Foliated groupoids and their infinitesimal data

Abstract

In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of IM-foliation on a Lie algebroid. The main result of this paper shows that if $G$ is a source simply connected Lie groupoid with Lie algebroid $A$, then there exists a one-to-one correspondence between multiplicative foliations on $G$ and IM-foliations on the Lie algebroid $A$.

arXiv commentWe have found out that one of the main results can already be found in the paper "A groupoid approach to quantization" by Eli Hawkins, Journal of symplectic geometry (2008). We were not aware of this paper while working on this problem. An adapted version of our paper will be uploaded here later

References in corpus (6)

• Higher vector bundles and multi-graded symplectic manifolds
• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Linear and multiplicative 2-forms
• Multiplicative Dirac structures on Lie groups
• Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces
• The leaf space of a multiplicative foliation

Cited by in corpus (6)

• Lie groupoids and the Frolicher-Nijenhuis bracket
• Infinitesimal objects associated to Dirac groupoids and their homogeneous spaces
• Multiplicative Forms and Spencer Operators
• Dorfman connections and Courant algebroids
• Multiplicative Connections and Their Lie Theory
• The leaf space of a multiplicative foliation