Yangians, quantum loop algebras, and abelian difference equations

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