The Yangian and the quantum loop algebra of a simple Lie algebra $g$
arise naturally in the study of the rational and trigonometric solutions of the Yang--Baxter equation, respectively. These algebras are deformations of the current algebra $g[s]$ and the loop algebra $g[z,z^{-1}]$ respectively.
The aim of this talk is to establish an explicit relation between the finite--dimensional representation categories of these algebras, as meromorphic braided tensor categories. The notion of meromorphic tensor categories was introduced by Y. Soibelman and finite--dimensional representations of Yangians and quantum loop algebras are among the first non--trivial examples of these.
The isomorphism between these two categories is governed by the monodromy of an abelian difference equation. Moreover, the twist relating the tensor products is a solution of an abelian version of the qKZ equations of Frenkel and Reshetikhin.
These results are part of an ongoing project, joint with V. Toledano Laredo.