Rank $n$ swapping algebra for ${\mathrm{PGL}}_n$ 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 $({\mathbb{K}}^n\times {\mathbb{K}}^{n\ast }{)}^r/\mathrm{GL}(n,\mathbb{K})$. 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 ${\mathcal{X}}_{{\mathrm{PGL}}_n,D_k}$ 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 ${\mathcal{T}}^{\prime }$ are compatible with each other under the flips.

Revised for submission, previous section 5 is removed, new added section 7, 36 pages, 12 figures

References in corpus (2)

• Ratio coordinates for higher Teichmüller spaces
• Rank $n$ swapping algebra for the $\mathrm{PSL}(n,\mathbb{R})$ Hitchin component