Yangians, quantum loop algebras and abelian difference equations

Abstract

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 πh), 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.