Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Question

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and $U(\mathfrak{sl}_2) = \langle E, F, H \rangle$. Is it possible to define a pairing between $\mathbb{C}[SL_2]$ and $U(\mathfrak{sl}_2)$ explicitly? For example, define $(a, E) = $, $(a, F)=$ and so on. Thank you very much.