The R-matrix of the affine Yangian

TL;DR

This paper constructs and proves the existence of two meromorphic R-matrices for representations of the affine Yangian, revealing their structure and relation through novel difference equations and higher order adjoint actions.

Contribution

It introduces a new construction of R-matrices for affine Yangians using irregular difference equations and a higher order affine Cartan action, with explicit factorization.

Findings

Existence of two meromorphic R-matrices for affine Yangian representations.

R-matrices are related by a unitary constraint and factorized as R(s)=R^+(s)R^0(s)R^-(s).

Both operators produce the same rational R-matrix on tensor products of highest-weight representations.

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)R^0(s)R^-(s), where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian R-matrix, 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 R^0(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.