非交换簇簇代数簇与局部系统的模空间
Noncommutative Cluster Varieties and Moduli Spaces of Local Systems
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中文总结 AI 辅助
本文构造了推广既有结果的非交换簇簇代数簇，定义了若尔当分裂群并分类其分次对，给出了双Bruhat胞的非交换簇结构，还证明了实代数群正结构的相关性质。
中文摘要 AI 辅助
在本文中，我们针对每个既约根系$R$和带标记曲面$S$构造了非交换簇簇代数簇$\u200b\mathcal{A}_{R,S}$，该构造同时推广了Fock-Goncharov、Li、Goncharov-Shen、Berenstein-Retakh、Goncharov-Kontsevich的簇簇代数簇，以及我们此前提出的多边形簇代数。此外，我们定义了一大类代数群，我们称之为若尔当分裂群。给定一个既约根系$R$和一族若尔当代数，群$G$的李代数通过将单个若尔当代数的Tits-Kantor-Koecher构造与分裂李代数的构造相统一来构建。\n若尔当分裂群的概念与其李代数的根系$R$分次密切相关。我们证明这些分次通常由标准抛物子代数$\mathfrak{p}_Θ$的选取诱导，并且我们通过仅依赖于单根集的子集$Θ\subset Δ$的条件，对$R$分次对$(G,Θ)$进行了分类。接下来，我们定义了$\mathcal{A}_{R,S}$的若尔当代数点，当$G$是$R$型若尔当分裂群时，这些点参数化了$S$上带有与陪集$G/U_Θ$相关的边界装饰的$G$-局部系统。当$S$是一个圆盘时，$\mathcal{A}_{R,S}$的点参数化了带装饰旗的构型。我们利用这一点在$G$的双$R$-Bruhat胞上给出了非交换簇结构，推广了Berenstein-Fomin-Zelevinsky的簇代数。当每个若尔当代数都是形式实的时，我们称$G$关于$Θ$具有正结构。这在$G$中定义了一个正半群。对于实代数群，具有正结构的对$(G,Θ)$恰好是那些容许Guichard-Wienhard所定义的正结构的对，并且我们给出了旗的正构型以及正表示的诸多性质的代数证明。
英文摘要
In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_Θ$ and we classify $R$-graded pairs $(G,Θ)$ via a condition depending only on the subset $Θ\subset Δ$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_Θ$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $Θ$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,Θ)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.
