# 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 $C^\infty$ on a differentiable (of class $C^\infty$) manifold $M$ as derivations of the algebra $F(M)$ of functions of class $C^\infty$ on $M$, then the Lie bracket of the fields $X$ and $Y$ is given by the formula

$$[X,Y]f=X(Yf)-Y(Xf),$$

where $f\in F(M)$. The totality of all vector fields of class $C^\infty$ on $M$ 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://www.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