TITLE: Why Drinfel'd-Jimbo-type quantum groups?
QUESTION [32 upvotes]: 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 $\mathcal{U}g$?

REPLY [9 votes]: 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?

No. The original motivations came from quantum inverse scattering method, as they are the natural place where the quantum R-matrices live in abstract form (universal R-element); in particular godo representations of quantum groups supply examples of concrete R-matrices. Thus at least one restricts to quasitriangular or coquasitriangular Hopf algebras (cf. nlab entry).
Later it appeared that apart of some deformed integrable models (spin chains and alike), most physics applications belong to quantum groups at root of unity. For example, the Chern-Simons theory which somebody mentions has a Hamiltonian reduction to WZNW model which has roughly speaking a quantum group symmetry. But the representation theory at the root of unity is related to the representation theory of affine Lie algebras. There are many aspects of this, including the hints from deep work of Kazhdan and Lusztig mentioned in other answers, but also hypergeometric pairing
A.Varchenko, "Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups," Advanced Series in Mathematical Physics, Vol. 21, World Scientific (1995)
which can be interpreted as evaluating cohomology cocycles on certain homology cycles; this picture is later extended in many directions. Big step is a deep work of then extremely young Bezrukavnikov, summarized in the joint book Factorizable sheaves and quantum groups with Finkelberg and Schechtman. More recently this is further understood by joint work of Lurie and Gaitsgory, I know of the published part by
Dennis Gaitsgory, Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.)  13  (2008),  no. 4, 617--659.
the newer part with even more exciting idea mentioned by Scott I did not see in any detailed form yet.
As far as constructing part of the quantum group (say just the Borel part) from Cartan data withouit explicit formulas but somehow a priori has with various degrees of success being attempted by various people. Luzstig, Nakajima and others take an appropriate variant of configuration spaces and look at certain perverse sheaves there (this kind of ideas is not unrelated to the business of Bezrukavnikov and others involving another category of (factorizable) sheaves to study quantum groups). Lyubashenko, Majid, Schauenburg and have tried starting with some simple braided tensor category and few other data to construct the quantum group; some categorical meaning of relations, though still complicated and with much input by hand is in Aguiar's thesis, allowing for some neat generalizations. Rosenberg (here) starts with only a braided tensor category and a finite family of distinguished objects to create a quantum group-like structure and claims without proof that the category of vector spaces with certain easy choice of braiding (imvolving formula q to the power involving Cartan matrix elements) gives Drinfel'd-Jimbo case; all the relations, including Serre relations are for free by general nonsense.