The leaf space of a multiplicative foliation

We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G\rr P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$. As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.

1. Introduction. Let G be a Lie group with Lie algebra g = T e G and multiplication map m : G × G → G. Then the tangent space T G of G is also a Lie group with unit 0 e ∈ g and multiplication map T m : T G × T G → T G. A multiplicative distribution S ⊆ T G is a distribution on G, i.e., S(g) := S ∩ T g G is a vector subspace of T g G for all g ∈ G, that is in addition a subgroup of T G. Since at each g ∈ G, S(g) is a vector subspace of T g G, the zero section of T G is contained in S. Thus, using T (g,h) m(0 g , v h ) = T h L g v h for any g, h ∈ G and v h ∈ T h G, where L g : G → G is the left translation by g, we find that the distribution S is left invariant. It follows that S is a smooth left invariant subbundle of T G defined by S(g) = s L (g), where s the vector subspace S(e) = S ∩ g of g. In the same manner, S is right invariant and we find thus that s is invariant under the adjoint action of G on g. Hence, s is an ideal in g and the subbundle S ⊆ T G is completely integrable in the sense of Frobenius. Its leaf N through the unit element e of G is a normal subgroup of G and since the leaf space G/S of S is equal to G/N , it inherits a group structure from G such that the projection G → G/N is a homomorphism of groups. If N is closed in G, the 314 M. JOTZ foliation by S is simple and the leaf space G/S = G/N is a Lie group such that the projection is a smooth surjective submersion. Similar discussions leading to this result can be found in [18,9]. If G⇒P is a Lie groupoid, its tangent space T G is, in the same manner as in the Lie group case, a Lie groupoid over the tangent space T P of the units. A distribution S ⊆ T G on G is multiplicative if it is a subgroupoid of T G⇒T P . The problem studied in this paper is hence natural: we ask how the simple fact stated above about the leaf space of a multiplicative distribution on a Lie group generalizes to Lie groupoids. We consider smooth multiplicative and involutive subbundles S ⊆ T G (that is, distributions of constant rank). The foliation of G by the leaves of S is then said to be multiplicative and the intersection T P ∩ S is automatically an involutive subbundle of T P . We show that the leaf space G/S inherits the source, target, unit inclusion and inversion maps of a groupoid over the space of leaves in P of T P ∩ S, that are all compatible with the projections G → G/S and P → P/(T P ∩ S). To be able to define a multiplication on the leaf space, we have to assume that a special family of vector fields spanning S on the Lie groupoid is complete in a sense that will be explained, and that a compatibility condition on the intersections of the leaves with the s-fibers is satisfied.
If the foliations S and S ∩ T P are simple, we get hence, under these conditions, the structure of a Lie groupoid on the leaf space G/S⇒P/(S ∩ T P ), such that the projection is a Lie groupoid morphism.
Using a partial S ∩ T P -connection on AG/(AG ∩ S) that is naturally induced by the multiplicative, involutive subbundle S ⊆ T G [6], we show that Γ(AG) is spanned as a C ∞ (P )-module by sections which left invariant images leave S invariant. We show that under (rather strong) conditions on these special left invariant vector fields that leave the foliation invariant, the completeness condition on S is not necessary and the compatibility condition that is necessary for the multiplication is satisfied.
We illustrate the theory by four examples. The first two examples (Examples 3.2 and 3.3) illustrate the result of the main theorem, and the third example (Example 3.16) shows that we recover as a special case the quotient of a Lie groupoid by a Lie group action by groupoid morphisms. This example suggests that our main theorem (Theorem 3.14) could be useful for discrete mechanics on groupoids (see Remark 3.17). The fourth example (Example 3.19, which is taken from [27]) is an example of a multiplicative foliation on a Lie groupoid that doesn't satisfy the necessary condition and which space of leaves fails hence to inherit a groupoid multiplication.
As an application, we show that if (G⇒P, D G ) is a Dirac groupoid such that the characteristic distribution T G ∩ D G has constant rank and its leaf space is simple and satisfies the required conditions, then the quotient Lie groupoid inherits the structure of a Poisson groupoid such that the quotient map is a forward Dirac map and a morphism of Lie groupoids. Again, this is a fact that is automatically true for an integrable Dirac Lie group if the leaf N through the unit e of the characteristic foliation is closed in the group ( [9]).
Notations and conventions. Let M be a smooth manifold. We will denote by X(M ) and Ω 1 (M ) the spaces of (local) smooth sections of the tangent and the cotangent bundle of M , respectively. For an arbitrary vector bundle E → M , the space of (local) sections of E will be written Γ(E). We will write Dom(σ) for the open subset of the smooth manifold M where the local section σ ∈ Γ(E) is defined.
, there exists a smooth vector field X ∈ X(M ) with values in F such that X(m) = v m . Note that in general, the rank of F (m) depends on m. If not, then F is simply a subbundle of T M .
