quality:correction
Lie derivative

A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by

:= L A B = − L B A. {\displaystyle:={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A.}

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way around, the group of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.