Chapter 1 : A brief introduction from geometric perspective

从微分几何视角对李群及李代数的简明引入

§1 Lie groups as manifolds

1.1 Endowed structures

  李群是一类兼具微分流形与群的结构的数学对象，且两者和谐相容。

Definition : A Lie Group is a $C^{\infty}$ manifold which is endowed with a group structure such that the two group operations,
multiplication
$\mu: G\times G\to G$, $\mu(g_{1},g_{2})=g_{1}\cdot g_{2}$
and inverse
$\tau: G\to G$, $\tau(g)=g^{-1}$
are both $C^{\infty}$.

  从微分流形的角度来看，李群被赋予的群结构使得其有许多到自身的微分同胚，如：

Definition : If $G$ is a Lie Group, any element $g\in G$ defines maps $L_{g}:G\to G$ and $R_{g}:G\to G$, called left translation and right translation, respectively by
$L_{g}(h)=g\cdot h$ and $R_{g}(h)=h\cdot g$
for each $h\in G$.

注意求逆运算 $\tau$ 也是一个微分同胚映射，且这三者具有关系：$L_{g}=\tau\circ R_{g}^{-1}\circ\tau$。

  当李群作用于其他流形，特别是其在等距范畴下的齐性空间时，其作为群在对称性、不变性上的性质及意义便更加明显。（李群本身显然也是自己的齐性空间之一）

  而其微分性质使得左不变向量场的集合与单位元处的切空间线性同构而具有泊松括号的封闭性，产生出李代数以及指数映射等结构。自然地提供了连续可微的参数化与群表示方法，可以说微分流形的局部坐标和线性性质与连续群的表示是息息相通的。

1.2 Examples

可以参考 Wikipedia 上的 Table of Lie groups | 本地镜像

§2 Lie algebras as sets of vector fields

2.1 Left-invariant vector fields

对于光滑流形之间的光滑映射，在原点和像点切空间的元素间能够诱导出相应的映射关系

Definition : If $G$ is a Lie Group, any element $g\in G$ defines maps $L_{g}:G\to G$ and $R_{g}:G\to G$, called left translation and right translation, respectively by
$L_{g}(h)=g\cdot h$ and $R_{g}(h)=h\cdot g$
for each $h\in G$.

2.2 The Lie algebra of a Lie group

2.3 Maurer-Cartan’s structure equation

§3 Exponential mapping unto flows

3.1 One-parameter subgroups of Lie groups

3.2 One-parameter groups of transformations generated by vector fields

3.3 The exponential mapping & some calculations

§4 Action of Lie groups on other manifolds

4.1 Definitions

4.2 Orbits and stabilizers

4.3 Homogeneous spaces and examples