Noncommutative Cluster Varieties and Moduli Spaces of Local Systems

Abstract

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 ${\mathfrak{p}}_{\Theta }$ and we classify $R$-graded pairs $(G,\Theta )$ via a condition depending only on the subset $\Theta \subset \Delta$ 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 $G/U_{\Theta }$ 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 $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta )$ 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.