Symmetries and reduction of multiplicative 2-forms

This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu \cite{FOR} on Poisson actions.


Introduction
This paper studies symmetries of closed multiplicative 2-forms on Lie groupoids as well as their infinitesimal counterparts. This study leads to an extension of some of the recent work of Fernandes-Ortega-Ratiu [11] on Poisson actions and their Hamiltonian lifts to symplectic groupoids to the framework of Dirac structures.
Consider a Lie group G acting on a Poisson manifold M by Poisson diffeomorphisms, and suppose that G is the source-simply-connected symplectic groupoid integrating M (see e.g. [7]). A key observation that can be traced back to [17] is that, even when the G-action on M does not admit a momentum map, it can always be lifted to a Hamiltonian G-action on G with a momentum map J : G → g * suitably compatible with the groupoid structure (in the sense of (3.6) below). 1 In [11], Fernandes, Ortega and Ratiu use this canonical momentum map on G as a tool for studying actions on M by Poisson diffeomorphisms. For example, when the action is free and proper, so that the quotient M/G is again a Poisson manifold, they show that the Marsden-Weinstein quotient J −1 (0)/G is a symplectic groupoid for M/G (though not necessarily the source-simply-connected one). The simplest instance of this result is when M is an arbitrary manifold (equipped with the zero Poisson structure), in which case one just recovers the well-known fact that any G-action on M , when lifted to T * M , is Hamiltonian with momentum map where u ∈ g and u M is the infinitesimal generator for the G-action on M . The reduction of T * M at level zero is T * (M/G), which is just the symplectic groupoid of M/G. Our goal in this note is to place some of these results from [11] in the broader context of Dirac structures [8], offering an alternative viewpoint based on methods from [2,3] and pointing out additional subtleties that arise in this generality.
Our starting point is [5], where the global objects associated with Dirac manifolds are identified; these objects are referred to as presymplectic groupoids, and their role in the theory of Dirac structures is entirely analogous to that of symplectic groupoids in Poisson geometry. A presymplectic groupoid is a Lie groupoid equipped with a closed multiplicative 2-form but, in contrast with symplectic groupoids, this 2-form may be degenerate (though in a "controlled" way). Let (G, ω) be a source-simplyconnected presymplectic groupoid integrating a Dirac structure L on M , and suppose that a Lie group G acts on M by Dirac diffeomorphisms. We observe that, just as in the case of Poisson actions, the action on M lifts to a Hamiltonian action on G, in the sense that there is a G-equivariant map J : G → g * satisfying (1.2) i u G ω = −d J, u , ∀u ∈ g, where u G is the infinitesimal generator associated with u for the G-action on G. Note, however, that condition (1.2) does not necessarily determine the infinitesimal action since ω may be degenerate. When the G-action on M is free and proper, the quotient M/G is a smooth manifold that inherits (with the aid of extra regularity conditions to be specified in Section 6) natural geometrical structures: on the one hand, a Marsden-Weinstein type reduction of G produces a Lie groupoid G red over M/G equipped with a closed multiplicative 2-form; on the other hand, the G-invariant Dirac structure L on M may be pushed forward to a Dirac structure L quot on M/G. Unlike the case of Poisson manifolds, however, G red is generally not a presymplectic groupoid for (M/G, L red ) (in the terminology of [5,Sec. 4], G red will be proven to be an over -presymplectic groupoid). As we will discuss, it may happen that the quotient (M/G, L quot ) does not admit any presymplectic groupoid at all; i.e., L quot may not be integrable as a Lie algebroid, despite the fact that L was. Our approach to Dirac structures and presymplectic groupoids is through the notion of IM 2-forms [5] (see also [1,2,3]), differently from [11] that uses path spaces [9] (see also [6,19]). At the end of the paper, we discuss how to reconcile both viewpoints.
The increasing role of Dirac structures in mechanical problems has been one of our motivations to revisit the work in [11] from this broader perspective. Even in the context of Poisson geometry, Dirac structures naturally appear in the stratified geometry of non-free Poisson actions, see Sections 2.3 and 4.3 of [11]. A more complete picture, that we leave for future work, should include proper actions on Dirac manifolds which are not necessarily free; Jotz, Ratiu and Sniatycki have begun this study in [12,13]. Another source of inspiration for the present note is the possibility of using Dirac structures as a tool for studying symplectic groupoids of Poisson homogeneous spaces [14], yet to be explored.
The paper is structured as follows: in Section 2, we recall IM 2-forms on Lie algebroids and present the geometric set-up that will be considered in the paper; Section 3 discusses the Hamiltonian properties of actions on Lie groupoids that lift symmetries of closed IM 2-forms; Section 4 describes the reduction of closed IM 2-forms, while Section 5 deals with its global version, i.e., reduction of closed multiplicative 2-forms; the particular situation of Dirac structures is treated in Section 6; in Section 7, we relate the approach in this paper to the viewpoint of path spaces in [11].
It is a great pleasure to dedicate this note to Tudor, whose work has been central in so many aspects of Poisson geometry and geometric mechanics, including many of the issues discussed here. We are thankful to him for his mathematical insights, unmatched enthusiasm, and constant support.
Let G be a Lie group with Lie algebra g. In this paper, an action of G on E is always assumed to be by vector-bundle automorphisms, i.e., defined by a group homomorphism G → Aut(E); any G-action on E naturally covers, and it is said to be a lift of, an action of G on M .
The space of infinitesimal automorphisms of E, denoted by Der(E), is the set of pairs (X, D), where X ∈ X(M ) is a vector field on M and D : Γ(E) → Γ(E) is a linear endomorphism such that For a one-parameter family of automorphisms Φ t in Aut(E), Φ 0 = Id, the corresponding infinitesimal automorphism is given by D(s) = d dt t=0 (Φ t ) * (s). The space Der(E) has a natural Lie algebra structure, given by commutators, for which the projection Der(E) → X(M ), (X, D) → X, is a Lie algebra homomorphism. Any G-action on E gives rise to a Lie algebra homomorphism g → Der(E) for which the composition g → Der(E) → X(M ) agrees with the infinitesimal counterpart of the G-action on M covered by the G-action on E.
Suppose G acts on M freely and properly; let p : M → M/G be the natural projection, which is a surjective submersion. Any lift of this action to a G-action on E determines a unique vector bundle (up to isomorphism) over M/G, denoted by such that the pull-back bundle p * (E/G) is naturally identified with E, and the pullback of sections identifies Γ(E/G) with the space of G-invariant sections Γ(E) G .

