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In our lecture, we have defined the representation of a Lie algebra $\mathfrak{g}$ on the space of linear maps $L(V,W)$ between two representations $V$ and $W$ of $\mathfrak{g}$ to be of the following form:

$$(X \cdot \phi)(v) := X \cdot (\phi(v)) - \phi(X \cdot v) $$

It is explained that this is the differentiated version of the group action of a group $G$ on the set of all functions $\mathcal{F}(X,Y)$ of two sets $X$ and $Y$ defined by

$$ (g\cdot \phi)(x) := g \cdot(\phi(g^{-1}\cdot x)) $$

I have two questions here:

Why do we differentiate it?

And more importantly: How does one differentiate it? With respect to what? I have been trying to calculate it but failed.

By the way, this is not a homework question, it is just not explained in the notes.

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    See [here](https://en.wikipedia.org/wiki/Lie_group–Lie_algebra_correspondence) for the general principle of differentiating for the Lie group and Lie algebra correspondence. For your question, a partial answer is given [here](https://mathoverflow.net/questions/235927/unclear-asymmetry-in-lie-algebra-module-structure-on-space-of-linear-transformat). – 2017-01-18
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    Thanks! Concerning the second link: I have no knowledge on categories and modules. Is there a way to understand it without? – 2017-01-18
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    Yes, the wikipedia article is enough, you can find some other examples of "differentiating", and then you know how to do it. For such an example see also [here](http://math.stackexchange.com/questions/16124/how-to-differentiate-a-homomorphism-between-two-lie-groups). – 2017-01-18
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    Ok thanks again! – 2017-01-18

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