Yangians, quantum loop algebras, and abelian difference equations

Abstract:

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 $\mathcal {O}$.