The pullback or restriction of a distribution F ⊆ T M to an embedded submanifold N of M will be written F | N . We say that the distribution F has constant rank on N if the pullback F | N is a vector bundle over N , i.e., if all its fibers have the same dimension.
Assume that F is an involutive subbundle of T M , hence completely integrable in the sense of Frobenius. We say that F , or the foliation defined by F , is simple if the leaf space M/F of the foliation is a smooth manifold such that the projection M → M/F is a smooth surjective submersion.
We write E × M F for the direct sum of two vector bundles Assume finally that M, N are smooth manifolds and f : M → N is a smooth surjective submersion. The tangent space to the f -fibers will be written 2. The Pontryagin groupoid of a Lie groupoid. The general theory of Lie groupoids and their Lie algebroids can be found in [14], [15]. We fix here first of all some notations and conventions.
A groupoid G with base P will be written G⇒P . The set P will be considered most of the time as a subset of G, that is, the unity 1 p will be identified with p for all p ∈ P . A Lie groupoid is a groupoid G on base P together with the structures of smooth Hausdorff manifolds on G and P such that the maps s, t : G → P are surjective submersions, and such that the object inclusion map : P → G, p → 1 p and the partial multiplication m : G × P G → G are all smooth.
Let g ∈ G, then the right translation by g is The left translation L g is defined in an analogous manner. Given a (local) bisection K of G⇒P , we write also R K for the right-translation. In this paper, the Lie algebroid of the Lie groupoid G⇒P is AG := T t P G, equipped with the anchor map T s| AG and the Lie bracket defined by the left invariant vector fields.
We give in the following the induced Lie groupoid structures on the tangent, cotangent and Pontryagin bundle of a Lie groupoid.
The tangent prolongation of a Lie groupoid. Let G⇒P be a Lie groupoid. Applying the tangent functor to each of the maps defining G yields a Lie groupoid structure on T G with base T P , source T s, target T t and multiplication T m : T (G× P G) → T G. The identity at v p ∈ T p P is 1 vp = T p v p . This defines the tangent prolongation T G⇒T P of G⇒P or the tangent groupoid associated to G⇒P . The cotangent Lie groupoid defined by a Lie groupoid. If G⇒P is a Lie groupoid, then there is also an induced Lie groupoid structure on T * G⇒ A * G = (T P ) • . The source mapŝ : T * G → A * G is given bŷ ) for all u s(g) ∈ A s(g) G, and the target mapt : T * G → A * G is given bŷ . This Lie groupoid structure was introduced in [1] and is explained for instance in [1], [20] and [14].
The "Pontryagin groupoid" of a Lie groupoid. If G⇒P is a Lie groupoid, there is hence an induced Lie groupoid structure on P G = T G × G T * G over T P × P A * G. We will write Tt for the target map

Ts for the source map
Ts : T G × G T * G → T P × P A * G and T , Ti, Tm for the embedding of the units, the inversion map and the multiplication of this Lie groupoid.
3. Multiplicative subbundles of the tangent space T G. We start with the definition of a multiplicative distribution on a Lie groupoid G⇒P . Later, we will mostly be interested in multiplicative subbundles of the tangent bundle T G. Yet, as we will see in Examples 3.2 and 3.3, our main result can already be valid in the more general setting of smooth integrable multiplicative distributions. Since the discussion about the Lie group case in the introduction also starts from (even non necessarily smooth) distributions, we prefer to give the general definition here. Definition 3.1. Let G ⇒ P be a Lie groupoid and T G⇒T P its tangent prolongation. A distribution S ⊆ T G is multiplicative if S is a (set) subgroupoid of T G⇒T P .
Note that this means in particular that T g s(v g ) ∈ S(s(g)) and T g t(v g ) ∈ S(t(g)) for all g ∈ G and v g ∈ S(g).

Examples.
The two examples in this subsection illustrate the general theory in the following sections. We study two completely integrable, multiplicative smooth distributions on special classes of Lie groupoids and show that their spaces of leaves inherit groupoid structures. In the general theory, we will have to assume that the distributions have constant rank, that the elements of a special family of vector fields spanning S are complete, and that a compatibility condition on the leaves and the multiplication is satisfied.
We will see in both examples that the leaf spaces inherit groupoid structures, although the studied distributions are here not necessarily subbundles of the tangent space. This shows that our main theorem can also hold for leaf spaces of singular distributions.
If Since the following diagram commutes,  We identify in the following Consider a completely integrable, smooth distribution F on M and a vector subspace W ⊆ R k . It is easy to see that S : commutes and (pr S , pr F ) is a morphism of groupoids. Here also, we get a Lie groupoid if and only if F is a simple involutive subbundle of T M , i.e., its leaf space is a smooth manifold and the projection is a smooth surjective submersion. In Example 3.3, the groupoid structure on the leaf space does not coincide with the quotient by the kernel N but rather with the quotient by the normal subgroupoid system (N, for all x ∈ R k and (m, n) ∈ R(pr F ). These two quotients are explained in more details in [8]. This illustrates hence the different definitions of normal subgroupoids objects in [12] and [14], and the fact that the notion of normal subgroupoid systems of groupoids is needed to generalize to groupoids the relation between the kernel of surjective homomorphisms and normal subgroups of a group (see [14] for more details).

