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 equals the Lie derivative .
Let X and Y be smooth vector fields on a smooth -manifold M. The Jacobi-Lie bracket or simply Lie bracket of X and Y, denoted is the unique vector field such that
where 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
where n is the dimension of M.
The Lie bracket of vector fields equips the real vector space (i.e., smooth sections of the tangent bundle of ) with the structure of a Lie algebra, i.e., [.,.] is a map from VV to V with the following properties
An immediate consequence of these properties is that for any .
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.
The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.