is a bijection.
Universal coverage group
Considering the universal coverage space of $G$, we can define the group structure on it, so that the following theorem is established.
Theorem 3. Let $G$ be a connected Lie group, there exist a simply connected Lie group $\widetilde{G}$ and a smooth covering map $\pi:\widetilde{G}\rightarrow G,$ and they are group homomorphisms. Call $\widetilde{G}$ the universal covering group of $G$. It is also unique in the sense of group isomorphism, preserving the covering map.
Li Ziqun
A Lie subgroup is a subgroup endowed with topology and smooth structure. It is a submanifold.
It is easy to show that a subgroup as an embedded submanifold is automatically a Plum subgroup.
Lemma 4. Assume $H\subset G$ is an open subgroup, then $H$ is an embedded plum subgroup, which is also closed and is the union of several connected components in $G$.
Open manifolds are automatically embedded submanifolds. Note that the coset $gH$ is always open,
Because it is the image of an open set under diffeomorphism $L_g$. Therefore $G\setminus H$ is the union of several open cosets,
Therefore $H$ is also closed, and the conclusion is obtained.
Proposition 5. Assume $G$ is a Lie group and $W\subset G$ is a unitary neighborhood, then $W$ generates the open subgroup in $G$. When $W$ is connected, the generated subgroup is also connected; when $G$ is connected, $W$ generates $G.$
Remember $W$ generated $H,$ then $\,\forall\,h \in H,$ $hW\subset H,$
Therefore, $h$ is surrounded by the neighborhood of the translated unit, so $H$ is open. When $W$ is connected,
Just prove that $\,\forall\,h\in H,$ $e,h$ roads are connected.
In fact, $h$ is the finite multiplication of the elements in $W$ and its inverse. We only need to prove that $h$ and $hg$ are connected by road,
$\,\forall\,h\in H,$ $g\in W.$ The conclusion can be drawn inductively.
However, $h,hg\in hW,$ $hW$ is the image of the connected set under $L_h$, which is also connected, and the proposition is proved.
Finally, if $G$ is connected, according to the previous lemma, $H$ is also closed, so $H$ is the whole space $G.$
It can be seen that the connected component $G_0$ containing unitary elements in the Lie group $G$ is very critical.
we also call it identity component.
It can be proved that it is a regular subgroup in $G$, and it is the only connected open subgroup.
Every connected component of $G$ is diffeomorphic with $G_0$.
From the previous conclusion that the homomorphic rank is constant and the rank theorem, we have:
Proposition 6. Let $F:G\rightarrow H$ be a homomorphism between Lie groups, then $\ker F$ is an properly embedded Lie subgroup, and the codimension is exactly $r(F).$
Proposition 7. If $F:G\rightarrow H$ is a Lie group monomorphism, then the image of $F$ has a unique smooth structure, such that $F(G)$ is a Lie subgroup, and $F:G\approx F(G)$ is a Lie group isomorphism.
$SL(n,\mathbb{R})$ is the total $n$ order real matrix with determinant $1$,
Called $n$ order special linear group.
It is the core under $\det GL(n,\mathbb{R})\rightarrow \mathbb{R}^\ast $,
Therefore, it is an properly embedded Lie subgroup. Since $\det$ is surjective,
Analyzing the rank, we get that $SL(n,\mathbb{R})$ is the $n^2-1$-dimensional submanifold. In the same way,
There are also complex special linear groups and similar conclusions.
We know that for smooth submanifolds, there is no obvious connection between closure and embeddedness.
However, there is a plum subgroup. Using topological language analysis, there are:
Theorem 8. Assume $G$ is a Lie group and $H\subset G$ is a Lie subgroup, then $H$ is closed in $G$ if and only if it is embedded.
According to this theorem, every embedded Plum subgroup is also automatically properly embedded.
Because the embedded submanifold is inversely compact if and only if it is closed.
Group action and equivariant mapping
group action
The group action of Lie groups on $M$ is one of the most critical applications of Lie groups.
Lie group $G$ on the Riemannian manifold $M$ left acting Refers to the mapping $G\times M\rightarrow M,$ that satisfies:
$$
g_1(g_2p)=(g_1g_2)p,\quad ep=p.
$$
Right action Similar definition. When the mapping is smooth,
called Smoothing effect. Generally, we will write the left-hand action as $\theta_g(p)L=gp,$
So the condition becomes:
$$
\theta_{g_1}\theta_{g_2}=\theta_{g_1g_2},\quad \theta_e=\mathrm{id}_M.
$$
The right action is similar. For smooth action, $\theta_g:M\approx M$ is diffeomorphism,
Take $\theta_{g^{-1} }$ as the inverse.
Referring to classical group theory, we have Orbit with Stabilizer concept,
and functional removable with freedom sex.
Some classic examples of group action are the natural actions of matrix groups on $\mathbb{R}^n$,
The left action, conjugate action, etc. of the Lie group on itself, as well as the deck changes acting on the covering space.
Proposition 9. For the covering map $\pi:E\rightarrow M,$, take the automorphism group $\operatorname{Aut}_\pi(E)$ of the discrete topology as a zero-dimensional Lie group, and act smoothly and freely on $E$.
Equivariant mapping
We call the map $F:M\rightarrow N$ with respect to the given $G$ - the group action is Equivariant , if for the left-hand action there is
$$
F(gp)=gF(p)
$$
The right function is similar.
Theorem 10. There is a smooth manifold $M,N$ and a Lie group $G.$. If $F:M\rightarrow N$ is a smooth map, the transitive smooth group action on $M$ and the smooth group action on $N$ are equivariant, then the rank of $F$ is a constant.
Perform differential tangent mapping on $\varphi_gF=F\theta_g$ to explain $\,\forall\,p,q\in M,$
$r(dF_p)=r(dF_q)$ is enough.
The equivariant rank theorem has an important application. Acting on the left $\theta,$
Definition$\theta^{(p)}:G\rightarrow M,$
$$
\theta^{(p)}(g)=\theta_g(p).
$$
called Orbit mapping, its image is orbit$Gp.$
Proposition 11. The orbit map $\theta^{(p)}:G\rightarrow M$ is smooth and has a constant rank, so the stable subgroup $G_p$ is an properly embedded Lie subgroup. If the action is free, the orbit $Gp$ is a submanifold in $M$.
The orbit map is smooth because it has the following decomposition:
$$
G\approx G\times\{p\}\hookrightarrow G\times M\xrightarrow{\theta}M
$$
In fact it is an equivariant map, $\theta^{(p)}(gh)=g\theta^{p}(h).$
Therefore, the conclusion can be obtained from the previous proposition. If the action is free, $\theta^{(p)}$ is an injective,
According to the global rank theorem, it is immersed, so the orbits are all submanifolds.
We can construct equivariant maps to calculate the dimensions of Lie groups.
The article was last updated on 2022-09-28 22:31:58
