Reference : Rank n swapping algebra for PGLn Fock--Goncharov X moduli space
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Physical, chemical, mathematical & earth Sciences : Mathematics
Rank n swapping algebra for PGLn Fock--Goncharov X moduli space
Sun, Zhe mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
Mathematische Annalen
Yes (verified by ORBilu)
[en] Poisson algebra homomorphism ; rank n swapping algebra ; Fock–Goncharov X moduli space
[en] The rank $n$ swapping multifraction algebra is a field of cross ratios up to $(n+1)\times (n+1)$-determinant relations equipped with a Poisson bracket, called the {\em swapping bracket}, defined on the set of ordered pairs of points of a circle using linking numbers. Let $D_k$ be a disk with $k$ points on its boundary. The moduli space $\mathcal{X}_{\operatorname{PGL}_n,D_k}$ is the building block of the Fock--Goncharov $\mathcal{X}$ moduli space for any general surface. Given any ideal triangulation of $D_k$, we find an injective Poisson algebra homomorphism from the rank $n$ Fock--Goncharov algebra for $\mathcal{X}_{\operatorname{PGL}_n,D_k}$ to the rank $n$ swapping multifraction algebra with respect to the Atiyah--Bott--Goldman Poisson bracket and the swapping bracket. Two such injective Poisson algebra homomorphisms related to two ideal triangulations $\mathcal{T}$ and $\mathcal{T}'$ are compatible with each other under the flips.
FP7 ; 246918 - HIGHTEICH - Higher Teichmüller-Thurston Theory: Representations of Surface Groups in PSL(n,R).
FnR ; FNR13242285 > Zhe Sun > > COmbinatorial and ALgebraic Aspects of Surface group representations. > 01/09/2018 > 31/08/2020 > 2017

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