The focus of this talk is on the quantum groups associated to a simple Lie algebra and a one–dimensional algebraic group, namely Yangians, quantum loop algebras and elliptic quantum groups. These correspond respectively to rational, trigonometric and elliptic solutions of the Yang–Baxter equation, and arise in their simplest guises as symmetries of the Heisenberg XXX, XXZ and XYZ lattice models.
In our earlier work, we constructed a (tensor) functor between finite–dimensional representations of Yangians and quantum loop algebras. The functor is governed by the monodromy of the abelian difference equations defined by the commuting fields of the Yangian. The aim of this talk is to extend the use of this principle to (a) give an intrinsic definition of elliptic quantum groups and (b) construct a functor between finite–dimensional representations of quantum loop algebras and elliptic quantum groups.
This talk is based on a joint work with V. Toledano Laredo.