Lie bracket

The Lie bracket is an anticommutative, bilinear, first order differential operator on vector fields. It may be defined either in terms of local coordinates or in a global, coordinate-free fashion. Though both defintions are prevalent, it is perhaps easier to formulate the Lie Bracket without the use of coordinates at all, as a commutator:

Definition (Global, coordinate-free) Suppose $X$ and $Y$ are vector fields on a smooth manifold $M$. Regarding these vector fields as operators on functions, the Lie bracket is their commutator:

$[X,Y](f)=X(Y(f))-Y(X(f)).$

Definition (Local coordinates) Suppose $X$ and $Y$ are vector fields on a smooth $n$-dimensional manifold $M$,suppose $(x^1,\mathrm{\dots },x^n)$ are local coordinates around some point $x\in M$,and suppose that in these local coordinates

$X(x)$$=$${X^i(x){\displaystyle \frac{\partial }{\partial x^i}}|}_x,$

$Y(x)$$=$${Y^i(x){\displaystyle \frac{\partial }{\partial x^i}}|}_x.$

Then the Lie bracket of the above vector fields is the locally defined vector field

$$[X,Y](x)={X^i\frac{\partial Y^j}{\partial x^i}\frac{\partial }{\partial x^j}|}_x-{Y^i\frac{\partial X^j}{\partial x^i}\frac{\partial }{\partial x^j}|}_x.$$

(The Einstein summation convention employed in the above equations —repeated indices are to be summed from the range 1 to $n$.)

Properties

Suppose $X,Y,Z$ are smooth vector fields on a smooth manifold $M$.

• $[X,Y]={ℒ}_{X}Y$ where ${ℒ}_{X}Y$ is the Lie derivative.

• $[\cdot ,\cdot ]$ is anti-symmetric and bi-linear.

• Vector fields on $M$ with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the Jacobi identity:

  $$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$$

• The Lie bracket is covariant with respect to changes of coordinates.