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 $M$ .

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$[X,Y]=\mathcal{L}_{X}Y$ where $\mathcal{L}_{X}Y$ is the Lie derivative.
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$[\cdot,\cdot]$ is anti-symmetric and bi-linear.
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Vector fields on $M$ with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the Jacobi identity:

$[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0.$
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The Lie bracket is covariant with respect to changes of coordinates.

Title	Lie bracket
Canonical name	LieBracket
Date of creation	2013-03-22 14:10:02
Last modified on	2013-03-22 14:10:02
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	10
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 53-00
Related topic	HamiltonianAlgebroids