Duals of Semisimple Poisson–Lie Groups and Cluster Theory of Moduli Spaces of G-local Systems
Duals of Semisimple Poisson–Lie Groups and Cluster Theory of Moduli Spaces of G-local Systems
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半单泊松-李群的对偶与G局部系统模空间的簇论
DOI：
10.1093/imrn/rnab094
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Li
中科院分区：
文献类型：
--
作者：
Li
我们从聚类理论的角度研究标准半单泊松李群${\rm G}$的对偶${\rm G}^\ast$。我们证明坐标环 $\mathcal{O}({\rm G}^\ast)$ 可以自然地嵌入到具有 Weyl 群作用的簇泊松代数中。我们证明 $\mathcal{O}({\rm G}^\ast)$ 承认一个具有正整数结构系数的自然基，并满足编织群作用的不变性。我们继续研究\cite{GS3}中引入的${\rm G}$局部系统的模空间$\mathscr{P}_{{\rm G},\mathbb{S}}$，并证明$\mathscr{P}_{{\rm G}, \mathbb{S}}$的坐标环与其底层的泊松代数簇重合。
We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $\mathcal{O}({\rm G}^\ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $\mathscr{P}_{{\rm G},\mathbb{S}}$ of ${\rm G}$-local systems introduced in \cite{GS3}, and prove that the coordinate ring of $\mathscr{P}_{{\rm G}, \mathbb{S}}$ coincides with its underlying cluster Poisson algebra.