4 added 1453 characters in body

Edit in light of David Ben-Zvi's answer: I was not quite speaking about this trichotomy, but rather mentionning the existence of elliptic categories leading to representations of the elliptic braid group, following the suggestion of David Roberts that one should view quantum groups by their braided monoidal category of modules. But the fact that $r$-matrices and the (dynamical) Yang Baxter equation comes into play is not really a coincidence.

Roughlys speaking, in the KZ world, rational $r$-matrices leads to monodromy representations of the braid group $B_n$, which can be used to construct braided monoidal categories. If you take the trivial $r$-matrice $r=0$ these representations factor through representations of $S_n$, so in some sense you recover the symmetric categories of module over a Lie group. For non trivial rational $r$-matrices, you get braided monoidal categories, and quantum groups by the Kohno--Drinfeld theorem. For trigonometric one, you get representations of the cylinder braid group, the associated categorical notion being that of braided module categorie (hence comodules over quantum groups). Finally, for elliptic solutions of the (dynamical) Yang-Baxter equation, you get elliptic categories. The associated algebraic notion seems to be strongly related with quantum analogs of the algebra of differential operators on a Lie group : http://arxiv.org/abs/0805.2766v2

I hop it makes my answer more clear :)

3 Small mistake corrected

Elliptic solutions of the classical Yang-Baxter equation (CYBE) exists only in the $\mathfrak sl_n$ case. In the general case, there are elliptic solutions of the dynamical CYBE, which is an algebraico-differential equation generalizing the CYBE. You can use such an elliptic dynamical $r$-matrix to define a flat connexion (the KZB connexion) over on the configuration space of $n$ points on the torus which leads to monodromy representations of the torus braid group.

This monodromy morphism can be expressed using the good old KZ associator, and the monodromy operators allows one to construct an exemple of an elliptic structure over a braided monoidal category. Elliptic structures plays the same role for the torus braid group as braided monoidal categories does for the usual braid group.

http://arxiv.org/abs/math/0702670

Edit: the basic reference concerning the anwser of Bruce and Elliptic quantum groups : http://arxiv.org/abs/hep-th/9412207 . Rmk: the dynamical CYBE is called modified CYBE in this paper.

Elliptic solutions of the classical Yang-Baxter equation (CYBE) exists only in the $\mathfrak sl_n$ case. In the general case, there are elliptic solutions of the dynamical CYBE, which is an algebraico-differential equation generalizing the CYBE. You can use such an elliptic dynamical $r$-matrix to define a flat connexion (the KZB connexion) over the torus which leads to monodromy representations of the torus braid group.