Lie bracket of vector fields

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

Each smooth vector field ${\displaystyle X:M\rightarrow TM}$ on a manifold ${\displaystyle M}$ may be regarded as a differential operator acting on smooth functions ${\displaystyle f(p)}$ (where ${\displaystyle p\in M}$ 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 ${\displaystyle f}$ at ${\displaystyle p}$ in the direction ${\displaystyle X(p)}$. In this way, each smooth vector field ${\displaystyle X}$ becomes a derivation on ${\displaystyle C^{\infty }(M)}$. Furthermore, any derivation on ${\displaystyle C^{\infty }(M)}$ arises from a unique smooth vector field ${\displaystyle X}$.

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:

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

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

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

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

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

Though the above definitions of Lie bracket are intrinsic (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 ${\displaystyle \partial _{i}={\tfrac {\partial }{\partial x^{i}}}}$ 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 ${\displaystyle X^{i},Y^{i}:M\to \mathbb {R} }$. Then the Lie bracket can be computed as:

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

If ${\displaystyle M}$ 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 (${\displaystyle \partial _{j}Y^{i}}$ and ${\displaystyle \partial _{j}X^{i}}$ 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 ${\displaystyle V=\Gamma (TM)}$ of all vector fields on ${\displaystyle M}$ (i.e., smooth sections of the tangent bundle ${\displaystyle TM\to M}$) with the structure of a Lie 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 a Lie 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 ${\displaystyle (\Phi _{t}^{Y}\Phi _{s}^{X})(x)=(\Phi _{s}^{X}\,\Phi _{t}^{Y})(x)}$ for all ${\displaystyle x\in M}$ and sufficiently small ${\displaystyle s}$, ${\displaystyle t}$.

This is a special case of the Frobenius integrability theorem.

For a Lie group ${\displaystyle G}$, 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 ${\displaystyle G}$. 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 usual commutator of matrices:

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

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.