## Lie Groups Seminar

The Yangian Yg and quantum loop algebra Uq(Lg) of a complex semisimple Lie algebra g share very many similarities, and were

long thought to have the same representations, though no precise relation between them existed until recently.

I will explain how to construct a functor from the finite-dimensional representations of Yg to those of Uq(Lg). The functor is entirely explicit, and governed by the monodromy of the abelian difference equations determined by the commuting fields of the Yangian. It yields a meromorphic, braided Kazhdan-Lusztig equivalence between finite-dimensional representations of Yg and U_q(Lg).

A similar construction yields a functor from representation of U_q(Lg) to those of the elliptic quantum group E_{q,t}(g) corresponding to g. This allows in particular a classification of irreducible finite-dimensional representations of E_{q,t}(g), which was previously unknown.

This is joint work with Sachin Gautam (Ohio State).