ﻻ يوجد ملخص باللغة العربية
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 introduce the rank $n$ swapping algebra, which is the quotient of the swapping algebra by the $(n+1)times(n+1)$ determinant relations. The main results are the well-definedness of the rank $n$ swapping algebra and the cross-ratio in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank $n$ swapping algebra generated by these cross-ratios to characterize the $operatorname{PSL}(n, mathbb{R})$ Hitchin component for a fixed $n>1$. We also show the relation between the rank $2$ swapping algebra and the cluster $mathcal{X}_{operatorname{PGL}(2,mathbb{R}),D_k}$-space.
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
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}(
In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy repr
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
Given a hypersurface $M$ of null scalar curvature in the unit sphere $mathbb{S}^n$, $nge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $Rr^{n+1}$ as a normal graph over a trunca