Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like any situation where you take some interesting geometric construction, characterize it in terms of some structure on the corresponding ring of functions, and then ask what happens when you no longer require such algebras to be commutative.

However, most of the people I know who study Hopf algebras study a very particular class of them often called 'quantum groups'. This term very often applies to much more general class of Hopf algebras, so for clarity I will call them Drinfel'd-Jimbo quantum groups, as wikipedia does. I am referring to a very specific q-deformation of the universal enveloping algebra of a semi-simple Lie algebra $g$ (the definition is long, so I will defer to wikipedia).

My question is, **what is so special about these Hopf algebras in particular?** Why not some other deformation of the Hopf algebra structure on $\mathcal{U}g$? I have seen some cool things you can do with these, such as characterizing crystal bases of modules; but are these the reasons people started studying these in the first place? Is there a natural reason why these are the 'best' deformations of $\mathcal{U}g$?