### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 063, 13 pages      arXiv:1610.01782      https://doi.org/10.3842/SIGMA.2017.063

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

Victor Mouquin
University of Toronto, Toronto ON, Canada

Received March 26, 2017, in final form August 01, 2017; Published online August 09, 2017

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 Ševera under an equivalence of categories between Poisson and quasi-Poisson spaces.

Key words: flat connections; Poisson Lie groups; $r$-matrices; quasi-Poisson spaces.

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