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

SIGMA 13 (2017), 063, 13 pages      arXiv:1610.01782

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

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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  1. Alekseev A., Kosmann-Schwarzbach Y., Manin pairs and moment maps, J. Differential Geom. 56 (2000), 133-165, math.DG/9909176.
  2. Alekseev A., Kosmann-Schwarzbach Y., Meinrenken E., Quasi-Poisson manifolds, Canad. J. Math. 54 (2002), 3-29, math.DG/0006168.
  3. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  4. Fock V.V., Rosly A.A., Flat connections and polyubles, Theoret. and Math. Phys. 95 (1993), 228-238.
  5. Fock V.V., Rosly A.A., Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix, in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 191, Amer. Math. Soc., Providence, RI, 1999, 67-86, math.QA/9802054.
  6. Li-Bland D., Ševera P., Quasi-Hamiltonian groupoids and multiplicative Manin pairs, Int. Math. Res. Not. 2011 (2011), 2295-2350, arXiv:0911.2179.
  7. Li-Bland D., Ševera P., Moduli spaces for quilted surfaces and Poisson structures, Doc. Math. 20 (2015), 1071-1135, arXiv:1212.2097.
  8. Lu J.-H., Mouquin V., Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras, Int. Math. Res. Not., to appear, arXiv:1504.06843.
  9. Massuyeau G., Turaev V., Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not. 2014 (2014), 1-64, arXiv:1205.4898.
  10. Meusburger C., Wise D.K., Hopf algebra gauge theory on a ribbon graph, arXiv:1512.03966.

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