Lie bracket
From Encyclopedia of Mathematics
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.
Comments
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.
How to Cite This Entry:
Lie bracket. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_bracket&oldid=31676
Lie bracket. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_bracket&oldid=31676
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article