The R-matrix of the affine Yangian (2309.02377v1)
Abstract: Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed as products of the form R(s)=R+(s)R0(s)R-(s), where R+(s) = R-_{21}(-s){-1}. The factor R0(s) is a meromorphic, abelian R-matrix, with a WKB-type singularity in h, and 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 g. The regularization of this difference equation gives rise to R0(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 of g on Y_hg. This action has no classical counterpart, and produces a system of linear equations from which R-(s) is recovered as the unique solution. Moreover, we show that both operators give rise to the same rational R-matrix on the tensor product of any two highest-weight representations.