3.2.
Properties of multiplicative subbundles of T G. In the following, the subbundle S ⊆ T G is always assumed to be smooth, even if this is not stated explicitely.
Lemma 3.5. Let G⇒P be a Lie groupoid and S ⊆ T G a multiplicative subbundle. Then the intersection S ∩ T P has constant rank on P . Since this is the set of units of S viewed as a subgroupoid of T G, the pair S⇒(S ∩ T P ) is a Lie groupoid.
The bundle S| P splitts as In the same manner, S s (g) = S s (t(g)) 0 g for all g ∈ G.
As a consequence, the intersections S ∩ T t G and S ∩ T s G have constant rank on G.
Proof. We start by showing that the intersection S| P ∩ T P is smooth. If p ∈ P and v p ∈ S(p) ∩ T p P , then we find a smooth section X of S defined at p such that X(p) = v p . The restriction of X to Dom(X) ∩ P is then a smooth section of S| P , and, since s is a smooth surjective submersion, and S is a subgroupoid of T G, the image T s(X| S ) is a smooth section of S| P ∩ T P . Futhermore, we have Since the intersection S ∩ T P is a smooth intersection of vector bundles over P , we know (for instance by Proposition 4.4 in [10]) that it is a vector bundle on P . In particular, it is the set of units of S viewed as a subgroupoid of T G.
Since S is a vector bundle on G, it has constant rank on G and in particular on P . For each p ∈ P , we can write From this follows the fact that S| P ∩ T t P G and S| P ∩ T s We show the other inclusion and the equality S s (g) = T t(g) R g (S s (t(g))) in the same manner.
Since S| P ∩ AG and S| P ∩ T s P G have constant rank on P , we get from this that S ∩ T t G and S ∩ T s G have constant rank on G.
Corollary 3.6. Let G⇒P be a Lie groupoid and S a multiplicative subbundle of T G. The induced maps T g s : Proof. The map T s : S/(S ∩T s G) → S ∩T P is a well-defined injective vector bundle homomorphism over s : G → P . Since for any g ∈ G, both vector bundles have the same rank, and the map is an isomorphism in every fiber. Thus, the claim follows.
The following corollary (see also a result in [13] about star-sections) will be used often in the following.
Corollary 3.7. Let G⇒P be a Lie groupoid and S ⊆ T G a multiplicative subbundle. LetX be a section of S ∩ T P defined on Dom(X) =:Ū ⊆ P . Then there exist We say that a vector field X ∈ X(G) is t-(respectively s-) descending if there existsX ∈ X(P ) such that X ∼ tX (respectively X ∼ sX ), that is, for all g ∈ Dom(X), we have T g tX(g) =X(t(g)).
Corollary 3.8. Let G⇒P be a Lie groupoid and S ⊆ T G a multiplicative subbundle. Then 1. S is spanned by local t-descending sections and 2. S is spanned by local s-descending sections.
Proof. Choose g ∈ G and smooth sectionsX 1 , . . . , Let M be a smooth manifold and F ⊆ T M a subbundle spanned by a family F of vector fields. If F is involutive, it is integrable in the sense of Frobenius and each of its leaves is an accessible set of F, i.e., the leaf L m of F through m ∈ M is the set (see [17], [21], [23], [22]). If the multiplicative subbundle S ⊆ T G is involutive, we get the following corollary which will be very useful in the next section.
Corollary 3.9. Let G⇒P be a Lie groupoid and S ⊆ T G an involutive multiplicative subbundle. Then S is completely integrable in the sense of Frobenius and its leaves are the accessible sets of each of the two following families of vector fields.
By Corollary 3.7, there exists for each sectionX of T P ∩ S defined on U ⊆ P a section of S that is defined on t −1 (U ) and t-related toX. We can find a family of spanning sections of S ∩ T P that are all complete, but the corresponding sections of S are then not necessarily complete. For the proof of our main theorem, we will have to assume that S ∩ T P is spanned by the following families of vector fields: ∃X ∈ Γ(S) such that X ∼ tX and X,X are complete, ∃X ∈ Γ(S) such that X ∼ sX and X,X are complete, Note that there exists a complete familyF t S if and only if there exists a complete familyF s S , since the vector fields in F t S are the inverses of the vector fields in F s S and vice versa. We say that S is complete if it has this property. For instance, if G⇒P is such that the target map t (and equivalently the source map s) is proper, then it is easy to see that any multiplicative subbundle S ⊆ T G is complete. Also, note that the multiplicative distributions in Examples 3.2 and 3.3 are complete.
