Shifted Yangians and polynomial R-matrices

TL;DR

This paper investigates the category of representations over shifted Yangians, establishing the polynomial nature of R-matrices, their functional relations, and implications for module structure and factorization in finite types.

Contribution

It proves the polynomiality of R-matrices, their functional relations, and the Jordan-Hölder property in the category O of shifted Yangian representations, extending results uniformly across finite types.

Findings

Tensor products of prefundamental and irreducible modules are cyclic or co-cyclic.

R-matrices are polynomial in the spectral parameter.

Irreducible modules factor through truncated shifted Yangians.

Abstract

We study the category O 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 co-cyclic. 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--H\"older property in the category O. We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian.