Comptes Rendus
Lie Algebras/Geometry
An algebra of observables for cross ratios
[Une algèbre d'observables pour les birapports]
Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 503-507.

Nous introduisons une algèbre de Poisson, l'algèbre d'échange, définie à l'aide de l'intersection des courbes dans le disque. Nous interprétons l'algèbre des multifractions – une sous-algèbre de l'algèbre des fractions de l'algèbre d'échange – comme une algèbre de fonctions sur l'espace des birapports et donc en particulier comme une algèbre de fonctions sur la composante de Hitchin ainsi que sur l'espace des SL(n,R)-opers d'holonomie triviale. Nous relions alors notre structure de Poisson à la structure de Poisson de Drinfel'd–Sokolov ainsi qu'à la structure symplectique d'Atiyah–Bott–Goldman.

We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra – called the algebra of multifractions – as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of SL(n,R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd–Sokolov structure and to the Atiyah–Bott–Goldman symplectic structure.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.03.012

François Labourie 1

1 Univ. Paris-Sud, laboratoire de mathématiques, CNRS, 91405 Orsay cedex, France
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François Labourie. An algebra of observables for cross ratios. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 503-507. doi : 10.1016/j.crma.2010.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.012/

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Cité par Sources :

Partially supported by the ANR program ETTT-ANR-09-BLAN-0116-01 and the ANR program RepSurfaces-ANR-06-BLAN-0311.

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