Universal lifting theorem and quasi-Poisson groupoids

Inglesias-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 -