In mathematics, the Lie derivative, named after Sophus Lie by W?adys?aw ?lebodzi?ski, evaluates the change of one vector field along the flow of another vector field. The Lie derivative 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 [A,B] = 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 diffeomorphism group of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory.

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