In the case of $sl_2$, there are three Lie bialgebra structures. We have three cobrackets $\delta: sl_2 \to \Lambda^2 sl_2$. Each $\delta$ can be written as $\delta=d r$ for some matrix $r$. Therefore all the Lie bialgebra structure on $sl_2$ are coboundary. Are all the Lie bialgebra structure on $sl_n$ coboundary for each $n$?
The compatibility between bracket and cobracket $\delta$ can be interpreted as $\delta$ being a $1$-cocycle. Then Whitehead's lemma implies the existence of such $r$.