Des structures de (quasi-)Poisson quadratiques sur l'algèbre de lacets pour la construction d'un système intégrable sur un espace de modules

Abstract : This thesis is a work on the moduli space $\mathscr M$ of flat connections
of the principal bundle $S\times G$ of a punctured Riemann sphere $S$ (with
$n\geq3$ boundary components), whose Lie group is $G=\GL{N,\C}$, and on
the loop algebra $\tilde\g=\gl{N,\C}(\!(\l^\mi)\!)$ simultaneously.

In a first time, we study a hierarchy of quadratic biderivations on
$\tilde\g$. In particular, thanks to the fusion processus introduced by
Alekseev, Kosmann-Schwarzbach and Meinrenken in 2002, we extract, among
them, a quasi-Poisson structure $\PB^Q_1$ on $\tilde\g$. This one
restricts to the subspace $\tilde\g_n=\set{\sum_{k=0}^nx^{[k]}\l^k}$.

We prove then a reduction result in the framework of a quasi-Poisson
biderivation. It allows us to equip with a genuine Poisson structure the quotient $\mathscr
A/G:=\set{\Id\l^n+\l Y(\l)+\Id|Y\in\tilde\g_{n-2}}/G$.

Knowing Beauville's integrable system on $\tilde\g_{n-2}/G$, we prove that
the family of functions $({\text{tr}} X^k(a))_{k\in\N,a\in\C}$ constitute
an integrable system on $\mathscr A/G$. The functions that we considere on
the moduli space $\mathscr M$ are the pull-back $(\mathscr
T^*{\text{tr}}X^k(a))_{k\in\N,a\in\C}$, where $\mathscr T:G^n\to\tilde\g_n$
is a quasi-Poisson morphism and a local diffeomorphism. We use these
properties of $\mathscr T$ to show that this family of functions constitute
an integrable system on $\mathscr M$.
Document type :
Theses
Mathematics [math]. Université de Poitiers, 2006. French


https://tel.archives-ouvertes.fr/tel-00114640
Contributor : Ariane Le Blanc <>
Submitted on : Friday, November 17, 2006 - 11:53:33 AM
Last modification on : Friday, November 17, 2006 - 12:17:45 PM

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Ariane Le Blanc. Des structures de (quasi-)Poisson quadratiques sur l'algèbre de lacets pour la construction d'un système intégrable sur un espace de modules. Mathematics [math]. Université de Poitiers, 2006. French. <tel-00114640>

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