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

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.