We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category $\mathcal{O}$ of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type $A$, we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type $A_1$, we identify remarkable relations in this quantum Grothendieck ring.

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量子格罗腾迪克环作为量子簇代数

我们为该类别的某个幺半群子类别定义并构造了一个量子格洛腾迪克环$\mathcal{○}$Hernandez-Jimbo 引入的量子环代数的表示。我们利用这一类格洛腾迪克环的簇代数结构，将量子格洛腾迪克环定义为一个量子簇代数。当基础简单李代数是类型时$\mathrm{一个}$，我们证明这个量子格洛腾迪克环包含相关量子仿射代数的有限维表示范畴的量子格洛腾迪克环。在类型${\mathrm{一个}}_1$，我们在这个量子格洛腾迪克环中发现了显着的关系。