# 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:

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

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

 $\displaystyle X(x)$ $\displaystyle=$ $\displaystyle X^{i}(x)\frac{\partial}{\partial x^{i}}\Big{|}_{x},$ $\displaystyle Y(x)$ $\displaystyle=$ $\displaystyle Y^{i}(x)\frac{\partial}{\partial x^{i}}\Big{|}_{x}.$

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

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

(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 $M$.

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

Title Lie bracket LieBracket 2013-03-22 14:10:02 2013-03-22 14:10:02 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Definition msc 53-00 HamiltonianAlgebroids