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. A generalization of the Lie bracket is the Lie derivative, which allows differentiation of any tensor field along the flow generated by X. The Lie bracket [X,Y] equals the Lie derivative of the vector Y (which is a tensor field) along X, and is sometimes denoted $\mathcal{L}_X Y$ (read "the Lie derivative of Y along X").

The Lie bracket is an R-bilinear operation and turns the set of all 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 theorem, and is also fundamental in the geometric theory for nonlinear control systems (Isaiah 2009, pp. 20–21, nonholonomic systems; Khalil 2002, pp. 523–530, feedback linearization).

Definitions

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

Vector fields as derivations

Each vector field X on a smooth manifold M may be regarded as a differential operator acting on smooth functions on M. Indeed, each smooth vector field X becomes a derivation on the smooth functions C^∞(M) when we define X(f) to be the element of C^∞(M) whose value at a point p is the directional derivative of f at p in the direction X(p). Furthermore, it is known that any derivation on C^∞(M) arises in this fashion from a uniquely determined smooth vector field X.

In general, the commutator $\delta_1\circ \delta_2 - \delta_2\circ\delta_1$ of any two derivations $\delta_1$ and $\delta_2$ is again a derivation. This can be used to define the Lie bracket of vector fields as follows.

The Lie bracket, [X,Y], of two smooth vector fields X and Y is the smooth vector field [X,Y] such that

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

Flows and limits

Let $\Phi^X_t$ 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

$[X, Y]_x := \lim_{t \to 0}\frac{(\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)} - Y_x}t = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\mathrm{d}\Phi^X_{-t}) Y_{\Phi^X_t(x)}$

or in terms of the Lie derivative

$[X, Y] = \mathcal{L}_X Y$

which is also equivalent to

$[X, Y]_x := \left.\frac12\frac{\mathrm{d}^2}{\mathrm{dt}^2}\right|_{t=0} (\Phi^Y_{-t} \circ \Phi^X_{-t} \circ \Phi^Y_{t} \circ \Phi^X_{t})(x) = \left.\frac{\mathrm{d}}{\mathrm{d} t}\right|_{t=0} (\Phi^Y_{-\sqrt{t}} \circ \Phi^X_{-\sqrt{t}} \circ \Phi^Y_{\sqrt{t}} \circ \Phi^X_{\sqrt{t}})(x)$

In coordinates

Though neither definition of the Lie bracket depends on a choice of coordinates, in practice one often wants to compute the bracket with respect to a coordinate system.

If we have picked a coordinate chart on M with local coordinate functions $\{x^i \}$, and we write $\partial_i = \frac{\partial}{\partial x^i}$ for the associated local basis for the tangent bundle, then the vector fields can be written as

$X=\sum_{i=1}^n X^i \partial_i$

and

$Y=\sum_{i=1}^n Y^i \partial_i$

with smooth functions $X^i:M\to\mathbb{R}$ and $Y^i:M\to\mathbb{R}$. Then the Lie bracket is given by

$[X,Y] := \sum_{i=1}^n\left(X(Y^i) - Y(X^i)\right) \partial_i = \sum_{i=1}^n \sum_{j=1}^n \left(X^j \partial_j Y^i - Y^j \partial_j X^i \right) \partial_i$

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

$[X,Y] := J_Y X - J_X Y$

where $J_Y$ and $J_X$ are the Jacobian matrices of $Y$ and $X$, respectively. These n-by-n matrices are multiplied by the n-vectors X and Y.

Properties

The Lie bracket of vector fields equips the real vector space $V=\Gamma(TM)$ of all vector fields on M (i.e., smooth sections of the tangent bundle $TM$ of $M$) with the structure of a Lie algebra, i.e., [·,·] is a map $V\times V\to V$ with the following properties

• $[\cdot,\cdot]$ is R-bilinear
• $[X,Y]=-[Y,X]\,$
• $[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.\,$ This is the Jacobi identity.

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

Furthermore, there is a "product rule" for Lie brackets. Given a smooth real-valued function f defined on M and a vector field Y on M, we have a new vector field fY, defined by multiplying the vector Y_x with the number f(x), at each point x∈M. The Lie bracket of X and fY is then given by

• $[X, fY] = X(f) Y + f [X,Y]$

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

We also have the following fact:

Theorem:

$[X,Y]=0\,$ iff the flows of X and Y commute locally, i.e. iff for every x∈M and all sufficiently small real numbers s, t we have $(\Phi^Y_t \Phi^X_s) (x) =(\Phi^X_{s}\, \Phi^Y_t)(x)$.

Examples

For a Lie group G, the corresponding Lie algebra is the tangent space at the identity, which can be identified with the left invariant vector fields on G. The Lie bracket of the Lie algebra is then the Lie bracket of the left invariant vector fields, which is also left invariant.

For a matrix Lie group, smooth vector fields can be locally represented in the corresponding Lie algebra. Since the Lie algebra associated with a Lie group is isomorphic to the group's tangent space at the identity, elements of the Lie algebra of a matrix Lie group are also matrices. Hence the Jacobi–Lie bracket corresponds to the usual commutator for a matrix group:

$[X,Y] = XY - YX$

where juxtaposition indicates matrix multiplication.

Applications

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

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 forms) is the Frölicher–Nijenhuis bracket.

References

• Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
• For generalizations to infinite dimensions.