带装饰旗的量子配置空间的代数研究

中文摘要 AI 辅助

设$G$为连通、单连通的复单代数群，$\boldsymbol{\textit{A}}_G=G/U^+$是其基础仿射空间，该空间中的元素称为带装饰旗。本文引入带装饰旗的量子配置空间$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G)$，并基于量子包络代数的表示理论开启对其的代数研究。我们的代数$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G)$是带$K$个装饰旗的配置空间$\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G$的量子类比，后者为标记曲面$\boldsymbol{\textit{A}}_{G,\boldsymbol{\textit{\u03a3}}}$（带装饰的扭曲$G$-局部系统的Fock–Goncharov模空间）提供局部构建块。我们确立了$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G)$的基本代数性质，如代表元的量子正规化、量子循环移位、量子Wilson线，其经典对应物是研究$\boldsymbol{\textit{A}}_{G,\boldsymbol{\textit{\u03a3}}}$的基础。此外，我们通过量子Wilson线将$\boldsymbol{\textit{O}}_q(G)$上的Berenstein–Zelevinsky量子簇结构迁移，构造出$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_4 \boldsymbol{\textit{A}}_G)$的量子种子，并证明在冻结变量局部化后，$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_4 \boldsymbol{\textit{A}}_G)$与对应的量子簇代数及其上簇代数一致；该量子种子的交换矩阵与Goncharov–Shen交换矩阵一致。我们还证明$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_4 \boldsymbol{\textit{A}}_G)$的量子种子可限制为$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_3 \boldsymbol{\textit{A}}_G)$的量子种子。最后，在一个自然猜想下，我们构造出$K\boldsymbol{\u2265}5$时$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G)$的量子种子，并证明对应的量子簇代数包含量子配置空间$\boldsymbol{\textit{O}}_q(\boldsymbol{\textit{Conf}}_K \boldsymbol{\textit{A}}_G)$。

英文摘要

Let $G$ be a connected, simply connected complex simple algebraic group and $\mathscr{A}_G=G/U^+$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ gives a quantum analogue of the configuration space $\mathrm{Conf}_K \mathscr{A}_G$ of $K$ decorated flags, which provides local building blocks for the Fock--Goncharov moduli space $\mathscr{A}_{G,Σ}$ of decorated twisted $G$-local systems on a marked surface $Σ$. We establish basic algebraic properties of $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of $\mathscr{A}_{G,Σ}$. Moreover, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ by transporting the Berenstein--Zelevinsky quantum cluster structure on $\mathcal{O}_q(G)$ via quantum Wilson lines, and prove that $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov--Shen exchange matrices. We also show that quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ restrict to those for $\mathcal{O}_q(\mathrm{Conf}_3 \mathscr{A}_G)$. Finally, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$, $K\geq 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$.

发表机构

• Tohoku University（东北大学）
• Institute of Science Tokyo（东京科学大学）

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补充信息

备注

v1, 92 pages; v2, 93 pages. Section 9 was updated, in particular, Conjecture 9.1 in v1 was proved. The domain property of quantum configuration spaces was shown in Propositions 3.5 and 3.7. Minor typos corrected