Lie bracket - Knowlpedia

A Lie bracket on a vector space $\mathfrak{g}$ over a field $k$ is a $k$ -bilinear map

[\,,\,]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}

such that:

Alternating: $[X,X]=0$ for all $X\in\mathfrak{g}$ .
Jacobi identity: for all $X,Y,Z\in\mathfrak{g}$ , $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$

Alternation and vector fields

Alternation implies $[X,Y]=-[Y,X]$ ; the converse holds when the field has characteristic different from $2$ .

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,

[X,Y](f) := X(Y(f)) - Y(X(f)), \qquad f\in C^\infty(M).

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

On $M=\mathbb{R}^2$ with coordinates $(x,y)$ , the coordinate vector fields commute: $\Big[\frac{\partial}{\partial x},\frac{\partial}{\partial y}\Big]=0.$

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

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