In the mathematical field ofdifferential topology, theLie bracket of vector fields, also known as theJacobi–Lie bracket or thecommutator of vector fields, is an operator that assigns to any twovector fields${\displaystyle X}$ and${\displaystyle Y}$ on asmooth manifold${\displaystyle M}$ a third vector field denoted${\displaystyle [X,Y]}$.

Conceptually, the Lie bracket${\displaystyle [X,Y]}$ is the derivative of${\displaystyle Y}$ along theflow generated by${\displaystyle X}$, and is sometimes denoted${\displaystyle {\mathcal{L}}_{X}Y}$ ("Lie derivative of Y along X"). This generalizes to theLie derivative of anytensor field along the flow generated by${\displaystyle X}$.

The Lie bracket is anR-bilinear operation and turns the set of allsmooth vector fields on the manifold${\displaystyle M}$ into an (infinite-dimensional)Lie algebra.

The Lie bracket plays an important role indifferential geometry anddifferential topology, for instance in theFrobenius integrability theorem, and is also fundamental in the geometric theory ofnonlinear control systems.^[1]

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 thefloat are flowing according to vector field${\displaystyle X}$, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field${\displaystyle Y}$. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.^[2]

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

Each smooth vector field${\displaystyle X:M\to TM}$ on a manifold${\displaystyle M}$ may be regarded as adifferential operator acting on smooth functions${\displaystyle f(p)}$ (where${\displaystyle p\in M}$ and${\displaystyle f}$ of class${\displaystyle C^{\mathrm{\infty }}(M)}$) when we define${\displaystyle X(f)}$ to be another function whose value at a point${\displaystyle p}$ is thedirectional derivative of${\displaystyle f}$ at${\displaystyle p}$ in the direction${\displaystyle X(p)}$. In this way, each smooth vector field${\displaystyle X}$ becomes aderivation on${\displaystyle C^{\mathrm{\infty }}(M)}$. Furthermore, any derivation on${\displaystyle C^{\mathrm{\infty }}(M)}$ arises from a unique smooth vector field${\displaystyle X}$.

In general, thecommutator${\displaystyle {\delta }_1\circ {\delta }_2-{\delta }_2\circ {\delta }_1}$ of any two derivations${\displaystyle {\delta }_1}$ and${\displaystyle {\delta }_2}$ is again a derivation, where${\displaystyle \circ }$ denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

${\displaystyle [X,Y](f)=X(Y(f))-Y(X(f))\text{ for all }f\in C^{\mathrm{\infty }}(M).}$

Let${\displaystyle {\mathrm{\Phi }}_t^X}$ be theflow associated with the vector field${\displaystyle X}$, and let${\displaystyle D}$ denote thetangent map derivative operator. Then the Lie bracket of${\displaystyle X}$ and${\displaystyle Y}$ at the point${\displaystyle x\in M}$ can be defined as theLie derivative:

${\displaystyle [X,Y{]}_x=({\mathcal{L}}_{X}Y{)}_x:=\underset{t\to 0}{lim}\frac{(\mathrm{D}{\mathrm{\Phi }}_{-t}^X)Y_{{\mathrm{\Phi }}_t^X(x)}-Y_x}{t}={\frac{\mathrm{d}}{\mathrm{d}t}|}_{t=0}(\mathrm{D}{\mathrm{\Phi }}_{-t}^X)Y_{{\mathrm{\Phi }}_t^X(x)}.}$

This also measures the failure of the flow in the successive directions${\displaystyle X,Y,-X,-Y}$ to return to the point${\displaystyle x}$:

${\displaystyle [X,Y{]}_x={\frac{1}{2}\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}|}_{t=0}({\mathrm{\Phi }}_{-t}^Y\circ {\mathrm{\Phi }}_{-t}^X\circ {\mathrm{\Phi }}_t^Y\circ {\mathrm{\Phi }}_t^X)(x)={\frac{\mathrm{d}}{\mathrm{d}t}|}_{t=0}({\mathrm{\Phi }}_{-\sqrt{t}}^Y\circ {\mathrm{\Phi }}_{-\sqrt{t}}^X\circ {\mathrm{\Phi }}_{\sqrt{t}}^Y\circ {\mathrm{\Phi }}_{\sqrt{t}}^X)(x).}$

Though the above definitions of Lie bracket areintrinsic (independent of the choice of coordinates on the manifold${\displaystyle M}$), in practice one often wants to compute the bracket in terms of a specific coordinate system${\displaystyle \{x^i\}}$. We write${\displaystyle {\mathrm{\partial }}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}}$ for the associated local basis of the tangent bundle, so that general vector fields can be written${\displaystyle X=\sum _{i=1}^n{X}^i{\mathrm{\partial }}_i}$and${\displaystyle Y=\sum _{i=1}^n{Y}^i{\mathrm{\partial }}_i}$for smooth functions${\displaystyle X^i,Y^i:M\to \mathbb{R}}$. Then the Lie bracket can be computed as:

${\displaystyle [X,Y]:=\sum _{i=1}^n\left(X(Y^i)-Y(X^i)\right){\mathrm{\partial }}_i=\sum _{i=1}^n\sum _{j=1}^n\left(X^j{\mathrm{\partial }}_j{Y}^i-Y^j{\mathrm{\partial }}_j{X}^i\right){\mathrm{\partial }}_i.}$

