Let $\mathfrak{g}$ be a Lie bialgebra and $\mathfrak{g}^*$ be its dual. My question is how to construct a bracket on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^*$ such that we obtain a Manin triple? My guess is that there will be the sum of the brackets coming from $\mathfrak{g}$ and $\mathfrak{g}^*$, but that there will be mixed terms too. Does somebody have any reference where this is explained explicitly?
Thank you!
For the Lie algebra brackets $(\mathbb{g},b)$ and $(\mathbb{g}^*,b')$ a skew-symmetric bracket on the vector space $\mathbb{l}:=\mathbb{g}\oplus \mathbb{g}^*$ is given by $$ [(x,\alpha), (y,\beta)]:=\left( b(x,y)+ad_{b'}^*(\alpha)y-ad_{b'}^*(\beta)x ,\, b'(\alpha,\beta)+ad_{b}^*(x)\beta-ad_{b}^*(y)\alpha \right), $$ where $ad_b(x)y=b(x,y)$, $ad_b^*(x)=ad_b(-x)^*\in End(\mathbb{g}^*)$, and similarly for $b'$. The result is as follows:
Theorem (Drinfeld): There is a natural bijective correspondence between Lie bialgebras $(\mathbb{g},b,b')$ and metrical Lie algebras $\mathbb{l}$ with Manin decomposition $\mathbb{l}=\mathbb{g}\oplus \mathbb{g}^*$.
Reference: Drinfeld: Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the Yang-Baxter equation.