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{{Short description|Operator in differential topology}}
{{Use dmy dates|date=December 2021}}
In the mathematical field of [[differential topology]], the '''Lie bracket of vector fields''', also known as the '''Jacobi&ndash;Lie bracket''' or the '''commutator of vector fields''', is an operator that assigns to any two [[vector field]]s <math>X</math> and <math>Y</math> on a [[smooth manifold]] <math>M</math> a third vector field denoted <math>[X,Y]</math>.

Conceptually, the Lie bracket <math>[X,Y]</math> is the derivative of <math>Y</math> along the [[vector flow|flow]] generated by <math>X</math>, and is sometimes denoted ''<math>\mathcal{L}_X Y</math>'' ("Lie derivative of Y along X"). This generalizes to the [[Lie derivative]] of any [[tensor field]] along the flow generated by <math>X</math>.

The Lie bracket is an '''R'''-[[bilinear operator|bilinear]] operation and turns the set of all [[Smoothness|smooth]] vector fields on the manifold <math>M</math> into an (infinite-dimensional) [[Lie algebra]].

The Lie bracket plays an important role in [[differential geometry]] and [[differential topology]], for instance in the [[Frobenius theorem (differential topology)|Frobenius integrability theorem]], and is also fundamental in the geometric theory of [[nonlinear control theory|nonlinear control systems]].<ref>{{harvnb|Isaiah|2009|pp=20&ndash;21}}, [[nonholonomic system]]s; {{harvnb|Khalil|2002|pp=523&ndash;530}}, [[feedback linearization]].</ref>

[[Vladimir Arnold|V. I. Arnold]] refers to this as the "fisherman derivative", as one can imagine being a fisherman, holding a fishing rod, sitting in a boat. Both the boat and the [[Fishing float|float]] are flowing according to vector field <math>X</math>, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field <math>Y</math>. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.<ref>{{Cite book |last=Arnolʹd |first=V. I. |title=Topological methods in hydrodynamics |last2=Khesin |first2=Boris A. |date=1999 |publisher=Springer |isbn=978-0-387-94947-5 |edition=Corr. 2. printing |series=Applied mathematical sciences |location=New York Berlin Heidelberg |pages=6}}</ref>

==Definitions==
There are three conceptually different but equivalent approaches to defining the Lie bracket:

===Vector fields as derivations===
Each smooth vector field <math>X : M \rightarrow TM</math> on a manifold <math>M</math> may be regarded as a [[differential operator]] acting on smooth functions  <math>f(p)</math> (where <math>p \in M</math> and <math>f</math> of class <math>C^\infty(M)</math>) when we define <math>X(f)</math> to be another function whose value at a point <math>p</math> is the [[directional derivative]] of <math>f</math> at <math>p</math> in the direction <math>X(p)</math>. In this way, each smooth vector field <math>X</math> becomes a [[Derivation (abstract algebra)|derivation]] on <math>C^{\infty}(M)</math>. Furthermore, any derivation on <math>C^{\infty}(M)</math> arises from a unique smooth vector field <math>X</math>.

In general, the [[commutator]] <math>\delta_1\circ \delta_2 - \delta_2\circ\delta_1</math> of any two derivations <math>\delta_1</math> and <math>\delta_2</math> is again a derivation, where <math>\circ</math> denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

:<math>[X,Y](f) = X(Y(f))-Y(X(f)) \;\;\text{  for all } f\in C^\infty(M).</math>

===Flows and limits===
Let <math>\Phi^X_t</math> be the [[Flow (mathematics)|flow]] associated with the vector field <math>X</math>, and let <math>D</math> denote the [[Pushforward (differential)|tangent map derivative operator]]. Then the Lie bracket of <math>X</math> and <math>Y</math> at the point <math>x\in M</math> can be defined as the [[Lie derivative]]:

