Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$,$U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp (\pi ih)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $\mathrm{\Gamma }$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$tothoseof$U_q(L{\mathfrak g})$\mathrm{.}Thefunctor$\Gamma$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h(\mathfrak{g})$ defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on $\mathrm{\Gamma }$ and show that, if $\mathrm{\mid }q\mathrm{\mid }\mathrm{\ne }1$, it yields an equivalence of meromorphic braided tensor categories, when $Y_h(\mathfrak{g})$ and $U_q(L\mathfrak{g})$ are endowed with the deformed Drinfeld coproducts and the commutative part of the universal $R$-matrix. This proves in particular the Kohno-Drinfeld theorem for the abelian $q$KZ equations defined by $Y_h(\mathfrak{g})$. The tensor structure arises from the abelian $q$KZ equations defined by a appropriate regularisation of the commutative $R$-matrix of $Y_h(\mathfrak{g})$.