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 (x1,…,xn) are local coordinates around some point x∈M, and suppose that in these local coordinates
X(x) | = | Xi(x)∂∂xi|x, | ||
Y(x) | = | Yi(x)∂∂xi|x. |
Then the Lie bracket of the above vector fields is the locally defined vector field
[X,Y](x)=Xi∂Yj∂xi∂∂xj|x-Yi∂Xj∂xi∂∂xj|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]=ℒXY where ℒXY is the Lie derivative
.
-
•
[⋅,⋅] 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 |
---|---|
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 |