# Chapter 1 : A brief introduction from geometric perspective

2022-10-27
2分钟阅读时长

## §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 $h\in G$.

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。

## §2 Lie algebras as sets of vector fields

### 2.1 Left-invariant vector fields

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 $h\in G$.