Machine-translated from Chinese.
Introduction to smooth manifolds by John M. Lee
定义
称李群$G$上向量场是左不变的, 若
$$
d(L_g)_{g'}(X_{g'})=X_{gg'},\quad \,\forall\,g,g'\in G.
$$
简单写可以记为$(L_g)_\ast X=X.$ 回忆$F_\ast [X,Y]=[F_\ast X,F_\ast Y],$ 我们有:
命题 1. 若$X,Y$是$G$上的光滑左不变向量场, 那么$[X,Y]$也是左不变的.
一个实向量空间上的李代数指它配备了一个李括号, 满足双线性性, 反称性,
以及Jacobi恒等式:
$$
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.
$$
我们有子李代数, 李代数同态(线性映射$A:$ $A[X,Y]=[AX,AY],$ 与李代数同构.
我们知道$\mathfrak{X}(M)$是一个李括号. 特别地由前面的命题,
光滑左不变向量场全体是一个子李括号. 称为$G$的李代数,
记为$\operatorname{Lie}(G).$
命题 2. 对李群同态$F:G\rightarrow H,$ $F^\ast $为李代数同态.
由于$L_g$是可迁的,
可以看出任意左不变向量场由$X_e\in T_eG$通过$(L_g)_\ast $决定.
单参数李子群
一个$G$的单参数李子群指一个李群同态 $\gamma:\mathbb{R}\rightarrow G.$
定理 3. $G$的单参数李子群恰是一个原点出发的左不变向量场的极大积分曲线.
因此对$X\in \operatorname{Lie}(G),$ 它决定了一个单参数李子群,
称为由$X$生成的单参数李子群.
定义指数映射:
$$
\exp X=\gamma(1),\quad X\in \operatorname{Lie}(G),
$$
$\gamma$为由原点出发的$X$的积分曲线. 容易得到$\gamma(s)=\exp sX.$
它满足很多类似指数函数的性质.
引理 4. $G$为连通李群, $H\subset G$为连通李子群. 那么$H$是正则子群当且仅当
$$
\exp(X)\exp(Y)\exp(-X)\in H,\quad \,\forall\,X\in\operatorname{Lie}(G),Y\in\operatorname{Lie}(H).
$$
文章最后更新于 2022-10-18 17:41:33
Introduction to smooth manifolds by John M. Lee
definition
The vector field on the Lie group $G$ is called left unchanged of, if
$$
d(L_g)_{g'}(X_{g'})=X_{gg'},\quad \,\forall\,g,g'\in G.
$$
Simply written as $(L_g)_\ast X=X.$ recalling $F_\ast [X,Y]=[F_\ast X,F_\ast Y],$ we have:
Proposition 1. If $X,Y$ is a smooth left-invariant vector field on $G$, then $[X,Y]$ is also left-invariant.
A Lie algebra on a real vector space means that it is equipped with a Lie bracket, satisfies bilinearity and inverse symmetry,
And the Jacobi identity:
$$
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.
$$
We have subLie algebras, Lie algebra isomorphisms (linear maps $A:$ $A[X,Y]=[AX,AY],$ are isomorphisms with Lie algebras.
We know that $\mathfrak{X}(M)$ is a Lie bracket. In particular, from the previous proposition,
The whole smooth left invariant vector field is a sub-Lie bracket. It is called $G$ Lie algebra,
Recorded as $\operatorname{Lie}(G).$
Proposition 2. The Lie group homomorphisms $F:G\rightarrow H,$ $F^\ast $ are Lie algebraic homomorphisms.
Since $L_g$ is removable,
It can be seen that any left invariant vector field is determined by $X_e\in T_eG$ through $(L_g)_\ast $.
One-parameter plum group
A one-parameter Lie group of $G$ refers to a Lie group homomorphism $\gamma:\mathbb{R}\rightarrow G.$
Theorem 3. The single-parameter Lie subgroup of $G$ is exactly the maximum integral curve of a left-invariant vector field starting from the origin.
Therefore for $X\in \operatorname{Lie}(G),$ it determines a one-parameter plum group,
is called the one-parameter plum group generated by $X$.
Define index mapping:
$$
\exp X=\gamma(1),\quad X\in \operatorname{Lie}(G),
$$
$\gamma$ is the integral curve of $X$ starting from the origin. It is easy to get $\gamma(s)=\exp sX.$
It satisfies many properties similar to exponential functions.
Lemma 4. $G$ is a connected Lie group, $H\subset G$ is a connected Lie subgroup. Then $H$ is a regular subgroup if and only if
$$
\exp(X)\exp(Y)\exp(-X)\in H,\quad \,\forall\,X\in\operatorname{Lie}(G),Y\in\operatorname{Lie}(H).
$$
The article was last updated on 2022-10-18 17:41:33
