The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular $r$-Matrix

TL;DR

This paper reinterprets the Fock-Rosly Poisson structure on moduli spaces of flat connections using Lie algebra actions and quasitriangular $r$-matrices, linking it to mixed product Poisson structures and quasi-Poisson geometry.

Contribution

It provides a new formulation of the Fock-Rosly Poisson structure within the framework of Lie algebra actions and quasitriangular $r$-matrices, connecting it to quasi-Poisson structures.

Findings

Fock-Rosly Poisson structure is a mixed product Poisson structure.

Establishes an equivalence between Poisson and quasi-Poisson structures.

Links the structure to Lie algebra actions and $r$-matrices.

Abstract

We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces.