Stable maps, Q-operators and category 𝒪

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 \mathcal {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 R -matrices in the category O \mathcal {O} and we establish that a large family of simple modules, including the prefundamental representations associated to Q Q -operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified Q Q ∗ QQ^* -systems in terms of the R R -matrices we construct.