I know that there is meant to be a one-to-one correspondence between Manin triples $(\mathfrak{p},\mathfrak{p_+},\mathfrak{p_-})$ and Lie bialgebra structures on $\mathfrak{p_+},$ but I cannot seem to prove the implication that Manin Triple implies Lie bialgebra.
I define $\beta$ to be the dual of the map $\mathfrak{p_-} \otimes \mathfrak{p_-} \to \mathfrak{p_-},$ and I aim to show that this is a $1$-cocycle, and that this implies that $\mathfrak{p_-} \cong \mathfrak{p_+}^*.$
Any suggestions?
If $(\mathfrak{p}, \mathfrak{p}_+, \mathfrak{p}_-)$ is a finite-dimensional Manin triple then $\mathfrak{p}_+$ can be made into a Lie bialgebra by letting the cocommutator map $\mathfrak{p}_+\rightarrow \mathfrak{p}_+\otimes \mathfrak{p}_+$ be dual to the map $\mathfrak{p}_- \otimes \mathfrak{p}_- \rightarrow \mathfrak{p}_-$, using the fact that the symmetric bilinear form on $\mathfrak{p}$ identifies $\mathfrak{p}_-$ with the dual of $\mathfrak{p}_+$.
For more details see the proof on page $12$ here.
