The R-matrix of the affine Yangian

We study the Yangian attached to an affine Lie algebra and prove that, for any two representations in category O, there exist two associated meromorphic R matrices. These two R matrices satisfy a unitarity relation and admit a factorized construction into three pieces: a positive factor, an abelian middle factor with two canonical choices, and a rational twisting factor, with the positive and negative factors linked by an explicit inversion and reversal rule. The proof introduces two new ideas. First, we formulate an irregular abelian additive difference equation whose difference operator is determined by the q Cartan matrix of the affine Lie algebra. After a suitable regularization, this equation produces the two middle abelian factors as exponentials of the two canonical fundamental solutions. Second, we develop a higher order analogue of the adjoint action of the affine Cartan subalgebra on the Yangian. This genuinely new action has no classical analogue and yields a linear system whose unique solution recovers the rational twisting factor. Finally, we show that the two meromorphic R matrices induce the same rational R matrix on the tensor product of any two highest weight representations.