2.2.
The geometric set-up. This paper will be concerned with the following geometric objects: (i) A smooth manifold M acted upon by a Lie group G. The notion of closed IM 2-form [5] in (ii) refers to a morphism of vector bundles µ : A → T * M , covering the identity map, satisfying In (iii), we consider the group of automorphisms of the Lie algebroid A, i.e., the subgroup of vector-bundle automorphisms Aut(A) defined by One can directly verify that any Φ ∈ Sym(A), covering ϕ ∈ Diff(M ), satisfies A central theme in this paper is transferring the geometric information in (i), (ii), (iii) to a Lie groupoid G integrating A. Our notation and conventions for a Lie groupoid G over M are as follows: the source and target maps are denoted by s and t, the set G (2) ⊂ G × G of composable pairs (g, h) is defined by the condition s(g) = t(h), and the multiplication is denoted by m : We denote by pr T : TM → T M and pr T * : Any Poisson structure π ∈ Γ(∧ 2 T M ) may be viewed as a Dirac structure via its graph, i.e., L = {(π ♯ (α), α) | α ∈ T * M }, where π ♯ : T * M → T M is given by π ♯ (α) = i α π; closed 2-forms may be viewed as Dirac structures analogously, as graphs of the associated bundle maps T M → T * M .
For a Dirac structure L on M , the vector bundle L → M inherits a Lie algebroid structure, with bracket on Γ(L) given by the restriction of [[·, ·]] and anchor given by restriction of pr T to L: Moreover, the map preserves L; equivalently, the following holds for all x ∈ M : In particular, ϕ is both a backward and forward Dirac map (see e.g. [5, Sec. 2.1] and references therein).
The situation that will concern us is that of a Lie group G acting, freely and properly, on a manifold M by diffeomorphisms preserving a Dirac structure L ⊂ TM . This fits into the framework of (i), (ii), (iii) if we set A = L, µ = µ L as in (2.15), and the G-action on A to be the natural lift of the G-action on M via (2.16) restricted to L. When M is a Poisson manifold, the Lie algebroid in question is naturally identified with T * M , in such a way that µ L is just the identity map; in this case, the information in (i), (ii), (iii) boils down to an action on M by Poisson diffeomorphisms, as considered in [11].
Although we are primarily interested in Dirac structures, it turns out that more general closed IM 2-forms naturally arise when one considers reduction by symmetries; so we will work in the more general context of (i), (ii), (iii) from the outset.

