Question on the definition of connection in Liviu.Nicolaescu's book In "Lectures on the Geometry of Manifolds",Liviu said that a connection on a principle $G$-bundle defined by an open cover $(U_{\alpha})$ and gluing cocycle $g_{\alpha\beta}:U_{\alpha\beta} \to G$ is a collection
    $$A_{\alpha}\in \Omega^{1}(U_{\alpha})\otimes \mathfrak{g}$$
    which satisfies the transition rules
    $$A_{\beta}(x)=g_{\alpha\beta}^{-1}(x)dg_{\alpha\beta}(x)+g_{\alpha\beta}^{-1}(x)A_{\alpha}(x)g_{\alpha\beta}(x)$$
I got confused with this formula,basically,does $g_{\alpha\beta}^{-1}(x)dg_{\alpha\beta}(x)$ really make sense? 
How to understand the product of an element in $G$ and an element in $\Omega^{1}(U_{\alpha})\otimes \mathfrak{g}$?
If it means the adjoint action of $G$ on $\mathfrak{g}$, what does the latter half formula mean?
Probably it's just a notation problem.Is the assumption that $G$ is a matrix Lie group necessary here?For a general principle $G$ bundle ,does the above formula make sense?
I always have to read books on my own,no one to ask,even my own "boss",so ,really appreciated for any helpful comments!
 A: Here $g_{\alpha\beta}^{-1}(x)dg_{\alpha\beta}(x)$ is used as a notation for the left logarithmic derivative of the smooth function $g_{\alpha\beta}:U_{\alpha\beta}\to G$ (which is fine for a matrix group). The result is then a one-form with values in the Lie algebra $\mathfrak g$ of $G$. The general definition of the left logarithmic derivative is that you use the ordinary tangent map $T_xg_{\alpha\beta}$ to map tangent vector at $x$ to tangent vectors to $G$ in $g_{\alpha\beta}(x)$ and then use an appropriate left translation to move them to the neutral element of $G$, thus obtaining an element of the Lie algebra. Explicitly:
$$
\delta g_{\alpha\beta}(x)(\xi)=T_{g_{\alpha\beta}(x)}\lambda_{g_{\alpha\beta}^{-1}(x)}\cdot T_x g_{\alpha\beta}\cdot\xi, 
$$
where I denote the left logarithmic derivative by $\delta$ and left translations in $G$ by $\lambda$. Alternatively, $\delta g_{\alpha\beta}$ is simply the pullback of the left Maurer-Cartan form on $G$ along the smooth function $g_{\alpha\beta}$. 
The second term in the formula indeed represents an adjoint action, it should be $Ad(g_{\alpha\beta}^{-1}(x))(A_{\alpha}(x))$ in the case of a general Lie group. 
