Yangians, quantum loop algebras, and abelian difference equations

Let \mathfrak{g} be a complex, semisimple Lie algebra, and Y_\hbar (\mathfrak{g}) and U_q(L\mathfrak{g}) the Yangian and quantum loop algebra of \mathfrak{g}. Assuming that \hbar is not a rational number and that q= e^{\pi i\hbar }, we construct an equivalence between the finite-dimensional representations of U_q(L\mathfrak{g}) and an explicit subcategory of those of Y_\hbar (\mathfrak{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_\hbar (\mathfrak{g}). Our results are compatible with q-characters, and apply more generally to a symmetrizable Kac-Moody algebra \mathfrak{g}, in particular to affine Yangians and quantum toroïdal algebras. In this generality, they yield an equivalence between the representations of Y_\hbar (\mathfrak{g}) and U_q(L\mathfrak{g}) whose restriction to \mathfrak{g} and U_q\mathfrak{g}, respectively, are integrable and in category \mathcal {O}.