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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, 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.

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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} $

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