The Meromorphic R-Matrix of the Yangian

Abstract

Let g be a complex semisimple Lie algebra and Yħ(g ) its Yangian. Drinfeld proved that the universal R-matrix ℛ(s) of Yħ(g ) gives rise to rational solutions of the QYBE on irreducible, finite-dimensional representations of Yħ(g ). This result was recently extended by Maulik–Okounkov to symmetric Kac–Moody algebras and representations arising from geometry. We show that rationality ceases to hold on arbitrary finite-dimensional representations, if one requires such solutions to be natural and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of Yħ(g ) does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. Each possesses an asymptotic expansion in s which has the same formal properties as ℛ(s), and therefore coincides with it by uniqueness. In particular, we give a constructive proof of the existence of ℛ(s). Our construction relies on the Gauss decomposition ℛ+(s) ⋅ ℛ0(s) ⋅ ℛ−(s) of ℛ(s). The divergent abelian term ℛ0 was resummed on finite-dimensional representations by the first two authors in Gautam and Toledano Laredo (Publ Math Inst Hautes Études Sci 125:267–337, 2017). In the present paper, we construct ℛ±(s), prove that they are rational on finite-dimensional representations, and that they intertwine the standard coproduct of Yħ(g ) and the deformed Drinfeld coproduct introduced in loc. cit.