Lie bracket

The Lie bracket is an anticommutative, bilinearPlanetmathPlanetmath, 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 coordinatesPlanetmathPlanetmath 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:


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 xM, 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


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


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

  • [X,Y]=XY where XY is the Lie derivativeMathworldPlanetmathPlanetmath.

  • [,] 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:

  • 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