Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra?
I can see that they are same as $k$-vector spaces. Now how is the quotient Hopf algebra defined? This needs to be understood first in order to argue whether they are isomorphic as Hopf algebras. Any suggestion in this regard would be warmly appreciated.
Thanks in advance.
Source $:$ A Guide to Quantum Groups by Chari and Pressley.