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

In differential geometry, if we have a differentiable tensor T of rank (p q) (i.e. a differentiable linear map of smooth sections, α, β, ... of the cotangent bundle T*M and **X**, **Y**, ... of the tangent bundle TM, T(α,β,...,**X**,**Y**,...) such that for any smooth functions f_{1},...,f_{p},...,f_{p+q}, T(f_{1}α,f_{2}β,...,f_{p+1}**X**,f_{p+2}**Y**,...)=f_{1}f_{2}...f_{p+1}f_{p+2}...f_{p+q}T(α,β,...,**X**,**Y**,...)) and a differentiable vector field (section of the tangent bundle) **A** , then the linear map

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_{A}T)(α,β,...,**X**,**Y**,...)≡∇_{A}T(α,β,...,**X**,**Y**,...)-∇_{T(-,β,...,X,Y,...)}**A**(α)-...+ T(α,β,...,∇_{X}**A**,**Y**,...)+...

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