Article (Scientific journals)
Rank n swapping algebra for PGLn Fock--Goncharov X moduli space
Sun, Zhe
2000In Mathematische Annalen
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Keywords :
Poisson algebra homomorphism; rank n swapping algebra; Fock–Goncharov X moduli space
Abstract :
[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.
Disciplines :
Mathematics
Author, co-author :
Sun, Zhe ;  University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
External co-authors :
no
Language :
English
Title :
Rank n swapping algebra for PGLn Fock--Goncharov X moduli space
Publication date :
2000
Journal title :
Mathematische Annalen
ISSN :
1432-1807
Publisher :
Springer, Heidelberg, Germany
Peer reviewed :
Peer Reviewed verified by ORBi
European Projects :
FP7 - 246918 - HIGHTEICH - Higher Teichmüller-Thurston Theory: Representations of Surface Groups in PSL(n,R).
FnR Project :
FNR13242285 - COmbinatorial and ALgebraic Aspects of Surface group representations, 2017 (01/09/2018-31/08/2020) - Zhe Sun
Funders :
CE - Commission Européenne [BE]
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