What is the group of group-like elements of a quantum group? A quantum group is not a group.
For example, the Drinfeld-Jimbo "quantum doubles" are Hopf algebras obtained by deforming the universal enveloping algebras of Lie algebras.
But in every Hopf algebra, there's a subset of group-like elements that satisfy
$$ \Delta(g) = g \otimes g $$
which form a group.
So there is a group hiding somewhere in the quantum group, just like there is a group hiding in the universal enveloping algebra (the latter is given by exponentiating the Lie algebra).
What is this group? Is it equal to the original Lie group, or is it deformed? Afaik Lie groups are rigid objects, so I don't see how the latter can be possible. But I'd like to confirm nevertheless.

I do not have a complete answer to the question. But the group of group-like elements of a quantum group might be deformed somehow.
I'm refering to Lemma 6.4.1 of V.Chari, A.Pressley A guide to quantum groups.
In the case of $\mathfrak{sl_2}$, if you take the h-adic version of quantum $\mathfrak{sl_2}$, $U_h(\mathfrak{sl_2})$, it is a $\mathbb{C}[[h]]$ Hopf algebra. 
Any elements $e^{h\lambda H}$, where $\lambda \in \mathbb{C}[[h]]$, is group like. (Example: $K$ in the standard quantum version $U_q( \mathfrak{sl_2})$)
Hence, we have some group like elements coming from the deformation of the universal enveloping algebra.
