Rank $n$ swapping algebra for $\operatorname{PGL}_n$ Fock-Goncharov $\mathcal{X}$ moduli space

TL;DR

This paper introduces a new Poisson algebra called the rank n swapping algebra, which models the PGL_n Fock-Goncharov moduli space and provides injective homomorphisms from its coordinates, compatible under triangulation flips.

Contribution

It constructs an injective Poisson algebra homomorphism from Fock-Goncharov coordinates to the rank n swapping algebra for disk surfaces, forming a foundation for general surfaces.

Findings

Established an injective Poisson algebra homomorphism for disk surfaces.

Demonstrated compatibility of homomorphisms under triangulation flips.

Provided a geometric model for the swapping algebra in the context of PGL_n moduli spaces.

Abstract

The {\em rank $n$ swapping algebra} is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of $(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$. For any ideal triangulation of $D_k$---a disk with $k$ points on its boundary, using determinants, we find an injective Poisson algebra homomorphism from the fraction algebra generated by the Fock--Goncharov coordinates for $\mathcal{X}_{\operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction algebra for $r=k\cdot(n-1)$ with respect to the (Atiyah--Bott--)Goldman Poisson bracket and the swapping bracket. This is the building block of the general surface case. Two such injective Poisson algebra homomorphisms related to two ideal triangulations $\mathcal{T}$ and $\mathcal{T}'$ are compatible with each other under the flips.