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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 7th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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]]. ## Related concepts * [[vector field]] * [[Lie derivative]] * [[Lie algebra]] <a href="https://ncatlab.org/nlab/revision/Lie+bracket+of+vector+fields/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Lie+bracket+of+vector+fields">current</a>

   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.

   Related concepts

   • vector field

   • Lie derivative

   • Lie algebra

   v1, current