As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has an alternative formulation of $U_q(\frak{sl}_2)$: One introduces an extra generator $G$, along with some extra relations, and extensions of the Hopf algebra maps. This new algebra is isomorphic to $U_q(\frak{sl}_2)$, yet well-defined at $q=1$. It is not equal to $U(\frak{sl}_2)$, but a double cover of $U(\frak{sl}_2)$.

What happens in the general $U_q(\frak{g})$ case? Again $U_q(\frak{g})$ is not well-defined since we the relation $$ [E_i,F_j] = \delta_{ij}\frac{K_i-K^{-1}}{q-q^{-1}}. $$ Judging by representation theory, there should exist a reformulation of $U_q(\frak{g})$ where we introduce $n$ generators $G_i$ (where $n$ is the rank of $\frak{g}$), and a set of new relations, which is well-defined when $q=1$. I would then guess that at $q=1$ we hav an $n$-fold cover of $U(\frak{g})$, which one consider as justification for the $2^n$ representations types of $U_q(\frak{g})$. Am I OK in reasoning here?