Lie Bracket of Vector Fields

In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or commutator of vector fields 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 . It is the specialization of the Lie derivative to the case of Lie differentiation of a vector field. Indeed, equals the Lie derivative .

It plays an important role in differential geometry and differential topology, 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).

A generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.