Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\mathfrak g^{\ast},$ namely for $f,g \in C^{\infty} (G)$ we define $:$ $$[(df)_1, (dg)_1] : = (d \{f,g \})_1.$$
Question $:$ Is this Lie algebra structure on $\mathfrak g^{\ast}$ induced from a cocommutator on $\mathfrak g$ which makes $\mathfrak g$ into a Lie bialgebra?
I think it's true but I couldn't able to figure out what the dual of this Lie bracket is. I need to check whether the dual map (which is supposed to be a cocommutator on $\mathfrak g$) is a $1$-cocycle or not. Could anyone please give me some suggestions in this regard?
Thanks for your time.
