Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y].

Conceptually, the Lie bracket [X, Y] is the derivative of Y along the flow generated by X, and is sometimes denoted

${displaystyle {mathcal {L}}_{X}Y}$

(“Lie derivative of Y along X”). This generalizes to the Lie derivative of any tensor field along the flow generated by X.

The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.

The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.^[1]

Definitions[edit]

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

Vector fields as derivations[edit]

Each smooth vector field

${displaystyle X:Mrightarrow TM}$

on a manifold M may be regarded as a differential operator acting on smooth functions

${displaystyle f(p)}$

(where

${displaystyle pin M}$

and

${displaystyle f}$

of class

${displaystyle C^{infty }(M)}$

) when we define

${displaystyle X(f)}$

to be another function whose value at a point

${displaystyle p}$

is the directional derivative of f at p in the direction X(p). In this way, each smooth vector field X becomes a derivation on C^∞(M). Furthermore, any derivation on C^∞(M) arises from a unique smooth vector field X.

In general, the commutator

${displaystyle delta _{1}circ delta _{2}-delta _{2}circ delta _{1}}$

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:

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

Flows and limits[edit]

Let

${displaystyle Phi _{t}^{X}}$

be the flow associated with the vector field X, and let D denote the tangent map derivative operator. Then the Lie bracket of X and Y at the point x ∈ M can be defined as the Lie derivative:

${displaystyle [X,Y]_{x} = ({mathcal {L}}_{X}Y)_{x} := lim _{tto 0}{frac {(mathrm {D} Phi _{-t}^{X})Y_{Phi _{t}^{X}(x)},-,Y_{x}}{t}} = left.{tfrac {mathrm {d} }{mathrm {d} t}}right|_{t=0}(mathrm {D} Phi _{-t}^{X})Y_{Phi _{t}^{X}(x)}.}$

This also measures the failure of the flow in the successive directions

${displaystyle X,Y,-X,-Y}$

to return to the point x:

${displaystyle [X,Y]_{x} = left.{tfrac {1}{2}}{tfrac {mathrm {d} ^{2}}{mathrm {d} t^{2}}}right|_{t=0}(Phi _{-t}^{Y}circ Phi _{-t}^{X}circ Phi _{t}^{Y}circ Phi _{t}^{X})(x) = left.{tfrac {mathrm {d} }{mathrm {d} t}}right|_{t=0}(Phi _{!-{sqrt {t}}}^{Y}circ Phi _{!-{sqrt {t}}}^{X}circ Phi _{!{sqrt {t}}}^{Y}circ Phi _{!{sqrt {t}}}^{X})(x).}$

In coordinates[edit]

Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold M), in practice one often wants to compute the bracket in terms of a specific coordinate system

${displaystyle {x^{i}}}$

. We write

${displaystyle partial _{i}={tfrac {partial }{partial x^{i}}}}$

for the associated local basis of the tangent bundle, so that general vector fields can be written

${displaystyle textstyle X=sum _{i=1}^{n}X^{i}partial _{i}}$

and

${displaystyle textstyle Y=sum _{i=1}^{n}Y^{i}partial _{i}}$

for smooth functions

${displaystyle X^{i},Y^{i}:Mto mathbb {R} }$

. Then the Lie bracket can be computed as:

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

If M is (an open subset of) Rⁿ, then the vector fields X and Y can be written as smooth maps of the form

${displaystyle X:Mto mathbb {R} ^{n}}$

and

${displaystyle Y:Mto mathbb {R} ^{n}}$

, and the Lie bracket

${displaystyle [X,Y]:Mto mathbb {R} ^{n}}$

is given by:

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

where

${displaystyle J_{Y}}$

and

${displaystyle J_{X}}$

are n × n Jacobian matrices (

${displaystyle partial _{j}Y^{i}}$

and

${displaystyle partial _{j}X^{i}}$

respectively using index notation) multiplying the n × 1 column vectors X and Y.

Properties[edit]

The Lie bracket of vector fields equips the real vector space

${displaystyle V=Gamma (TM)}$

of all vector fields on M (i.e., smooth sections of the tangent bundle

${displaystyle TMto M}$

) with the structure of a Lie algebra, which means [ • , • ] is a map

${displaystyle Vtimes Vto V}$

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 f on M and a vector field Y on M, we get a new vector field fY by multiplying the vector Y_x by the scalar f(x) at each point x ∈ M. Then:

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

  

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

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

Theorem:

${displaystyle [X,Y]=0,}$

iff the flows of X and Y commute locally, meaning

${displaystyle (Phi _{t}^{Y}Phi _{s}^{X})(x)=(Phi _{s}^{X},Phi _{t}^{Y})(x)}$

for all x ∈ M and sufficiently small s, t.

This is a special case of the Frobenius integrability theorem.

Examples[edit]

For a Lie group G, the corresponding Lie algebra

${displaystyle {mathfrak {g}}}$

is the tangent space at the identity

${displaystyle T_{e}G}$

, which can be identified with the vector space of left invariant vector fields on 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

${displaystyle gin Gsubset M_{ntimes n}(mathbb {R} )}$

, each tangent space can be represented as matrices:

${displaystyle T_{g}G=gcdot T_{I}Gsubset M_{ntimes n}(mathbb {R} )}$

, where

${displaystyle cdot }$

means matrix multiplication and I is the identity matrix. The invariant vector field corresponding to

${displaystyle Xin {mathfrak {g}}=T_{I}G}$

is given by

${displaystyle X_{g}=gcdot Xin T_{g}G}$

, and a computation shows the Lie bracket on

${displaystyle {mathfrak {g}}}$

corresponds to the usual commutator of matrices:

${displaystyle [X,Y] = Xcdot Y-Ycdot X.}$

Applications[edit]

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

Generalizations[edit]

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 forms) is the Frölicher–Nijenhuis bracket.

References[edit]

• “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”, Springer-Verlag, Berlin, Heidelberg, New York, 1993, Springer-Verlag{{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.
• Lewis, Andrew D., Notes on (Nonlinear) Control Theory (PDF)^{[permanent dead link]}
• Warner, Frank (1983) [1971], Foundations of differentiable manifolds and Lie groups, New York-Berlin: Springer-Verlag, ISBN 0-387-90894-3