The generalized quantum group $\mathcal{U}(\epsilon)$ of type $A$ is an affine analogue of quantum group associated to a general linear Lie superalgebra $\mathfrak{gl}_{M|N}$. We prove that there exists a unique $R$ matrix on tensor product of fundamental type representations of $\mathcal{U}(\epsilon)$ for arbitrary parameter sequence $\epsilon$ corresponding to a non-conjugate Borel subalgebra of $\mathfrak{gl}_{M|N}$. We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible $\mathcal{U}(\epsilon)$-modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type $A_{M-1}^{(1)}$ or $A_{N-1}^{(1)}$.

中文翻译：

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A 类广义量子群的 R 矩阵

类型为 $A$ 的广义量子群 $\mathcal{U}(\epsilon)$ 是与一般线性李超代数 $\mathfrak{gl}_{M|N}$ 相关联的量子群的仿射类比。我们证明，对于对应于 $\mathfrak 的非共轭 Borel 子代数的任意参数序列 $\epsilon$，在 $\mathcal{U}(\epsilon)$ 的基本类型表示的张量积上存在唯一的 $R$ 矩阵{gl}_{M|N}$。我们给出了它的谱分解的明确描述，然后作为一个应用，构造了一个有限维不可约 $\mathcal{U}(\epsilon)$-模的族，这些模具有与通常仿射的 Kirillov-Reshetikhin 模同构的子空间输入 $A_{M-1}^{(1)}$ 或 $A_{N-1}^{(1)}$。