Lie bracket

The “multiplication” of vectors in a Lie algebra. It must satisfy the following properties.

• Bilinearity: $[aX+bY,Z]=a[X,Z]+b[Y,Z]$ (and also in the second component.)
• Antisymmetry: $[X,Y]=-[Y,X]$
• Jacobi Indentity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$

Examples of Lie bracket of $\text{Lie}(G)$

For the Lie algebra of a Lie group, the Lie bracket can be defined in many equivalent ways.

Using vector fields

Let $G$ be a Lie group, and let ${\mathfrak{X}}_L(G)$ denote the $\mathbb{R}$ vector space of left-invariant vector field on $G$.

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

is a vector space isomorphism.

The Lie bracket on vector fields is given by the commutator bracket

$$[X,Y](f)=XY(f)-YX(f)$$

Thus, we can find the Lie bracket of $x,y\in \mathfrak{g}$ using $\phi$ such that

$$[x,y]=\phi ([{\phi }^{-1}(x),{\phi }^{-1}(y)])$$

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.

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