# Stable maps, Q-operators and category O

## TL;DR

This paper introduces algebraic stable maps in the category O of quantum affine algebras, leading to new R-matrices and insights into module commutativity and categorified QQ*-systems.

## Contribution

It constructs algebraic stable maps on tensor products in category O, proving their invertibility and rational dependence, and applies these to derive new R-matrices and module relations.

## Key findings

- Constructed invertible algebraic stable maps depending rationally on spectral parameters.
- Derived new R-matrices in the category O.
- Established generic commutativity of a large family of simple modules.

## Abstract

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R-matrices in the category O and we establish that a large family of simple modules, including the prefundamental representations associated to Q-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified QQ*-systems in terms of the R-matrices we construct.