Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal enveloping algebra of $\mathfrak{g}$. Then, by the quantum duality principle, $U_h(\mathfrak{g})$ is a quantization of $\mathscr{C}(G^*)$.

Question

Is it possible to explicitly give an isomorphism between $U_h(\mathfrak{g})$ and $\mathscr{C}_h(G^*)$ in a concrete Poisson Lie group $G$? $G$ can be anything, but I would like to know more about the case $G=SL(2,\mathbb{C})$ if possible.