nLab foliation of a Lie groupoid

Idea

A generalization of the notion of foliation of a smooth manifold from manifolds to Lie groupoids.

Definition

One of several equivalent definitions of a (regular) foliation of a smooth manifold is

Definition

A regular foliation of a smooth manifold $X$ is a wide sub-Lie algebroid of its tangent Lie algebroid, hence a Lie algebroid $\mathcal{P}$ over $X$ with injective anchor map

$$\array{ \mathcal{P} &\hookrightarrow & T X \\ \downarrow && \downarrow \\ X &=& X } \,.$$

In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.

Definition

A (regular) foliation of a Lie groupoid $\mathcal{G}_\bullet$ is a sub-Lie algebroid-groupoid of the tangent Lie algebroid-groupoid which is wide

$$\array{ \mathcal{P} &\hookrightarrow& T \mathcal{G}_1 \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0 } \,.$$

Examples

(…)

Related concepts

• foliation of a Lie algebroid

References

Maybe the first discussion of foliations of Lie groupoids appears in

• Eli Hawkins, A Groupoid Approach to Quantization (arXiv:math.SG/0612363)

Related discussion is in

• Cristian Ortiz, Multiplicative Dirac structures (arXiv:1212.0176)