:<math>[X, Y]_x \ =\ (\mathcal{L}_X Y)_x \ :=\ \lim_{t \to 0}\frac{(\mathrm{D}\Phi^X_{-t}) Y_{\Phi^X_t(x)} \,-\, Y_x}t 
\ =\ 
\left.\tfrac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\mathrm{D}\Phi^X_{-t}) Y_{\Phi^X_t(x)} .</math>

This also measures the failure of the flow in the successive directions <math>X,Y,-X,-Y</math> to return to the point <math>x</math>:

:<math>[X, Y]_x \ =\ \left.\tfrac12\tfrac{\mathrm{d}^2}{\mathrm{d}t^2}\right|_{t=0} (\Phi^Y_{-t} \circ \Phi^X_{-t} \circ \Phi^Y_{t} \circ \Phi^X_{t})(x) 
\ =\  \left.\tfrac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\Phi^Y_{\!-\sqrt{t}} \circ \Phi^X_{\!-\sqrt{t}} \circ \Phi^Y_{\!\sqrt{t}} \circ \Phi^X_{\!\sqrt{t}})(x) .</math>

===In coordinates===
Though the above definitions of Lie bracket are [[Differential geometry#Intrinsic versus extrinsic|intrinsic]] (independent of the choice of coordinates on the manifold <math>M</math>), in practice one often wants to compute the bracket in terms of a specific coordinate system <math>\{ x^i \}</math>. We write <math>\partial_i = \tfrac{\partial}{\partial x^i}</math> for the associated local basis of the tangent bundle, so that general vector fields can be written <math>\textstyle X=\sum_{i=1}^n X^i \partial_i</math>and <math>\textstyle Y=\sum_{i=1}^n Y^i \partial_i</math>for smooth functions <math>X^i, Y^i:M\to\mathbb{R}</math>.  Then the Lie bracket can be computed as:

:<math>[X,Y] := \sum_{i=1}^n\left(X(Y^i) - Y(X^i)\right) \partial_i = \sum_{i=1}^n \sum_{j=1}^n \left(X^j \partial_j Y^i - Y^j \partial_j X^i \right) \partial_i .</math>

If <math>M</math> is (an open subset of) <math>\mathbb{R}^n</math>, then the vector fields <math>X</math> and <math>Y</math> can be written as smooth maps of the form <math>X:M\to\mathbb{R}^n</math> and <math>Y:M\to\mathbb{R}^n</math>, and the Lie bracket <math>[X,Y]:M\to\mathbb{R}^n</math> is given by:
:<math>[X,Y] := J_Y  X - J_X Y</math>

where <math>J_Y</math> and <math>J_X</math> are <math>n\times n</math> [[Jacobian matrix|Jacobian matrices]] (<math>\partial_jY^i</math> and <math>\partial_jX^i</math> respectively using index notation) multiplying the <math>n\times 1</math> column vectors <math>X</math> and <math>Y</math>.

==Properties==

The Lie bracket of vector fields equips the real vector space <math>V=\Gamma(TM)</math> of all vector fields on <math>M</math> (i.e., smooth sections of the tangent bundle <math>TM\to M</math>) with the structure of a [[Lie algebra]], which means [ • , • ]  is a map <math>V\times V\to V</math> with:
*'''R'''-[[Bilinear map|bilinearity]]
*Anti-symmetry, <math>[X, Y] = -[Y, X]</math>
*[[Jacobi identity]], <math>[X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0 .</math>

An immediate consequence of the second property is that <math>[X, X] = 0</math> for any <math>X</math>.

Furthermore, there is a "[[product rule]]" for Lie brackets. Given a smooth (scalar-valued) function <math>f</math> on <math>M</math> and a vector field <math>Y</math> on <math>M</math>, we get a new vector field <math>fY</math> by multiplying the vector <math>Y_x</math> by the scalar <math>f(x)</math> at each point <math>x\in M</math>. Then:

*<math> [X, fY] \ =\  X\!(f)\, Y \,+\, f\, [X,Y] ,</math>

where we multiply the scalar function <math>X(f)</math> with the vector field <math>Y</math>, and the scalar function <math>f</math> with the vector field <math>[X,Y]</math>.
This turns the vector fields with the Lie bracket into a [[Lie algebroid]].

