Let $(\mathcal{G}, \mathcal{G}^\*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^\*$ and $\mathcal{G}$ are
\begin{align}
& [t^a, t^b]\_\* = C\_c^{ab} t\_c, \\
& [t\_a, t\_b] = f\_{ab}^c t\_c,
\end{align}
respectively.


Let $r = r^{ab} t\_a \otimes t\_b \in \mathcal{G} \otimes \mathcal{G}$ be a classical r-matrix and assume that $\delta: \mathcal{G} \to \mathcal{G} \wedge \mathcal{G}$ is given by $\delta(X) = [X \otimes 1 + 1 \otimes X, r]$, $X \in \mathcal{G}$. 


In [the paper](https://arxiv.org/pdf/hep-th/9310068.pdf), it is said that the structure constant on $\mathcal{G}^\*$ is $C\_{c}^{ab} = f\_{cd}^b r^{ad} + f\_{cd}^a r^{db}$.


How to show that the structure constant on $\mathcal{G}^\*$ is $C\_{c}^{ab} = f\_{cd}^b r^{ad} + f\_{cd}^a r^{db}$?


Thank you very much.



How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$?
It is important to note that the literature often uses the shorthand $$[X\otimes 1+1\otimes X, Y\otimes Z]=(ad\_X\otimes 1+1\otimes ad\_X)(Y\otimes Z)=[X,Y]\otimes Z+X\otimes [Y,Z]$$ where $1$ is the identity map. We compute $C^{ab}\_c=\langle [t^a,t^b]\_\*,t\_c\rangle$ as follows. Recall that there is implied summation over repeated indices.


$\begin{align}\langle [t^a, t^b]\_\*,t\_c\rangle &= \langle t^a \otimes t^b ,[t\_c\otimes 1 + 1 \otimes t\_c,r]\rangle\\
 &=\langle t^a \otimes t^b , [t\_c\otimes 1 + 1 \otimes t\_c,r^{de}t\_d\otimes t\_e ]\rangle\\
&=\langle t^a \otimes t^b , r^{de}([t\_c,t\_d]\otimes t\_e + t\_d\otimes [t\_c,\otimes t\_e ])\rangle\\
&=\langle t^a \otimes t^b , r^{de}(f\_{cd}^f t\_f\otimes t\_e + f\_{ce}^gt\_d\otimes t\_g)\rangle\\
&= r^{de}f\_{cd}^f\langle t^a \otimes t^b , t\_f\otimes t\_e\rangle + r^{de}f\_{ce}^g\langle t^a \otimes t^b ,t\_d\otimes t\_g\rangle\\
&= r^{de}f\_{cd}^f\delta^a\_f\delta^b\_e + r^{de}f\_{ce}^g\delta^a\_d\delta^b\_g\\
&= r^{db}f\_{cd}^a+ r^{ae}f\_{ce}^b\\
&= r^{db}f\_{cd}^a+ r^{ad}f\_{cd}^b\\
\end{align}
$


