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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 7th 2023

Created:

## 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 $uv-vu$ 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.