Vanishing of the Lie bracket of <math>X</math> and <math>Y</math> means that following the flows in these directions defines a surface embedded in <math>M</math>, with <math>X</math> and <math>Y</math> as coordinate vector fields:

'''Theorem:'''  <math>[X,Y]=0\,</math> iff the flows of <math>X</math> and <math>Y</math> commute locally, meaning <math>(\Phi^Y_t \Phi^X_s) (x) =(\Phi^X_{s}\, \Phi^Y_t)(x)</math> for all <math>x\in M</math> and sufficiently small <math>s</math>, <math>t</math>.

This is a special case of the [[Frobenius integrability theorem]].

== Examples ==
For a [[Lie group]] <math>G</math>, the corresponding [[Lie algebra]] <math>\mathfrak{g}</math> is the tangent space at the identity <math>T_eG</math>, which can be identified with the vector space of [[left invariant]] vector fields on <math>G</math>. The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation <math>[\,\cdot\,,\,\cdot\,]: \mathfrak g \times \mathfrak g\to \mathfrak g</math>.

For a matrix Lie group, whose elements are matrices <math>g \in G \subset M_{n\times n}(\mathbb{R})</math>, each tangent space can be represented as matrices: <math>T_{g}G = g\cdot T_I G \subset M_{n\times n}(\mathbb{R})</math>, where <math>\cdot</math> means matrix multiplication and <math>I</math> is the identity matrix. The invariant vector field corresponding to <math>X\in \mathfrak{g}=T_IG</math> is given by <math>X_g = g\cdot X\in T_gG</math>, and a computation shows the Lie bracket on <math>\mathfrak g</math> corresponds to the usual [[Commutator#Ring theory|commutator]] of matrices:

:<math>[X,Y] \ =\ X\cdot Y - Y\cdot X .</math>

==Generalizations==
As mentioned above, the [[Lie derivative]] can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to [[vector-valued differential form]]s) is the [[Frölicher–Nijenhuis bracket]].

==References==

{{reflist}}
{{reflist|group=note}}

* {{springer|title=Lie bracket|id=p/l058550}}
* {{citation|last=Isaiah|first=Pantelis|title=Controlled parking [Ask the experts]|journal=IEEE Control Systems Magazine|year=2009|volume=29|issue=3|pages=17&ndash;21, 132|doi=10.1109/MCS.2009.932394|s2cid=42908664}}
* {{citation
 | last = Khalil
 | first = H.K.
 | author-link = Hassan K. Khalil
 | year = 2002
 | edition = 3rd
 | url = http://www.egr.msu.edu/~khalil/NonlinearSystems/
 | isbn = 0-13-067389-7
 | title = Nonlinear Systems
 | publisher=[[Prentice Hall]]
 | location = Upper Saddle River, NJ}}
* {{Citation|author=Kolář, I., Michor, P., and Slovák, J.|title=Natural operations in differential geometry|location=Berlin, Heidelberg, New York |url=http://www.emis.de/monographs/KSM/index.html|publisher=Springer-Verlag|year=1993|isbn=3-540-56235-4}} Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
* {{Citation|author=Lang, S.|title=Differential and Riemannian manifolds|publisher=Springer-Verlag|year=1995|isbn=978-0-387-94338-1}} For generalizations to infinite dimensions.
*{{Citation | last =Warner | first = Frank | title = Foundations of differentiable manifolds and Lie groups | orig-year = 1971 | year = 1983 | publisher=Springer-Verlag | location = New York-Berlin | isbn = 0-387-90894-3 }}

{{Manifolds}}

[[Category:Bilinear maps]]
[[Category:Differential geometry]]
[[Category:Differential topology]]
[[Category:Riemannian geometry]]