We study the dual $\textrm{G}^{\ast }$ of a standard semisimple Poisson–Lie group $\textrm{G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ can be naturally embedded into a quotient algebra of a cluster Poisson algebra with a Weyl group action. The coordinate ring $\mathcal{O}(\textrm{G}^{\ast })$ admits a natural basis, which has positive integer structure coefficients and satisfies an invariance property under a braid group action. We continue the study of the moduli space $\mathscr{P}_{\textrm{G},{{\mathbb{S}}}}$ of $\textrm{G}$-local systems introduced in [ 16] and prove that the coordinate ring of $\mathscr{P}_{\textrm{G}, {{\mathbb{S}}}}$ coincides with its underlying cluster Poisson algebra.

中文翻译：

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G-局域系统模空间的半单泊松李群对偶和聚类理论

我们从聚类理论的角度研究标准半单泊松李群 $\textrm{G}$ 的对偶 $\textrm{G}^{\ast }$。我们展示了坐标环 $\mathcal{O}(\textrm{G}^{\ast })$ 可以自然地嵌入到具有 Weyl 群作用的集群 Poisson 代数的商代数中。坐标环 $\mathcal{O}(\textrm{G}^{\ast })$ 承认自然基，它具有正整数结构系数，并满足辫群作用下的不变性。我们继续研究 [16] 中介绍的 $\textrm{G}$-local 系统的模空间 $\mathscr{P}_{\textrm{G},{{\mathbb{S}}}}$ 和证明$\mathscr{P}_{\textrm{G}, {{\mathbb{S}}}}$ 的坐标环与其底层簇泊松代数重合。