Nakahara defines the connection one-form on the principle bundle as follows:
I'm a little confused about the motivation for $g^{-1}$ in the latter equation.
Roughly speaking, we can think of $R_{g*}X$ as of $Xg$ (true only for matrix groups, of course). Now, assume $A\in g$. Then:
\begin{gather} \omega(A) \cong A \end{gather} (since we have an isomorphism between the Lie algebra and the vertical tangent subspace).
But \begin{gather} \omega(R_{g*}A) = g^{-1} \omega(A) g = g^{-1} A g \ncong R_{g*}A = Ag \end{gather}
What am I missing? So far, it seems like the consistent definition would be just $R^*_g\omega=\omega g$.
UPDATE
One of my colleagues has suggested the following. Applying $\omega$ to a vector involves smth pulling it back to the tangent space of unit element, which produces $g^{-1}$ on the left.