# Lie derivative and representations of a Lie algebra

I'm reading a book on integrable systems and am trying to understand Lie groups.

The author states a property I cannot understand: Let me define the protagonists: L is the Lie derivative, m is an element of a Lie group and X and Y are in its Lie algebra. The dot, unless I'm mistaken, refers to a certain action of X on m.

I can understand the first and the last equality, the problem lies in the second equality. Why is it so? Is it true?

Well, this comes from probably an older definition of a Lie Algebra, where someone defines for a given Lie Group $G$ its associated Lie Algebra, to be given by $$\mathfrak{g}= \{ A \in M_{n \times n}(\mathbf{C}) \thinspace | \thinspace exp(tA) \in G, \thinspace \forall t \in \mathbf{R}\}.$$
So if you have an action of the Lie group $G$ on a set $V$ (or vector space, module etc.), then this action defines a new one on the same $V$ of the the Lie Algebra this time, given as follows $$\mathfrak{g} \times V \rightarrow V, \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(X, u) \mapsto X.u= \frac{d}{dt}(exp(tX).u)_{t=0}.$$ You can check on your own that this is indeed a new action. What you see in the book is essentially this thing, applied into the corresponding Lie Bracket.