We study now the left invariant vector fields that leave the multiplicative subbundle S invariant. The following lemma is a result that is shown in [6].
Lemma 3.10. Let G⇒P be a Lie groupoid and S ⊆ T G an involutive multiplicative subbundle. Then the Bott connection, i.e., the partial S-connection on T G/S, induces a flat partial S ∩ T P -connection ∇ on AG/(S ∩ AG) → P with the following property. If a ∈ Γ(AG) is such that its class in Γ (AG/(S ∩ AG)) is ∇-parallel, then the left invariant vector field a l satisfies a l , Γ(S) ⊆ Γ(S).
Note that (5) is automatically satisfied for any a ∈ Γ(AG∩S) since S is involutive and a l is a section of S ∩ T t G (see Lemma 3.5). We can hence say that a section a ∈ Γ(AG) is ∇-parallel if its classā in Γ(AG/(AG ∩ S)) is ∇-parallel.
Note also that (5) implies that the flow of a l leaves the subbundle S ⊆ T G invariant. This fact is well-known, see for instance the appendix of [6] for a proof of it.
Corollary 3.11. Let G⇒P be a Lie groupoid and S ⊆ T G an involutive multiplicative subbundle. Then AG is spanned by the following family of (local) sections Proof. We show that there exists for each point p ∈ P a local frame of ∇-parallel sections for AG defined on a neighborhood of p, where ∇ is the flat partial S ∩ T Pconnection as in the preceding lemma. Let m := dim(P ) and r := rank(AG). Choose a foliated chart domain U centered at p and described by coordinates (x 1 , . . . , x m ) such that the first k among them define the local integral subman- We denote by l the rank of AG/(AG ∩ S). Choose a 1 , . . . , a l ∈ Γ(AG) such thatā 1 , . . . ,ā l ∈ Γ(AG/(AG ∩ S)) is a basis frame for AG/(AG ∩ S) on U . We consider this frame at points of Σ ∩ U and constructᾱ 1 , . . . ,ᾱ l ∈ Γ(AG/(AG ∩ S)) as follows. If q ∈ U , φ(q) = (x 1 , . . . , x m ), then we find a path c : ) is the parallel translate ofā i (q ) along c at time 1 by the T P ∩ S-partial connection ∇. Since U is simply connected and ∇ is flat, parallel translation is independent of the chosen path (see, for example, [4]), hence the sectionsᾱ i are ∇-parallel sections of AG/(AG ∩ S) and form a pointwise basis of AG/(AG ∩ S) on U . Choose representatives α 1 , . . . , α l ∈ Γ(AG) forᾱ 1 , . . . ,ᾱ l . Take α l+1 , . . . , α r to be a frame of AG ∩ S over U . Then α 1 , . . . , α r is a frame of AG over U composed of ∇-parallel sections.
3.3. The leaf space of an involutive multiplicative subbundle of T G. Let G⇒P be a Lie groupoid and S ⊆ T G an involutive multiplicative subbundle. Then S is completely integrable in the sense of Frobenius. Let pr : G → G/S be the projection to the space of leaves of S.
It is easy to check that the intersection S ∩ T P is an involutive subbundle of T P and hence itself also completely integrable. Let P/S be the space of leaves of S ∩ T P in P and pr • the projection pr • : P → P/S. Note that by a theorem in [7], S is involutive if and only if S ∩ T P is involutive and S ∩ AG is a subalgebroid of AG. Let P/S be the space of leaves of S ∩ T P in P and pr • the projection pr • : P → P/S. For g, h ∈ G, we will write g ∼ S h if g and h lie in the same leaf of S and [g] := {h ∈ G | h ∼ S g} for the leaf of S through g ∈ G. By the following proposition, we can use the same notation for the equivalence relation defined by the foliation by S ∩ T P on P . We will write [p] • for the leaf of S ∩ T P through p ∈ P .
Note that g and h ∈ G lie in the same leaf of S if they can be joined by finitely many flow curves of vector fields lying in F t S or F s S (see Corollary 3.9). For simplicity, if g ∼ S h, we will often assume without loss of generality that g and h can be joined by one such integral curve.

M. JOTZ
We will see that if S is complete with some additional properties, then the space G/S inherits a groupoid structure over P/S. If the foliations are simple, we will get a Lie groupoid G/S⇒P/S. We start by showing that the structure maps , s, t, i always induce maps on G/S. Proof.
