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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 , the Lie bracket of vector fields and can be defined in several ways.

   As commutator of derivations

   Since derivations of smooth functions are vector fields, we can identify and with the corresponding derivations .

   Taking the commutator of these derivations produces another derivation, which is denoted by , and which can be identified with a vector field on .

   As a Lie derivative

   Alternatively, we can set

   where denotes the Lie derivative of a vector field.

   Properties

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

   Related concepts

   • vector field

   • Lie derivative

   • Lie algebra

   v1, current