### An Algebra of Observables for Cross Ratios

Labourie, F *(Paris-Sud 11)*

Monday 14 March 2011, 15:00-16:00

Satellite

#### Abstract

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 for classical Teichmüller spaces and Hitchin components.