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In the mathematical field of differential topology, 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.

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 $X$.

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.

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

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

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Last updated on Tuesday September 09, 2008 at 21:17:05 PDT (GMT -0700)

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Last updated on Tuesday September 09, 2008 at 21:17:05 PDT (GMT -0700)

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