6.1 Basic setup for Lie group

TLDR

Define Lie group and its associated Lie algebra. Also define Lie subgroup, which is a little subtle.

Preliminary: about differential manifold

Definition of $C^k$-manifold

The definition of $C^k$-manifolds is omitted. You can see here.

Rmk.

• A $C^0$-manifold may not admit a $C^1$-structure, and a $C^0$-manifold may admit more than one $C^1$-structure.
• When $k\ge 1$, a $C^k$-manifold admits a $C^{\infty }$-structure. (Whitney)
• A $C^{\infty }$-manifold admits a unique $C^{\omega }$-structure.
• Every $C^k$-Lie group has a unique $C^{\omega }$-structure w.r.t. which the group operations are $C^{\omega }$. (when $k\ge 2$, easy to prove. $k=0$ highly non-trivial)

Rmk. $C^{\omega }$ means analytic, which is defined here. Note that there exists a function which is smooth but not analytic: $f(x)=e^{-1/x}$.

Immersion, submersion and embedding

Recall $f:M\to N$ is a $C^{\infty }$-map between $C^{\infty }$-manifolds.

• $f$ is said to be an immersion (reps. submersion) at $p\in M$ if

$$T_{p}f:T_{p}M\to T_{f(p)}N$$

is injective (reps. surjective).

• $f$ is called an immersion (reps. submersion) if it is an immersion (reps. submersion) at every $p\in M$.
• $f$ is called embedding if it is a one-to-one immersion and $M\to f(M)$ is a homeomorphism when $f(M)$ is equipped with the subspace topology.
• another definition of embedding: A subset $S$ of a $C^{\infty }$-manifold $M$ is called a regular subset if $\forall p\in S$, there is a coordinate neighborhood $(U,\phi )$ of $p$ in the maximal atlas of $M$ such that $U\cap S$ is defined by vanishing of $(n-k)$-coordinate functions, say $x_k,\cdots ,x_n$. When $S$ is a regular subset,

$$\{U\cap S:(x_1,\cdots ,x_k)\}$$

automatically give $S$ a $C^{\infty }$-manifold structure s.t. $S\hookrightarrow M$ is an embedding.

Rmk. I think the definition of wiki is more precisely: For a immersed submanifold $S$ of $M$, the subset $S$ is not a submanifold of $M$, in the subset topology.

Lie group and Lie subgroup

Definitions

Lie group is a differential manifold with group structure.

Lie group

A $C^k$ Lie group $(k=0,1,\cdots ,\infty ,\omega )$ is an abstract group $G$ together with a $C^k$-manifold structure s.t. the group operations $G\times G\to G,(x,y)\mapsto xy$ and $G\to G,x\to x^{-1}$ are $C^k$-maps.

Lie subgroup

A Lie subgroup $H$ of a Lie group $G$ is a subgroup $H$ together a Lie group structure s.t. the inclusion map $i:H\to G$ is an immersion. A Lie subgroup is called an embedding (or regular) Lie subgroup if $i:H\to G$ is an embedding.

Rmk. (exercise) Let $G$ be a Lie group. If $H\subseteq G$ is a subgroup and a regular sub-manifold, then $H$ is an embedding Lie group.

Later, we will prove that an embedding Lie subgroup is closed. Conversely, any closed subgroup of a Lie group is an embedding Lie subgroup. See here.

Lie group homomorphism

Let $G,H$ be Lie groups.

• A map $\phi :G\to H$ is called a Lie group homomorphism if it is smooth and is a group homomorphism.
• A Lie group homomorphism $\phi :G\to H$ is called an Lie group isomorphism if it is invertible and the inverse ${\phi }^{-1}:H\to G$ is also a Lie group homomorphism.

Famous example

Here is a famous example which shows difference between subgroup and regular subgroup.

Lie algebra of Lie group

Definition

Here are some observations:

• $G$ is a connected Lie group of dimension $n$. Define the tangent bundle $TG={\bigsqcup }_{g\in G}{T}_{g}G$. Note that there is a diffeomorphism

$$\begin{array}{rl}G\times T_{e}G& \to TG,\\ (g,X)& \mapsto (T_e{L}_g)X.\end{array}$$

So the tangent bundle on $G$ is trivial due to the group structure of $G$.

• A vector field $X$ on $G$ is called left-invariant if

$$X_g=(T_e{L}_g)X_e.$$

• In general, given a vector field $X$ on a smooth manifold $M$ and $p\in M$, there exist a solution curve $\gamma :(a,b)\to M$ s.t. $\gamma (t_0)=p$ and ${\gamma }^{\prime }(t)=X_{\gamma (t)}$ for $t\in (a,b)$.

associated Lie algebra

Let $G$ be a connected Lie group. The Lie algebra $g$ associated to $G$ is $g:=$ left-invariant vector fields on $G\simeq T_{e}G$. And the Lie bracket is defined by

$$[X,Y]f=X(Yf)-Y(Xf),f\in C^{\infty }(M).$$

Lie algebra describes non-commutativity of vector fields

Suppose $G$ is a (smooth) manifold $M$, one can attach a Lie algebra

$$\mathfrak{X}(M):=\{\text{smooth vector fields on }M\}$$

where $[X,Y]$ is defined by $[X,Y]f=X(Yf)-Y(Xf)$. When $M=G$ is a Lie group, $TG\simeq G\times T_{e}G$ with $(d{L}_{g}v)\leftarrow (g,v)$. Thus

$$\begin{array}{rl}\mathfrak{X}(G)& \simeq C^{\infty }(G,\mathbb{R}){\otimes }_{\mathbb{R}}{T}_{e}G,\\ f{X}_v& \leftarrow f\otimes v\end{array}$$

where $X_v$ is the left invariant vector field $(X_v{)}_g=d{L}_g(v)$. Thus choose a basis $\{v_1,\cdots ,v_n\}$ of $T_{e}G$, then each vector field on $G$ can be expressed uniquely as $X={\sum }_{i=1}^n{f}_i{X}_{v_i}$.

If $Y={\sum }_{i=1}^n{g}_i{X}_{v_i}$ is another smooth vector field, then

$$[X,Y]=[\sum f_i{X}_{v_i},\sum g_j{X}_{v_j}]=\cdots +\sum _{i,j}{f}_i{g}_j[X_{v_i},X_{v_j}]$$

where $\cdots$ is in the form of $\sum h_i{X}_{v_i}$ and describes non-commutativity of functions by product rule. Thus the major non-commutativity of $\mathfrak{X}(G)$ comes from the Lie algebra structure of $T_{e}G$.

Next question

Then our question is, how do we use the structure of $\mathrm{Lie}G$ to detect the Lie group $G$? It will be answered in 6.2 Relate Lie algebra with Lie group.