# Universal lifting theorem and quasi-Poisson groupoids

David Inglesias-Ponte; Camille Laurent-Gengoux; Ping Xu

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 3, page 681-731
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topInglesias-Ponte, David, Laurent-Gengoux, Camille, and Xu, Ping. "Universal lifting theorem and quasi-Poisson groupoids." Journal of the European Mathematical Society 014.3 (2012): 681-731. <http://eudml.org/doc/277554>.

@article{Inglesias2012,

abstract = {We prove the universal lifting theorem: for an $\alpha $-simply connected and $\alpha $-connected Lie groupoid $\Gamma $ with Lie algebroid $A$, the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $\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 $(D,G)$ associated to a Manin quasi-triple $\partial ,g,\hbar $ induces a quasi-Poisson groupoid on the transformation groupoid $G\times D/G \Rightarrow D/G$. Its momentum map corresponds exactly with the $D/G$-momentum map of Alekseev and Kosmann-Schwarzbach.},

author = {Inglesias-Ponte, David, Laurent-Gengoux, Camille, Xu, Ping},

journal = {Journal of the European Mathematical Society},

keywords = {Lie algebroids and groupoids; multiplicative multivector fields; Lie bialgebroids; Poisson and symplectic groupoids; Manin pairs; Hamiltonian spaces; universal lifting theorem; Manin triple; momentum maps; Universal lifting theorem; Lie groupoids; Lie algebroids; Poisson groupoids; Manin triple; momentum maps},

language = {eng},

number = {3},

pages = {681-731},

publisher = {European Mathematical Society Publishing House},

title = {Universal lifting theorem and quasi-Poisson groupoids},

url = {http://eudml.org/doc/277554},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Inglesias-Ponte, David

AU - Laurent-Gengoux, Camille

AU - Xu, Ping

TI - Universal lifting theorem and quasi-Poisson groupoids

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 3

SP - 681

EP - 731

AB - We prove the universal lifting theorem: for an $\alpha $-simply connected and $\alpha $-connected Lie groupoid $\Gamma $ with Lie algebroid $A$, the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $\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 $(D,G)$ associated to a Manin quasi-triple $\partial ,g,\hbar $ induces a quasi-Poisson groupoid on the transformation groupoid $G\times D/G \Rightarrow D/G$. Its momentum map corresponds exactly with the $D/G$-momentum map of Alekseev and Kosmann-Schwarzbach.

LA - eng

KW - Lie algebroids and groupoids; multiplicative multivector fields; Lie bialgebroids; Poisson and symplectic groupoids; Manin pairs; Hamiltonian spaces; universal lifting theorem; Manin triple; momentum maps; Universal lifting theorem; Lie groupoids; Lie algebroids; Poisson groupoids; Manin triple; momentum maps

UR - http://eudml.org/doc/277554

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.