The meromorphic R-matrix of the Yangian

Abstract

Let g be a complex semisimple Lie algebra and Yg its Yangian. Drinfeld proved that the universal R-matrix of Yg gives rise to rational solutions of the quantum Yang-Baxter equations on irreducible, finite-dimensional representations of Yg. This result was recently extended by Maulik-Okounkov to symmetric Kac-Moody algebras, and representations arising from geometry. We show that this rationality ceases to hold for arbitrary finite-dimensional representations, at least if one requires such solutions to be natural with respect to the representation and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of Yg does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. We show that each possesses an asymptotic expansion as s tends to infinity, which has the same formal properties as Drinfeld's R(s), and therefore coincides with the latter by uniqueness. In particular, we give an alternative, constructive proof of the existence of the universal R-matrix of Yg. Our construction relies on the Gauss decomposition R^+(s)R^0(s)R^-(s) of R(s). The divergent abelian term R^0 was resummed on finite-dimensional representations by the first two authors in arXiv:1403.5251. The main ingredient of the present paper is the construction of R^+(s) and R^-(s). We prove that they are rational functions on finite-dimensional representations, and that they intertwine the standard coproduct of Yg and the deformed Drinfeld coproduct introduced in arXiv:1403.5251.