ﻻ يوجد ملخص باللغة العربية
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 $r=kcdot(n-1)$ 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.
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for $operatorname{PSL}(n, mathbb{R})$ for any $n>1$ by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introd
The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(mathbb{K}^n times mathbb{K}^{n*})^r/operatorname{GL}(n,mathbb{K})$ is its geometric mode. In this
We induce a Poisson algebra ${cdot,cdot}_{mathcal{C}_{n,N}}$ on the configuration space $mathcal{C}_{n,N}$ of $N$ twisted polygons in $mathbb{RP}^{n-1}$ from the swapping algebra cite{L12}, which is found coincide with Faddeev-Takhtajan-Volkov algebr
We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $mathcal{N}geq 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces
In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $Phi_mathcal{T}$ for a distinguished collection $mathcal{W}_{3, widehat{S}}$ of $mathrm{SL}_3$-webs on a finite-type punctured surface $widehat{S}$, depending on