PlanetMath: Lie bracket

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Lie bracket

(Definition)

The Lie bracket is an antisymmetric, 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 and are vector fields on a smooth manifold . 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 and are vector fields on a smooth -dimensional manifold , suppose $(x^1,\ldots, x^n)$ are local coordinates around some point $x\in M$ , and suppose that in these local coordinates

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

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

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

(The Einstein summation convention employed in the above equations -- repeated indices are to be summed from the range 1 to

Properties

Suppose

are smooth vector fields on a smooth manifold

$[X,Y]=\mathcal{L}_XY$ where $\mathcal{L}_XY$ is the Lie derivative.
$[\cdot,\cdot]$ is anti-symmetric and bi-linear.
Vector fields on with the Lie bracket is a Lie algebra. That is to say, the Lie bracket satisfies the Jacobi identity:
$\displaystyle [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]]=0.$
The Lie bracket is covariant with respect to changes of coordinates.

"Lie bracket" is owned by rspuzio. [ full author list (3) | owner history (2) ]

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Cross-references: changes of coordinates, Jacobi identity, Lie algebra, anti-symmetric, Lie derivative, range, indices, equations, Einstein summation convention, point, smooth, functions, operators, smooth manifold, commutator, coordinates, local coordinates, terms, vector fields, differential operator, first order, bilinear, antisymmetric
There are 23 references to this entry.

This is version 6 of Lie bracket, born on 2004-02-16, modified 2006-10-14.
Object id is 5591, canonical name is LieBracket.
Accessed 8413 times total.

Classification:

AMS MSC:

53-00 (Differential geometry :: General reference works )

Pending Errata and Addenda

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