The Meromorphic R-Matrix of the Yangian

Abstract

Let \(\mathfrak {g}\) be a complex semisimple Lie algebra and \(Y_{\hbar }(\mathfrak {g})\) its Yangian. Drinfeld proved that the universal R-matrix \(\mathcal {R}(s)\) of \(Y_{\hbar }(\mathfrak {g})\) gives rise to rational solutions of the \(\operatorname {QYBE}\) on irreducible, finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\). 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 \(Y_{\hbar }(\mathfrak {g})\) 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 \(\mathcal {R}(s)\), and therefore coincides with it by uniqueness. In particular, we give a constructive proof of the existence of \(\mathcal {R}(s)\). Our construction relies on the Gauss decomposition \(\mathcal {R}^+(s)\cdot \mathcal {R}^0(s)\cdot \mathcal {R}^-(s)\) of \(\mathcal {R}(s)\). The divergent abelian term \(\mathcal {R}^0\) 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 \(\mathcal {R}^{\pm }(s)\), prove that they are rational on finite-dimensional representations, and that they intertwine the standard coproduct of \(Y_{\hbar }(\mathfrak {g})\) and the deformed Drinfeld coproduct introduced in loc. cit.