The commutator of vector fields (cf. Vector field on a manifold) on a differentiable manifold. If one interprets vector fields of class on a differentiable (of class ) manifold as derivations of the algebra of functions of class on , then the Lie bracket of the fields and is given by the formula
where . The totality of all vector fields of class on is a Lie algebra with respect to the Lie bracket.
The Lie bracket of two vector fields can also be viewed as the Lie derivative of one vector field in the direction of the other.
Lie bracket. A.L. Onishchik (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lie_bracket&oldid=11467