The solution of the classical Yang-Baxter equation induces a Lie bialgebra We know that the classical Yang-Baxter equation is
$$[r_{12}, r_{13}]+[r_{12}, r_{23}]+[r_{13}, r_{23}]=0,\quad(*)$$
and we have the following Theorem.
Theorem: Let $\mathcal{G}$ be a Lie algebra and $r\in \mathcal{G}\otimes \mathcal{G}.$ Then the map $\delta: \mathcal{G}\rightarrow
\mathcal{G}\otimes \mathcal{G}$ defined by Eq. $\delta(X)=X.r$ induces a Lie bialgebra structure on $\mathcal{G}$ if and only if the following two conditions are satisfied (for any $x\in \mathcal{G}$).


*

*$(\mathrm{ad}(x)\otimes \mathrm{id} +\mathrm{id}\otimes \mathrm{ad}(x))(r+r^{21})=0;$

*$(\mathrm{ad}(x)\otimes \mathrm{id} \otimes \mathrm{id} +\mathrm{id} \otimes \mathrm{ad}(x)\otimes \mathrm{id}+\mathrm{id} \otimes \mathrm{id} \otimes \mathrm{ad}(x))([r_{12}, r_{13}]+[r_{12}, r_{23}]+[r_{13}, r_{23}])=0.$
The theorem shows that when $r\in \mathcal{G}\otimes \mathcal{G},$ induces a Lie bialgebra structure if and only if satisfying two conditions.
My question is that how to induce the lie bialgebra when $r\in \mathcal{G}\otimes \mathcal{G}$ is the solution of the equation (*), can you give me some references? Thank you very much. 

Simply take $\delta(x)=ad_x(r)$.
First condition says that the image of $\delta$ is contained in the subspace of anti-symmetric tensor.
The cocycle condition is trivial because $\delta$ is a coboundary.
Condition 2 translates into co-Jacobi. This requires some pacient, but no trick. Write $r=r_1+r_2$ with $r_1$ Anti symmetric and $r_2$ symmetric. Condition 2 is equivalent to the fact that $ [r_1,r_1]\in \Lambda^3g$ is an invariant tensor. When $\delta(x)=ad_x(r)$ (this is equal to $ad_x(r_1)$ because $r_2$ is ad invariant), the invariance of this element is tautologically co-Jacobi condition.
