From Yangians to quantum loop algebras via abelian difference equations

The finite-dimensional representations of the Yangian $Y_h(g)$ and quantum loop algebra $U_q(Lg)$ of a complex, semisimple Lie algebra have long been known to share many similar features. Assuming that $q$ is not a root of unity, I will explain how to construct an equivalence of categories between finite-dimensional representations of $U_q(Lg)$ and an explicit subcategory of finite-dimensional representations of $Y_h(g)$. This equivalence is governed by the monodromy of an additive, abelian difference equation, and can be upgraded to a meromorphic tensor equivalence.
This is joint work with Sachin Gautam, and is based on: arXiv: 1310.7318 and arXiv: 1012.3687.