An algebra of observables for cross ratios (Q974009)

This paper introduces a Poisson algebra called the swapping algebra and studies a subalgebra interpreted as an algebra of observables on the space of cross ratios. Let \(P\) be a subset of the circle and \(Z(P)\) be the swapping algebra, defined by using the intersection of the curves. The author proves that the vector space \(B(P)\) generated by multifractions is a Poisson algebra. He then shows how this algebra can be seen as an algebra of observables, that is, an algebra of functions on a space. Finally, the author relates the Poisson structure on \(B(P)\) with two classical Poisson structures, namely, the Drinfeld-Sobolov structure on the space of \(\text{SL}(n,\mathbb R)\)-opers and the Atiyah-Bott-Goldman symplectic structure on the character variety of a surface group in \(\text{SL}(n,\mathbb R)\).