Universal lifting theorem and quasi-Poisson groupoids

  • David Iglesias-Ponte

    (CSIC-UAM-UCM-UC3M), Madrid, Spain
  • Camille Laurent-Gengoux

    Universidade de Coimbra, Portugal
  • Ping Xu

    University of Luxembourg, Luxembourg


We prove the universal lifting theorem: for an α\alpha-simply connected and α\alpha-connected Lie groupoid Γ\Gamma with Lie algebroid AA, the graded Lie algebra of multi-differentials on AA is isomorphic to that of multiplicative multi-vector fields on Γ\Gamma. As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases.
The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair (D,G)(D, G) associated to a Manin quasi-triple (d,g,h)(\mathfrak d, \mathfrak g, \mathfrak h) induces a quasi-Poisson groupoid on the transformation groupoid G×D/GD/GG\times D/G\rightrightarrows D/G. Its momentum map corresponds exactly with the D/GD/G-momentum map of Alekseev and Kosmann-Schwarzbach.

Cite this article

David Iglesias-Ponte, Camille Laurent-Gengoux, Ping Xu, Universal lifting theorem and quasi-Poisson groupoids. J. Eur. Math. Soc. 14 (2012), no. 3, pp. 681–731

DOI 10.4171/JEMS/315