  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.
