An example: $\mathfrak{sl}(2,\mathbb C)$
Let us consider the Lia algebra $\mathfrak g=\mathfrak{sl}(2,\mathbb C)$, with basis $$H=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \ X=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \ Y=\begin{pmatrix}0&0\\1&0\end{pmatrix}$$ and commutation relations, $$[H,X]=2X,\ [H,Y]=-2Y,\ [X,Y]=H.$$ The dual vector space $\mathfrak g^*$ has the dual basis $H^*, X^*, Y^*$, where, by definition, $\langle H^*,H\rangle=1$, $\langle X^*,X\rangle=1$, $\langle Y^*,Y\rangle=1$, and all other duality brackets are $0$. We shall consider the following commutation relations on $\mathfrak g^*$, $$[H^*,X^*]_{\mathfrak g^*}=\frac14X^*, \ [H^*,Y^*]_{\mathfrak g^*}=\frac14Y^*, \ [X^*,Y^*]_{\mathfrak g^*}=0.$$
My question is: how to get the final equation?
Thank you very much!
This seems to be $1.1$ here, in the book by Basil Grammaticos and K.M. Tamizhmani. The commutation relations are chosen such that $\mathfrak{g}\bowtie \mathfrak{g}^{*}$ is a Lie algebra having both $\mathfrak{g}$ and $\mathfrak{g}^*$ as subalgebras. The Lie bracket $[,]_{\mathfrak{g}^*}$ is recovered on page $117$ in an explicit way. In general, Lie bialgebra structures on $\mathfrak{sl}_2(\Bbb{C})$ are parametrised by the $3$-dimensional vector space $\Lambda^2(\mathfrak{sl}_2(\Bbb{C}))$.