$\operatorname{PGL}_n$ Fock-Goncharov $\mathcal{X}$模空间的秩$n$交换代数
Rank $n$ swapping algebra for $\operatorname{PGL}_n$ Fock-Goncharov $\mathcal{X}$ moduli space
摘要：
{ \em秩$n$交换代数}是一个Poisson代数，它定义在圆上的点的有序对的集合上，使用连接数，其几何模型由$(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$的某个子空间给出.对于$D_k$ --一个在其边界上具有$k$点的圆盘的任意理想三角剖分，利用行列式，我们找到了从$\mathcal{X}_{\operatorname{PGL}_n,D_k}$的Fock-Goncharov坐标生成的分数代数到$r=k\cdot(n-1)$的秩$n$交换重分数代数关于（Atiyah--Bott--）Goldman Poisson括号和交换括号的单射Poisson代数同态.这是一般表面情况的构建块。与理想三角剖分$\mathcal{T}$和$\mathcal{T}'$有关的两个这样的单射Poisson代数同态在翻转下是相容的。
英文摘要：
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.
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