标题： 从量子特征簇中获得的$U_q(\mathfrak{sl_n})$的聚类实现 显示英文标题

标题： A cluster realization of $U_q(\mathfrak{sl_n})$ from quantum character varieties

摘要： 我们构造了一个量子群$U_q(\mathfrak{sl}_{n+1})$到与带标记的带有穿孔圆盘上的框架$PGL_{n+1}$-局部系统模空间相关的量子簇代数$\mathbf{L}_n$的单射代数同态。 我们得到了在与带标记的双穿孔圆盘相关的相应量子簇代数中$U_q(\mathfrak{sl}_{n+1})$的余乘积的描述，并用对应于将一个穿孔绕另一个穿孔旋转的半德恩扭转的映射类群元素来表达$R$-矩阵的作用。 因此，我们通过由$R$矩阵共轭得到的代数自同构作为显式的簇变换序列来实现$U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$的代数自同构，并推导出$R$矩阵关于簇单项式的量子对数的更精细分解。 显示英文摘要

摘要： We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We obtain a description of the coproduct of $U_q(\mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(\mathfrak{sl}_{n+1})^{\otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials.