Lie bracket

A Lie bracket on a real vector space $\mathfrak{g}$ is a bilinear map

such that:

1. Alternating / skew-symmetry: $[X,X]=0$ for all $X\in\mathfrak{g}$ (equivalently $[X,Y]=-[Y,X]$).
2. Jacobi identity: for all $X,Y,Z\in\mathfrak{g}$, $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$

In differential geometry, there is a canonical Lie bracket on the space of vector fields on a smooth manifold $M$: for vector fields $X,Y$ define $[X,Y]$ by its action on smooth functions,

This produces another vector field and turns the space of vector fields into a Lie algebra.

For a Lie group $G$, the Lie bracket on the Lie algebra of $G$ is obtained by restricting the vector-field bracket to left-invariant vector fields and evaluating at the identity.

Examples

1. On $M=\mathbb{R}^2$ with coordinates $(x,y)$, the coordinate vector fields commute:

   $$\Big[\frac{\partial}{\partial x},\frac{\partial}{\partial y}\Big]=0.$$

2. On $M=\mathbb{R}$ with coordinate $x$, let $X=x\frac{\partial}{\partial x}$ and $Y=\frac{\partial}{\partial x}$. Then

   $$[X,Y] = -\,\frac{\partial}{\partial x}.$$

3. In the matrix Lie algebra $\mathfrak{gl}(2,\mathbb{R})$ (the Lie algebra of $GL(2,\mathbb{R})$), with bracket $[A,B]=AB-BA$, take

   $$A=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad B=\begin{pmatrix}0&0\\1&0\end{pmatrix}.$$

   Then

   $$[A,B]=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$