Shifted Yangians and Polynomial $R$-Matrices

We study the category \mathcal{O}^{\mathrm{sh}} of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or cocyclic. This implies the existence and uniqueness of morphisms, the R -matrices, for such tensor products. We prove the R -matrices are polynomial in the spectral parameter, and we establish functional relations for the R -matrices. As applications, we prove the Jordan–Hölder property in the category \mathcal{O}^{\mathrm{sh}} . We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.