1. This is a standard fact about foliations that are compatible with submanifolds. 2. If g and h ∈ G are in the same leaf of S, we find by Corollary 3.9 smooth s-descending vector fields X 1 , . . . , X k ∈ Γ(S) and t 1 , . . . , t k ∈ R such that where φ i is the flow of the vector field X i for each i = 1, . . . , k. There exist then smooth vector fieldsX 1 , . . . ,X k ∈ Γ(S ∩ T P ) such that X i ∼ sXi , and hence, ifφ i is the flow of the vector fieldX i , s • φ i =φ i • s for i = 1, . . . , k. We compute then which shows that s(h) and s(g) lie in the same leaf of S ∩ T P . The map [s] : G/S → P/S, [g] → [s(g)] • is consequently well-defined. We show in the same manner, but using this time the family (1), that [t] : G/S → P/S, [g] → [t(g)] • is well-defined (note that we don't use here the completeness of F t S ). 3. If g ∼ S h, then there exists without loss of generality one smooth section X ∈ Γ(S) and σ ∈ R such that g Hence, we have shown that the structure maps , s, t and i project to well-defined maps on P/S and G/S and the following diagrams commute. / / G/S On the existence of an induced multiplication. Assume that the leaf space G/S has a groupoid structure over P/S such that the projection (pr, pr • ) is a groupoid morphism. Then: Thus, the multiplication on G/S ⇒ P/S has to be defined as follows.
(6) For this to be well-defined, we will have to assume that the following conditions are satisfied.
For all g ∈ G, p ∈ P such that p ∼ S s(g), there exists h ∈ G such that h ∼ S g and s(h) = p.
For all g ∈ G and p := t(g) : In Example 3.19 (taken from [27, Example 6]), we have a Lie groupoid with a multiplicative foliation such that (8) is not satisfied, and the leaf space does not inherit a multiplication.
Condition (8) is satisfied for instance if all the intersections [g] ∩ t −1 (t(g)) have one connected component and are hence the leaves of the involutive vector bundle T t G ∩ S. This is the case if the Lie groupoid is a Lie group, and also in Examples 3.2 and 3.3.
We will show later that, under some (rather strong) regularity conditions on the family A S , (7) and (8) hold (see Proposition 3.18).
In the next lemma, we prove that (7) is always satisfied if S is complete.
Lemma 3.13. Let G⇒P be a Lie groupoid and S ⊆ T G be a complete multiplicative involutive subbundle of T G. If g ∈ G and t(g) ∼ S p ∈ P , then there exists h ∈ G such that g ∼ S h and t(h) = p. In the same manner, if s(g) ∼ S p ∈ P , then there exists h ∈ G such that g ∼ S h and s(h) = p. That is, condition (7) is satisfied.
Proof. Choose a vector fieldX ∈F t S and σ ∈ R such that p = φX σ (t(g)). We find then a t-descending vector field X ∼ tX defined at g. SinceX and X can be taken complete by hypothesis, the integral curve of X starting at g is defined at σ and we have t(φ X τ (g)) = φX τ (t(g)) for all τ ∈ [0, σ]. Set h := φ X σ (g), then h ∼ S g and t(h) = φX σ (t(g)) = p.
We can now formulate our main theorem and complete its proof. Remark 3.15. Note that in general, the induced groupoid G/S⇒P/S is not a Lie groupoid. As in the easier Lie group case, we get a Lie groupoid if the involutive subbundles S ⊆ T G and S ∩ T P ⊆ T P are simple.
Note that if a Lie group G with Lie algebra g is simply connected, then any ideal s ⊆ g integrates automatically to a closed, normal subgroup ( [3]). Thus, the space of leaves of a multiplicative subbundle S := s L = s R ⊆ T G is always a Lie group. In the general case of a t-simply connected Lie groupoid, it is easy to see that the leaf space of a multiplicative foliation is not necessarily a manifold. Take for instance any manifold as a groupoid over itself (and hence t-simply connected) and any non-simple foliation (that is automatically multiplicative) on the manifold.
Proof of Theorem 3.14. Assume that G/S ⇒ P/S is a groupoid such that (pr, pr • ) is a groupoid morphism and choose g ∈ G. If h ∈ [g] ∩ s −1 (s(g)), then we have The converse inclusion in (8) can be checked in the same manner.
Conversely, assume that (8) holds. We show that the multiplication (6) is welldefined, i.e., that is doesn't depend on the choice of the representatives g, h. Using Theorem 3.12, the groupoid axioms are then easily verified to hold and the projection (pr, pr • ) is a groupoid morphism by definition of the structure maps of G/S ⇒ P/S.

Choose [g] and [h] ∈ G/S such that [s]([g]) = [t]([h]). Then we have [s(g)] = [t(h)] by definition of [s] and [t]
and, by Lemma 3.13, there exists g ∈ [g] such that s(g ) = t(h). For simplicity, assume that the chosen representative g already satisfies this condition.
Assume that g, g are two such choices, i.e., g ∈ [g] and s(g ) = t(h) = s(g). By (8), we have and hence g h ∼ S g h. That is, (6) doesn't depend on the choice of the representative g of [g]. By symmetry, (6) doesn't depend on the choice of h. Example 3.16. Consider a t-connected Lie groupoid G⇒P and let a connected Lie group H act freely and properly on G⇒P by Lie groupoid morphisms. Let Φ : H × G → G be the action. That is, for all h ∈ H, the map Φ h : G → G is a groupoid morphism over the map φ h := Φ h | P : P → P . Let V ⊆ T G be the vertical space of the action, i.e., V(g) = {ξ G (g) | ξ ∈ h} for all g ∈ G, where h is the Lie algebra of H.
