We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra $\dot{U}(\mathfrak{g}[t])$. This extends a result of Beliakova, Habiro, Lauda, and Webster who showed that the trace of the categorified quantum group $\dot{\mathcal{U}}^*(\mathfrak{g})$ is isomorphic to $\dot{U}(\mathfrak{g}[t])$, and the trace of a cyclotomic quotient of $\dot{\mathcal{U}}^*(\mathfrak{g})$, which categorifies a single irreducible for the quantum group, is isomorphic to a Weyl module for $\dot{U}(\mathfrak{g}[t])$. We use a deformation argument based on Webster's technique of unfurling 2-representations.

中文翻译：

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张量积代数的迹去分类

我们表明，在 ADE 类型中，Webster 对量子群不可约量的张量积进行分类的迹与当前代数 $\dot{U}(\mathfrak{g}[t]) 的 Weyl 模的张量积同构$. 这扩展了 Beliakova、Habiro、Lauda 和 Webster 的结果，他们表明分类量子群 $\dot{\mathcal{U}}^*(\mathfrak{g})$ 的迹与 $\dot{ 同构U}(\mathfrak{g}[t])$，以及 $\dot{\mathcal{U}}^*(\mathfrak{g})$ 的分圆商迹，它为量子群，同构于 $\dot{U}(\mathfrak{g}[t])$ 的 Weyl 模。我们使用基于 Webster 展开 2-表示技术的变形参数。