nLab Lie bracket of vector fields

Definition

Given a smooth manifold $X$, the Lie bracket of vector fields $u$ and $v$ can be defined in several ways.

As commutator of derivations

Since derivations of smooth functions are vector fields, we can identify $u$ and $v$ with the corresponding derivations $C^\infty(X)\to C^\infty(X)$.

Taking the commutator $u v-v u$ of these derivations produces another derivation, which is denoted by $[u,v]$, and which can be identified with a vector field on $X$.

As a Lie derivative

Alternatively, we can set

$$[u,v]=\mathcal{L}_u v=-\mathcal{L}_v u,$$

where $\mathcal{L}$ denotes the Lie derivative of a vector field.

Properties

The real vector space of vector fields on $X$ equipped with the Lie bracket forms a Lie algebra.

Related concepts

• Schouten bracket

• vector field

• Lie derivative

• Lie algebra