We check that V ⊆ T G is multiplicative. Choose ξ G (g) ∈ V(g). Then we have and in the same manner T g s (ξ G (g)) ∈ V(s(g)) ∩ T s(g) P.
This shows that V ∩ T s G = V ∩ T t G = {0} in this example.
If ξ G (g) ∈ V(g) and η G (g ) ∈ V(g ) are such that then we have s(g) = t(g ) =: p and ξ P (p) = η P (p), which implies ξ = η since the action is free. We get then The inverse of ξ G (g) is then ξ G (g −1 ) for all ξ ∈ h and g ∈ G.
Choose g ∈ G. Then we have and hence x = e because the action of H on P is assumed to be free. As a consequence, Condition (8) is satisfied in a trivial manner. It follows also from the considerations above that the vector fields ξ G are tand s-descending to ξ P . Hence, V is spanned by complete vector fields, that are t-related to complete vector fields.
Hence, we recover from Theorem 3.14 the fact that the quotient G/H⇒P/H has the structure of a Lie groupoid such that the projections pr : G → G/H, pr • : P → P/H form a Lie groupoid morphism.
Remark 3.17. In discrete mechanics, the velocity space T Q of a configuration space Q is replaced by Q × Q. The lift of a Lie group action by a Lie group H on the configuration space is then naturally replaced by the diagonal action of H on Q × Q. It is observed by [25] that the Lagrangian formalism for discrete mechanics can be generalized using groupoids. This idea has been developped by [16] and [5]. In this setting, Lie group actions preserving the geometric structure should be Lie group actions by Lie groupoid morphisms as in the previous example. This suggests that Theorem 3.14 could be very useful for reduction by distributions in the context of discrete mechanics and discrete mechanics on Lie groupoids.
We show now that under additional conditions on the family A S , (7) and (8) are satisfied if the Lie groupoid is t-connected. We will assume that: • The elements of A S are globally defined.
• The images of the elements of A S under the anchor are complete. This implies that the corresponding left invariant vector fields on G are also complete (see [14,Theorem 3.6.4]).
Proposition 3.18. Let G⇒P be a t-connected Lie groupoid and S ⊆ T G an involutive, multiplicative subbundle. Assume that A S is a family of global sections of AG, with complete images under the anchor. Then (7) and (8) are satisfied and there is hence an induced groupoid structure on the leaf space G/S⇒P/S, such that (pr, pr • ) is a groupoid morphism.
Proof. Since AG is spanned by A S and G⇒P is t-connected, each element h of G can be written with n ∈ N, a 1 , . . . , a n ∈ A S and t 1 , . . . , t n ∈ R. Without loss of generality, we can assume in the following that n = 1 and we set a 1 =: a, t 1 =: t. That is, h = Exp(ta)(t(h)). By hypothesis, for all t ∈ R, the bisection Exp(ta) is globally defined on P . Equation (5) implies that the flow of a l , which is equal to R Exp(·a) , leaves S invariant. As a consequence, if g ∼ S g, then R Exp(τ a) (g ) ∼ S R Exp(τ a) (g) for all τ .
We end this paragraph with an example of a multiplicative foliation which leaf space is not a groupoid. Example 3.19 ([27]). Let P be the trivial circle bundle over the open 2-disk D, but with one point removed in the fiber over 0 ∈ D. We write P =Ŝ 1 → D , wherê S 1 p denotes the circle for all non-zero p ∈ D, whileŜ 1 0 is the circle with a point removed. Note that π 1 (P ) = Z, generated by any of the circle fibers. It is easy to see that the universal cover of P isP = (D × R) \ ({0} × Z). We writeP =R → D as a bundle over D, whereR p = R for non-zero p ∈ D andR 0 = R \ Z.
Consider the Lie groupoid G :=R× ZR ⇒ P , where the action of Z = π 1 (P ) is the diagonal action by deck transformations. Let π be the projectionR ×R →R × ZR , (p 1 , x 1 , p 2 , x 2 ) coordinates onR ×R and (p, x) coordinates on P . The subbundle of T (R×R) spanned by ∂ x1 and ∂ x2 is easily seen to project under π to a multiplicative subbundle S of T G. The intersection S P := S ∩ T P is the span of ∂ x and we find P/S P D.
