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 $\mathcal {O}$ 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 $R$-matrices in the category $\mathcal {O}$ 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.

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中文翻译：

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稳定的地图、Q 算子和类别𝒪

摘要：受与 quiver 变体相关的 Maulik-Okounkov 稳定映射的启发，我们定义并构建了任意非扭曲量子仿射代数的 Borel 子代数类别 $\mathcal {O}$ 中表示的张量积的代数稳定映射。我们的表示理论构造是基于对 Cartan-Drinfeld 子代数的作用的研究。我们证明了代数稳定映射是可逆的并且合理地依赖于谱参数。作为一个应用程序，我们在 $\mathcal {O}$ 类别中获得了新的 $R$-矩阵，并且我们建立了一个大系列的简单模块，包括与 $Q$-算子相关的基本表示，通常可以交换为Cartan-Drinfeld 子代数。我们还根据我们构建的$R$-矩阵建立了分类的$QQ^*$-系统。

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