If${\displaystyle M}$ is (an open subset of)${\displaystyle {\mathbb{R}}^n}$, then the vector fields${\displaystyle X}$ and${\displaystyle Y}$ can be written as smooth maps of the form${\displaystyle X:M\to {\mathbb{R}}^n}$ and${\displaystyle Y:M\to {\mathbb{R}}^n}$, and the Lie bracket${\displaystyle [X,Y]:M\to {\mathbb{R}}^n}$ is given by:

${\displaystyle [X,Y]:=J_{Y}X-J_{X}Y}$

where${\displaystyle J_Y}$ and${\displaystyle J_X}$ are${\displaystyle n\times n}$Jacobian matrices (${\displaystyle {\mathrm{\partial }}_j{Y}^i}$ and${\displaystyle {\mathrm{\partial }}_j{X}^i}$ respectively using index notation) multiplying the${\displaystyle n\times 1}$ column vectors${\displaystyle X}$ and${\displaystyle Y}$.

The Lie bracket of vector fields equips the real vector space${\displaystyle V=\mathrm{\Gamma }(TM)}$ of all vector fields on${\displaystyle M}$ (i.e., smooth sections of the tangent bundle${\displaystyle TM\to M}$) with the structure of aLie algebra, which means [ • , • ] is a map${\displaystyle V\times V\to V}$ with:

• R-bilinearity
• Anti-symmetry,${\displaystyle [X,Y]=-[Y,X]}$
• Jacobi identity,${\displaystyle [X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.}$

An immediate consequence of the second property is that${\displaystyle [X,X]=0}$ for any${\displaystyle X}$.

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

• ${\displaystyle [X,fY]=X(f)Y+f[X,Y],}$

where we multiply the scalar function${\displaystyle X(f)}$ with the vector field${\displaystyle Y}$, and the scalar function${\displaystyle f}$ with the vector field${\displaystyle [X,Y]}$.This turns the vector fields with the Lie bracket into aLie algebroid.

Vanishing of the Lie bracket of${\displaystyle X}$ and${\displaystyle Y}$ means that following the flows in these directions defines a surface embedded in${\displaystyle M}$, with${\displaystyle X}$ and${\displaystyle Y}$ as coordinate vector fields:

Theorem:${\displaystyle [X,Y]=0}$ iff the flows of${\displaystyle X}$ and${\displaystyle Y}$ commute locally, meaning${\displaystyle ({\mathrm{\Phi }}_t^Y{\mathrm{\Phi }}_s^X)(x)=({\mathrm{\Phi }}_s^X{\mathrm{\Phi }}_t^Y)(x)}$ for all${\displaystyle x\in M}$ and sufficiently small${\displaystyle s}$,${\displaystyle t}$.

This is a special case of theFrobenius integrability theorem.

For aLie group${\displaystyle G}$, the correspondingLie algebra${\displaystyle \mathfrak{g}}$ is the tangent space at the identity${\displaystyle T_{e}G}$, which can be identified with the vector space ofleft invariant vector fields on${\displaystyle G}$. The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation${\displaystyle [\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}}$.

For a matrix Lie group, whose elements are matrices${\displaystyle g\in G\subset M_{n\times n}(\mathbb{R})}$, each tangent space can be represented as matrices:${\displaystyle T_{g}G=g\cdot T_{I}G\subset M_{n\times n}(\mathbb{R})}$, where${\displaystyle \cdot }$ means matrix multiplication and${\displaystyle I}$ is the identity matrix. The invariant vector field corresponding to${\displaystyle X\in \mathfrak{g}=T_{I}G}$ is given by${\displaystyle X_g=g\cdot X\in T_{g}G}$, and a computation shows the Lie bracket on${\displaystyle \mathfrak{g}}$ corresponds to the usualcommutator of matrices:

${\displaystyle [X,Y]=X\cdot Y-Y\cdot X.}$

As mentioned above, theLie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (tovector-valued differential forms) is theFrölicher–Nijenhuis bracket.

• "Lie bracket",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Isaiah, Pantelis (2009), "Controlled parking [Ask the experts]",IEEE Control Systems Magazine,29 (3):17–21, 132,doi:10.1109/MCS.2009.932394,S2CID 42908664
• Khalil, H.K. (2002),Nonlinear Systems (3rd ed.), Upper Saddle River, NJ:Prentice Hall,ISBN 0-13-067389-7
• Kolář, I., Michor, P., and Slovák, J. (1993),Natural operations in differential geometry, Berlin, Heidelberg, New York: Springer-Verlag,ISBN 3-540-56235-4{{citation}}: CS1 maint: multiple names: authors list (link) Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
• Lang, S. (1995),Differential and Riemannian manifolds, Springer-Verlag,ISBN 978-0-387-94338-1 For generalizations to infinite dimensions.
• Warner, Frank (1983) [1971],Foundations of differentiable manifolds and Lie groups, New York-Berlin: Springer-Verlag,ISBN 0-387-90894-3

• Bilinear maps
• Differential geometry
• Differential topology
• Riemannian geometry

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