The leaves of S almost coincide with the fibers of the natural projection onto D×D: for (p 1 , p 2 ) ∈ D×D, the leaf L [p1,x1,p2,x2] of S through g = [p 1 , x 1 , p 2 , x 2 ] ∈ G is: Thereby, for x ∈ R, [x] is the biggest integer smaller than x. In other words, the leaf through [p 1 , x 1 , p 2 , x 2 ] is a cylinder if p 1 = 0 and p 2 = 0, a rectangle if either p 1 or p 2 vanishes, and, over (0, 0) ∈ D × D, we have the quotient of (R \ Z) × (R \ Z) by the diagonal Z-action, which consists of countably many leaves. Hence the leaf space is D × D, where the latter denotes the non-Hausdorff manifold obtained from D × D replacing (0, 0) with a copy of Z.
Is it easy to see that, as shown in Theorem 3.12, there are well-defined "source" and "target" maps D × D → D such that commutes, but the multiplication does not descend to the quotient. Indeed, consider (0, p) ∈ D × D where p is non-zero. A preimage under pr is [(0, µ 1 ), (p, x)] where µ 1 ∈ R \ Z and x ∈ R are arbitrary. Similarly, we consider (p, 0) ∈ D × D and as a preimage we pick [(p, x), (0, µ 2 )] where again µ 2 ∈ R \ Z is arbitrary. Now multiplying these two elements of G we obtain [(0, µ 1 ), (0, µ 2 )]. The value of its projection under pr depends on the concrete choice of µ 1 and µ 2 . This shows that D × D does not have an induced groupoid structure. In the same manner, one can see that (8) is not satisfied.
The Lie algebroid of G⇒P is the projection under T π of the restriction to ∆R of the subbundle {0} × TR of T (R ×R) = TR × TR. The vector fields on the second copy of D and ∂ x2 are π-related in this manner to ∇-parallel sections of AG. It is easy to see using points as above that the conditions for Proposition 3.18 are not satisfied here.
Tangent and cotangent spaces of the quotient. We end this section with the following lemma, which will be useful for the proof of Theorem 4.4. The proof is a straightforward computation that is left to the reader. Lemma 3.20. In the setting of Theorem 3.14, if G/S⇒P/S is a Lie groupoid and (pr, pr • ) a pair of smooth surjective submersions, choose v [g] ∈ T [g] (G/S) and v [h] ∈ T [h] (G/S) such that T [g] [s]v [g] [h] . Then we can assume without loss of generality that s(g) = t(h). If v g ∈ T g G and v h ∈ T h G are such that T g pr v g = v [g] and henceŝ (T g pr) * α [g] =t (T h pr) * α [h] and we have 4. An application. We use Theorem 3.14 to study how a theorem in [18] (see also [9]) about the shape of Dirac Lie groups generalizes to Dirac groupoids. The Pontryagin bundle P M = T M × M T * M of a smooth manifold M is endowed with the non-degenerate symmetric fiberwise bilinear form of signature (dim M, dim M ) given by for all m ∈ M , v m , w m ∈ T m M and α m , β m ∈ T * m M . An almost Dirac structure (see [2]) on M is a Lagrangian vector subbundle D ⊂ P M . That is, D coincides with its orthogonal relative to (9) and so its fibers are necessarily dim M -dimensional. The distribution G 0 = ∪ m∈M G 0 (m) is not necessarily smooth.
The almost Dirac structure D is a Dirac structure if its set of sections is closed under the Courant-Dorfman bracket, i.e., for all (X, α), (Y, β) ∈ Γ(D). The class of Dirac structures generalizes the one of Poisson manifolds in the sense of the following example. Let G⇒P be a Lie groupoid. Recall that the Pontryagin bundle of G has then a Lie groupoid structure over T P × P A * G.  19]). An (almost) Dirac groupoid is a Lie groupoid G ⇒ P endowed with an (almost) Dirac structure D G such that D G ⊆ T G × G T * G is a Lie subgroupoid. The (almost) Dirac structure is then said to be multiplicative.
If G is a Lie group, i.e., with P = {e}, an (almost) Dirac structure D G on G is multiplicative if the multiplication m : (G × G, D G ⊕ D G ) → (G, D G ) is a forward Dirac map. It is shown in [18], [9], that the induced distribution G 0 on G is multiplicative, hence left and right invariant and equal to G 0 = g L 0 = g R 0 for some ideal g 0 ⊆ g in the Lie algebra g of G. Thus, G 0 is automatically an involutive subbundle of T G for any multiplicative almost Dirac structure on the Lie group G. The leaf N of G 0 through e is then a normal subgroup of G and if it is closed, the quotient G/G 0 = G/N is a Lie group such that the projection q : G → G/N is a smooth surjective submersion. If (G, D G ) is a Dirac Lie group and N ⊆ G is a closed subgroup, the quotient G/N inherits a Poisson structure π such that q : (G, D G ) → (G/N, π) is a forward Dirac map and (G/N, π) is a Poisson Lie group. Furthermore, (G/N, π) is a Poisson homogeneous space of the Dirac Lie group (G, D G ) ( [9]).
