Well, i'm having problems with the follow construction of Arnold's book "Topological Methods of Hydrodynamics", pg. 51:
"The group $G$ acts naturally on itself by left translations, as well as by right ones. The left and right shifts commute with each other. Hence, right-invariant vector (or covector) fields are taken to right-invariant ones under left translations, while left-invariant fields are sent to left-invariant ones by right translations. Extend every covector on $G$, i.e., an element $\alpha_{g}$ of the cotangent bundle $T^{*}G$ at $g$ ∈ $G$, to the right-invariant section (covector field) $\alpha$ on the group. Define the action of this covector αg on the phase space $T^{*}G$ as follows. First add to every covector in $T^{*}G$ at $h$ the value of the right-invariant section $\alpha$ at $h$. Then apply the left shift of the entire phase space $T^{*}G$ by $g$."
In algebraic language , i understood the following, the covector in the end is
$$ L_{g}^{*} (\alpha(h) + \beta_{h})$$ where $\alpha(h) = R_{g^{-1}}^{*}(\alpha_{g})$ a extension of covector $\alpha_{g}$ to entire space and $\beta_{h}$ is any vector of cotangent space at $h$. But the operation only makes sense if we have $h$: $$ L_{h}^{*}( \alpha(h) + \beta_{h})$$
If someone could clarify.... Thanks !