Lie algebra of a Lie group

Every Lie group has a Lie algebra associated with it that is compatible with the Lie group structure. This can be taken as a definition/ theorem.

Definition

For a Lie group $G$, we define $T_{e}G=\text{Lie}(G)$ as the Lie algebra of $G$. It is generally denoted $\mathfrak{g}$.

Various constructions

The Lie algebra as a vector space is fairly easy to understand. It is simply the tangent space at the identity element of $G$. However, this does not tell us how the Lie bracket acts on the Lie algebra. These are some constructions of the Lie bracket on $\mathfrak{g}$.

Using vector fields

Let $G$ be a Lie group, and let $\mathfrak{X}(G)$ denote the $\mathbb{R}$ vector space of vector fields on $G$. (For more info on this, look here).

However, $\mathfrak{X}(G)$ is not the Lie algebra of the Lie group $G$. It is too big, so instead we can use the subspace of left-invariant vector fields.

The evaluation map

$$\begin{array}{rl}\phi :{\mathfrak{X}}_L(G)& ⟶T_{e}G=\mathfrak{g}\\ X& ⟼X_e\end{array}$$

is a vector space isomorphism.

Proof

todo

So thus, we can think of a left-invariant vector field $X\in {\mathfrak{X}}_L(G)$ as an element of the Lie algebra of $G$.

Using representations

The adjoint representation of the Lie algebra gives an equivalent definition of the Lie bracket.

$$[x,y]={\text{ad}}_x(y)\forall x,y\in \mathfrak{g}$$

Using the Lie derivative

The Lie derivative gives yet another way to (equivalently) define the Lie bracket.

$$[X,Y]={\mathcal{L}}_{X}Y.$$