Lie bracket of vector fields

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See also: Lie derivative

In the mathematical field of differential geometry, the Lie bracket of vector fields or Jacobi–Lie bracket is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]. It is closely related to, and sometimes also known as, the Lie derivative. In particular, the bracket $[X, Y]$ equals the Lie derivative $\mathcal{L}_X Y$ .

It plays an important role in differential geometry and differential topology, and is also fundamental in the geometric theory for nonlinear control systems.

1 Definition
2 Examples
3 Applications
4 References

[edit] Definition

Let X and Y be smooth vector fields on a smooth $n$ -manifold M. The Jacobi-Lie bracket or simply Lie bracket of X and Y, denoted $[X, Y]$ is the unique vector field such that

$\mathcal{L}_{[X,Y]} = \mathcal{L}_X \circ \mathcal{L}_Y - \mathcal{L}_Y \circ \mathcal{L}_X$

where $\mathcal{L}_X$ is the Lie derivative with respect to the vector field X. For a finite-dimensional manifold M we can define the Jacobi-Lie bracket in local coordinates as

$[X,Y]^i= \sum_{j=1}^n \left (X^j \frac {\partial Y^i}{\partial x^j} \right ) - \left ( Y^j \frac {\partial X^i}{\partial x^j} \right )$

where n is the dimension of M.

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

[.,.] 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 these properties is that $[X, X] = 0$ for any $X$ .

[edit] Examples

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] = X Y - Y X

where juxtaposition indicates matrix multiplication.

[edit] Applications

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

[edit] References

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 978-0387943381. For generalizations to infinite dimensions.
Lewis, Andrew D.. Notes on (Nonlinear) Control Theory.