  Let g be a complex, semisimple Lie algebra, and Y_h(g) and U_q(Lg) the
Yangian and quantum loop algebra of g. Assuming that h is not a rational number
and that q=exp(i \pi h), we construct an equivalence between the
finite-dimensional representations of U_q(Lg) and an explicit subcategory of
those of Y_h(g) defined by choosing a branch of the logarithm. This equivalence
is governed by the monodromy of the abelian additive difference equations
defined by the commuting fields of Y_h(g). Our results are compatible with
q-characters, and apply more generally to a symmetrisable Kac-Moody algebra g,
in particular to affine Yangians and quantum toroidal algebras. In this
generality, they yield an equivalence between the representations of Y_h(g) and
U_q(Lg) whose restriction to g and U_q(g) respectively are integrable and in
category O.

Quantum Algebra;Algebraic Geometry;Representation Theory