Derived from Lie

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In mathematics, one derived from Lie is a derivation in the algebra of differential functions on a differentiable variety $scriptstyle {mathcal {M}}$ , whose definition can be extended to the tensorial algebra of the variety. We then get what in differential topology is called tensorial derivation: implementation ${displaystyle scriptstyle mathbb {R} }$ -linear on the set of type tensioners (r,s), which preserves the tensorial type and satisfies the rule of the product of Leibniz and which conmutes with the contractions.

To define the derivative of Lie on the set of tensors of type (r,s) it will suffice to define its action on functions and on vector fields: Thus, if X is a differentiable field of vectors, the derivative of Lie with respect to X is defined as the only tensor derivative such that:

${displaystyle {mathcal {L}}_{X}f=X(f).}$ for any differential function f.
${displaystyle {mathcal {L}}_{X}Y=[X,Y].}$ for every different field Y. Where [,] is Lie's corchete.

The derivative thus defined will automatically satisfy the cited properties of a tensor derivation:

the product rule

{displaystyle {mathcal {L}}_{X}(Sotimes T)=({mathcal {L}}_{X}S)otimes T+Sotimes ({mathcal {L}}_{X}T).}

It will switch with the contractions.

The vector space of all Lie derivatives in M forms an infinite dimensional Lie algebra with respect to the Lie bracket.

Although less common, it also denotes the derivative of Lie de $Y$ regarding a field $X$ Like ${displaystyle X^{L}Y}$ . This notation, sometimes cleaner than the previous one because it avoids subscripts, comes from Professor Juan Bautista Sancho Guimerá.

Derivative of Lie of tensor fields

In differential geometry, if we have a differentiable tensor T of rank (p, q) (that is, a linear function of differentiable sections, α, β,... of the cotangent bundled T*M and X, Y,... of the tangent bundled TM,

T(α,β...,X,Y ,...)

Such that for any differentiable functions

f₁...,f_p...,f_p+q, T(f₁α,f₂β...,f_p+1X,f_p+2Y,...) = f₁f₂... f_p+1f_{p +2}... f_p+q T(α, β..., X, Y ,...)) and a vector field (section of the tangent bundle) A differentiable, then the linear function:

(£_AT)(α, β,..., X, Y,...) ≡ ∇_A T(α, β,..., X, Y,...) - ∇_{T(-,β...,X,Y,...)}A (α)-... + T(α, β...,∇_XA,Y,...)+...

is independent of the connection ∇ being used, as long as it is free of torsion, and is, in fact, a tensor.

This tensor is called the Lie derivative of T with respect to A.

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