The fact that the characteristic distribution of an almost Dirac Lie group is always multiplicative is a special case of the fact that the characteristic distribution of an almost Dirac groupoid is always multiplicative, as shows the next proposition. Proof. Choose a composable pair (g, h and hence Tt(v g , 0 g ) ∈ D G (t(g)), Ts(v g , 0 g ) ∈ D G (s(g)), (v g , 0 g ) −1 ∈ D G (g −1 ) and (v g , 0 g ) (v h , 0 h ) ∈ D G (g h). Since (v g , 0 g ) −1 = (v −1 g , 0 g −1 ) and (v g , 0 g ) (v h , 0 h ) = (v g v h , 0 g h ), this shows that T g s(v g ) ∈ G 0 (s(g)), T g t(v g ) ∈ G 0 (t(g)), v −1 g ∈ G 0 (g −1 ) and v g v h ∈ G 0 (g h).
Yet, the situation in the general case of an (almost) Dirac groupoid is more involved than in the Lie groups case. First of all, the induced distribution G 0 has not automatically constant rank on G anymore. If it is smooth, it is in general not completely integrable in the sense of Stefan and Sussmann, unless D G is a Dirac structure. For instance, each manifold can be seen as a (trivial) groupoid over itself (i.e., with t = s = Id M ) and any (almost) Dirac manifold can thus be seen as an (almost) Dirac groupoid, which will, in general not satisfy these conditions. Thus, trivial Dirac groupoids yield already many examples of Dirac groupoids for which G 0 is not an involutive subbundle of T G. (The class of Dirac groupoids described in the next theorem seems hence to be a small class of examples.) If G 0 associated to a Dirac groupoid (G⇒P, D G ) is assumed to be a vector bundle on G, then we are in the same situation as in the group case.
We know by Theorem 3.14 that we have to assume that G 0 is complete and satisfies (8) to ensure that the leaf space G/G 0 inherits a multiplication map and hence to show that there exists a Poisson groupoid structure on the quotient G/G 0 .
The next theorem shows that, under these conditions, we obtain the same result as the one that is already known in the group case.
Theorem 4.4. Let (G⇒P, D G ) be a Dirac groupoid. Assume that G 0 is a subbundle of T G, that it is complete and that (8) holds. If the leaf spaces G/G 0 and P/G 0 have smooth manifold structures such that the projections are submersions, then there is an induced multiplicative Poisson structure on the Lie groupoid G/G 0 ⇒ P/G 0 , such that the projection pr : G → G/G 0 is a forward Dirac map.
Proof. Recall that since D G is assumed to be a Dirac structure, the subbundle G 0 ⊆ T G is involutive. Since G 0 is multiplicative by Proposition 4.3 and all the hypotheses for Theorem 3.14 are hence satisfied, we get that G/G 0 ⇒P/G 0 has the structure of a Lie groupoid such that if pr : G → G/G 0 and pr • : P → P/G 0 are the projections, then (pr, pr • ) is a Lie groupoid morphism.
We have D G ∩(T G× G G 0 • ) = D G ∩(T G× G P 1 ) = D G and [Γ(G 0 × G {0}), Γ(D G )] ⊆ Γ(D G ) because D G is Dirac. Hence, by a result in [26] (see also [11]), we find that the Dirac structure pushes-forward to the quotient G/G 0 . The (almost) Dirac structure pr(D G ) is given by [g] , α [g] ) ∈ P G/G0 ([g]) ∃v g ∈ T g G such that (v g , (T g pr) * α [g] ) ∈ D G (g) and T g pr v g = v [g]    for all g ∈ G. The fact that pr(D G ) is a Dirac structure follows D G being a Dirac structure. If (v [g] , 0) ∈ pr(D G )( [g]), there exists v g ∈ T g G such that T g pr v g = v [g] and (v g , 0) ∈ D G (g). But then we get v g ∈ G 0 (g) and hence v [g] = T g pr v g = 0 [g] . This shows that the characteristic distribution G 0 associated to the Dirac structure pr(D G ) is trivial, and since it is integrable, pr(D G ) is the graph of the vector bundle homomorphism T * (G/G 0 ) → T (G/G 0 ) associated to a Poisson bivector on G/G 0 . that [t] : (G/G 0 , pr(D G )) → (P/G 0 , {· , ·} P/G0 ) is a forward Dirac map. It is easy to check that the map pr • : (P, D P ) → (P/G 0 , {· , ·} P/G0 ) is then also a forward Dirac map, i.e., D {· ,·} P /G 0 is the forward Dirac image of D P under pr • .
Remark 4.7. In the Lie group case, the Poisson Lie group (G/N, q(D G )) associated to a Dirac Lie group (G, D G ) satisfying the necessary regularity assumptions was also a Poisson homogeneous space of the Dirac Lie group. In general, the Poisson groupoid associated to the Dirac groupoid is not a Poisson homogeneous space of the Dirac groupoid since the quotient G/G 0 is not a homogeneous space of the Lie groupoid G⇒P .