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I want a reference to the definition of the Lie derivative of a smooth function $f:G \to \mathbb R$ on a Lie group $G$ in the direction of an element $\theta$ of the Lie algebra $\mathfrak G$.

I can see in many places the definition of the Lie derivative with respect to a vector field, and I also understand that an element of a Lie algebra $\theta$ can be considered as a left (or right) invariant vector field whose value at the identity is $\theta$.

But I would really like a sweet, short reference that combines both of these into one definition, or at least states this explicitly that this can be done.

What I really want is something accessible to engineers with a weak math background. As an example, think of $SO(n)$ as embedded in $\mathbb R^{n^2}$, and suppose $f:SO(n) \to \mathbb R$. Then the 'Lie derivative' of $f$ in the direction of a an anti-symmetric matrix $A$ at $Q$ is $$ \mathcal L_A(Q) = \frac d{dt} f\big(Q(I+tA)\big) \Bigg|_{t=0} .$$

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    $\begingroup$ Lie derivative along a vector field is by definition a map from tensor fields to tensor fields, and so in particular for scalar fields (as an argument) is determined by a vector field (as a parameter). If you are interested in the derivative in a given direction at a single point, you should take a look at any textbook on differential geometry, for this is not specific to Lie groups and algebras. $\endgroup$ Jan 17 at 15:11
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    $\begingroup$ @AndreiSmolensky I added more to my question clarifying what I really want. (I do know that for a scalar field that the Lie derivative is equivalent to a directional derivative, as you say.) $\endgroup$ Jan 17 at 18:27

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What about the formula $(D_X\varphi)(g)=\frac{\rm d}{{\rm d}t}\Bigl\vert_{t=0}\varphi(g\exp_G(tX))$ for $\varphi\in C^\infty(G)$, $g\in G$, and $X\in \mathfrak g$, where $G$ is a Lie group with its Lie algebra $\mathfrak g$? See for instance Eq. (5) in Ch. II of the book by S.Helgason, "Differential geometry, Lie groups, and symmetric spaces".

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    $\begingroup$ I think he wants a reference from a published book or paper. $\endgroup$
    – Ben McKay
    Jan 17 at 19:01
  • $\begingroup$ Right! I now added a reference. $\endgroup$ Jan 17 at 19:50
  • $\begingroup$ @DanielBeltita I now have the book in from of me. But I am unable to find this formula. Can you help me look for it? $\endgroup$ Jan 17 at 22:23
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    $\begingroup$ Sure, that formula is in Ch. II at the beginning of the subsection on Taylor's formula and the differential of the exponential mapping, at page 104 in the 1979 edition. $\endgroup$ Jan 18 at 3:48
  • $\begingroup$ I think engineers with a weak mathematics background will not be familiar with the content of pages 1-103 of Helgason's book. $\endgroup$
    – Ben McKay
    Jan 18 at 7:18

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