[Une algèbre d'observables pour les birapports]
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 -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 -opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd–Sokolov structure and to the Atiyah–Bott–Goldman symplectic structure.
Publié le :
@article{CRMATH_2010__348_9-10_503_0,
author = {Labourie, Fran\c{c}ois},
title = {An algebra of observables for cross ratios},
journal = {Comptes Rendus. Math\'ematique},
pages = {503--507},
publisher = {Elsevier},
volume = {348},
number = {9-10},
year = {2010},
doi = {10.1016/j.crma.2010.03.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.03.012/}
}
TY - JOUR AU - Labourie, François TI - An algebra of observables for cross ratios JO - Comptes Rendus. Mathématique PY - 2010 SP - 503 EP - 507 VL - 348 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.03.012/ DO - 10.1016/j.crma.2010.03.012 LA - en ID - CRMATH_2010__348_9-10_503_0 ER -
Labourie, François. An algebra of observables for cross ratios. Comptes Rendus. Mathématique, Tome 348 (2010) no. 9-10, pp. 503-507. doi : 10.1016/j.crma.2010.03.012. http://www.numdam.org/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.
