nLab foliation of a Lie algebroid

Idea

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

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 algebroid $A$ is a sub-double Lie algebroid of the tangent double Lie algebroid which is wide over $A$

$$\array{ \mathcal{P} &\hookrightarrow& T A \\ \downarrow && {}^{\mathllap{d p_A}}\downarrow & \searrow^{\mathrlap{p_{T A}}} \\ \mathcal{P}_0 &\hookrightarrow& T X && A \\ && & \searrow & \downarrow^{\mathrlap{p_A}} \\ && && X } \,.$$

Proposition

Foliations of a Lie algbroid $A \to X$ according to def. are in natural bijection to the following data:

1. an ordinary foliation $\mathcal{P}_0 \hookrightarrow T X$

2. $\mathcal{P}_1 \hookrightarrow A$ a sub-vector bundle

   (this is the joint kernel $\mathcal{P}_1 = ker(p_{T A}) \cap ker(d p_{A})$ naturally identified as a subspace of $A$)

3. $\nabla$ a flat connection on the quotient bundle $A/\mathcal{P}_1$ partially defined over vector fields in $\mathcal{P}_0$

   (this is the induced linear foliation of the total space $A$ regarded as a horizontal-subspace distribution)

such that over every open subset $U \hookrightarrow X$

1. The sections of $A$ that become $\nabla$-constant in $A/\mathcal{P}_1$ form a sub-Lie algebra of $\Gamma_U(A|_U)$ (an integrable distribution of subspaces);

2. the sections of $\mathcal{P}_1$ are a Lie ideal inside this sub-Lie algebra;

3. the image of $\mathcal{P}_1$ under the anchor map is in $\mathcal{P}_0$;

4. the quotiented anchor map $A/\mathcal{P}_1 \to T X / \mathcal{P}_0$ intertwines $\nabla$ with the $\mathcal{P}_0$-Bott connection.

This is (EH, theorem 7.2).

Remark

According to prop. the notion of foliation of a Lie algebroid generalizes the notion of ideal system of a Lie algebroid (Higgins-Mackenzie, Mackenzie): A foliation as in def. comes from an ideal system in this sense precisely of $\mathcal{P}_0$ is a simple foliation (the quotient map exists in smooth manifolds and is a surjective submersion) and the holonomy of $\nabla$ is trivial.

Examples

Example

Let $\mathcal{G}_\bullet$ be a Lie groupoid and let

$$\array{ \mathcal{P} &\hookrightarrow& T \mathcal{G}_1 \\ \downarrow\downarrow && {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}} \\ \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0 }$$

be a foliation of a Lie groupoid, regarded as an internal groupoid in Lie algebroids. Then applying Lie differentiation yields a foliation of the Lie algebroid $Lie(\mathcal{G}_\bullet)$.

Related concepts

• foliation of a Lie groupoid

• exterior differential system

Referemces

Maybe the first discussion of foliations of Lie algebroids appears in

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

Ideal systems of Lie algebroids have been introduced and studied in

• Philip Higgins, Kirill Mackenzie, Algebraic constructions in the category of Lie algebroids . J. Algebra

  129 (1990), no. 1, 194–230.MR1037400

• Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge (2005)

Related discussion is in

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