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 in the `direction' generated by X. It is a special case of the Lie derivative, which allows to form the derivative of any tensor field in the direction generated by X. Indeed, [X,Y] equals the Lie derivative .
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).
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 of any two derivations and 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
Flows and limits
or in terms of the Lie derivative
which is also equivalent to
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 , and we write for the associated local basis for the tangent bundle, then the vector fields can be written as
with smooth functions and . Then the Lie bracket is given by
If M is (an open subset of) Rn, then the vector fields X and Y can be written as smooth maps of the form and , and the Lie bracket is given by
where and are the Jacobian matrices of and , respectively. These n-by-n matrices are multiplied by the n-vectors X and Y.
The Lie bracket of vector fields equips the real vector space of all vector fields on M (i.e., smooth sections of the tangent bundle of ) with the structure of a Lie algebra, i.e., [·,·] is a map from to with the following properties
- is R-bilinear
- This is the Jacobi identity.
An immediate consequence of the second property is that for any .
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 Yx with the number f(x), at each point x∈M. The Lie bracket of X and fY is then given by
(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:
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 .
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:
where juxtaposition indicates matrix multiplication.
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.
- Hazewinkel, Michiel, ed. (2001), "Lie bracket", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Isaiah, Pantelis (2009), "Controlled parking [Ask the experts]", IEEE Control Systems Magazine 29 (3): 17–21, 132, doi:10.1109/MCS.2009.932394
- 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 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