$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

TL;DR

This paper proves that for certain Lie algebras and surfaces, the cluster algebra equals its upper cluster algebra, using functions generated by Wilson lines, and relates skein algebras to these cluster algebras.

Contribution

It establishes the equality of cluster and upper cluster algebras for specific cases and connects skein algebras with cluster algebras via Wilson line functions.

Findings

$ ext{A}= ext{U}$ for $ ext{g}$-cluster algebras from moduli spaces.

Skein algebras are isomorphic to cluster algebras for $ ext{sl}_2$, $ ext{sl}_3$, and $ ext{sp}_4$.

Function rings are generated by Wilson line matrix coefficients.

Abstract

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\Sigma$ with at least two marked points and no punctures, we prove that the cluster algebra $\mathscr{A}_{\mathfrak{g},\Sigma}$ associated with the pair $(\mathfrak{g},\Sigma)$ coincides with the upper cluster algebra $\mathscr{U}_{\mathfrak{g},\Sigma}$. The proof is based on the fact that the function ring $\mathcal{O}(\mathcal{A}^\times_{G,\Sigma})$ of the moduli space of decorated twisted $G$-local systems on $\Sigma$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras $\mathscr{S}_{\mathfrak{g}, \Sigma}[\partial^{-1}]$ [Muller,IY23,IY22] for $\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3,$ or $\mathfrak{sp}_4$ are isomorphic to the cluster algebras $\mathscr{A}_{\mathfrak{g}, \Sigma}$.