For a Lie algebra $\frak{g}$, we will recall that its universal enveloping algebra $U(\frak{g})$ has a Hopf algebra structure given by $$ \Delta(x) := 1 \otimes x + x \otimes 1, ~~~ \epsilon(x) = 0, ~~ S(x) = -x. $$
Now if $\frak{g}$ is a compact semi-simple Lie algebra, then its enveloping has another Hopf algebra structure, which deforms to give the famous Drinfeld--Jimbo algebras (see here for example).
What I would to know is: Does there exist any relationship between these two Hopf algebra structures? Moreover, does there exist some deformation of the first Hopf algebra to a noncommutative quantum group?
