The R-matrix of the affine Yangian

Abstract

Let $\mathfrak{g}$ be an affine Lie algebra with associated Yangian $Y_h(\mathfrak{g})$. We prove the existence of two meromorphic $R$--matrices associated to any pair of representations of $Y_h(\mathfrak{g})$ in the category $\mathcal{O}$. They are related by a unitary constraint and constructed as products of the form $\mathcal R^{\uparrow/\downarrow}(s)=\mathcal R^+(s)\cdot\mathcal R^{0,\uparrow/\downarrow}(s)\cdot\mathcal R^-(s)$, where $\mathcal R^+(s) = \mathcal R^-_{21}(-s)^{-1}$. The factors $\mathcal R^{0,\uparrow/\downarrow}(s)$ are meromorphic, abelian $R$--matrices, with a WKB--type singularity in $\hbar$, and $\mathcal R^-(s)$ is a rational twist. Our proof relies on two novel ingredients. The first is an irregular, abelian, additive difference equation whose difference operator is given in terms of the $q$--Cartan matrix of $\mathfrak g$. The regularisation of this difference equation gives rise to $\mathcal R^{0,\uparrow/\downarrow}(s)$ as the exponentials of the two canonical fundamental solutions. The second key ingredient is a higher order analogue of the adjoint action of the affine Cartan subalgebra $\mathfrak h\subset\mathfrak g$ on $Y_h(\mathfrak g)$. This action has no classical counterpart, and produces a system of linear equations from which $\mathcal R^-(s)$ is recovered as the unique solution. Moreover, we show that both $\mathcal R^{\uparrow/\downarrow}(s)$ give rise to the same rational $R$--matrix on the tensor product of any two highest--weight representations.