Momentum maps and lifted Hamiltonian actions
Let us consider the geometric set-up described in (i), (ii), (iii) of Section 2.2. Let J can : T * M → g * be the momentum map of the canonical lifting of the G-action on M to T * M , given by (1.1), and let Proof. Note that J A : A → g * is a morphism of vector bundles, because both µ : The remaining condition to be checked is as an equality in C ∞ (M, g * ). Pairing with v ∈ g, and using (1.1) and (2.5), we get The right-hand-side of the previous equation agrees with Using (2.4), (2.9), as well as the infinitesimal equivariance of µ (2.11), we obtain On the other hand and swapping a and b gives Comparing with (3.4), we see that (3.3) equals which immediately implies that (3.2) holds.
Suppose that A is integrable and G is the source-simply-connected Lie groupoid integrating it. We now make use of the Lie algebroid/groupoid version of Lie's second theorem (see e.g. the appendix of [16]). Since the G-action on A is by Lie-algebroid automorphisms, it has a natural lift to a G-action on G by Lie-groupoid automorphisms. Similarly, since J A : A → g * is a Lie algebroid morphism, it integrates to a Lie groupoid morphism where g * is now viewed as a groupoid over a point (i.e., an abelian group with respect to addition); i.e., J satisfies for (g, h) ∈ G (2) . We also know from [5] (c.f. [1,2,3]) that the closed IM 2-form µ uniquely integrates to a closed 2-form ω in G which is multiplicative, in the sense that (3.7) m * ω = pr * 1 ω + pr * 2 ω, where pr 1 , pr 2 : G × G → G are the natural projections and m : G (2) → G is the multiplication on G. The relationship between ω and µ is given by Note that the notion of being multiplicative, defined by condition (3.7), makes sense for differential forms of any degree; in particular, (3.6) says that J is a multiplicative (g * -valued) function.
Proposition 3.2. The map J : G → g * is G-equivariant and satisfies (3.9) i where v G is the infinitesimal generator defined by v for the G-action on G.
Proof. Let us verify the G-equivariance of J : G → g * . Each σ ∈ G defines an automorphism φ σ of A as well as its global counterpart Φ σ , which is an automorphism of G.
The fact that G acts on G by Lie-groupoid automorphisms implies that the infinitesimal generators of this action are multiplicative vector fields (i.e., each v G : G → T G is a groupoid morphism, see e.g. [15]). It follows that the 1-form i v G ω is multiplicative. Since J is also multiplicative (see (3.6)), so is −d J, v . To show that they are the same, it suffices (see e.g. [1,2]) to check that these 1-forms agree on elements a ∈ A x = ker(ds)| . On the other hand, using that As in [11,Thm. 3.3(ii)], one can check that there exists a map j : M → g * such that J = s * j − t * j if and only if µ(a), u M = −dj u (ρ(a)), for all u ∈ g, a ∈ A, where j u (x) = j(x), u . The map j should be seen as a momentum map for the G-action on M preserving the closed IM 2-form µ : A → T * M .

Reduction of closed IM 2-forms
We keep considering the set-up in (i), (ii), (iii) of Section 2.2, and J A : A → g * as in (3.1). Let us denote by K ⊆ T M the distribution tangent to the G-orbits on M , and by K • ⊆ T * M its annihilator.  Proof. Since the G-action on J −1 A (0) is by Lie-algebroid automorphisms, the space Γ(J −1 A (0)) G is closed under the Lie bracket, and the identification We define an anchor ρ red : A red → T (M/G) by the condition (4.2) ρ red (a) = dp(ρ(p * a)), for a ∈ Γ(A red ); note that this expression is well-defined (by the compatibility between ρ and the G-action, see (2.7)) and C ∞ (M/G)-linear. For a, b ∈ Γ(A red ) and f ∈ C ∞ (M/G), the Leibniz identity follows from where we used the natural identification K • /G ∼ = T * (M/G). It is a direct verification that the conditions (2.4), (2.5) for µ imply the same conditions for µ red .

Global reduction
We now discuss the global reduction associated with the data in (i), (ii), (iii) of Section 2.2. Let us assume that J −1 A (0) ⊆ A has constant rank, so it is a Lie subalgebroid of A, see Lemma 4.1. We assume that A is an integrable Lie algebroid and that G is the source-simply-connected Lie groupoid integrating it; let J : G → g * be the multiplicative function integrating J A , as in Section 3. We say that 0 ∈ g * is a clean value for J if J −1 (0) is a submanifold and ker(dJ)| g = T g J −1 (0) for all g ∈ J −1 (0). Lemma 5.1. If 0 is a clean value for J, then J −1 (0) is a Lie subgroupoid of G over M whose Lie algebroid is J −1 A (0). Proof. Lets,t : J −1 (0) → M be the restrictions of s and t to J −1 (0). In order to verify thats is a surjective submersion (t can be treated analogously), note that Since M ⊆ J −1 (0), as a consequence of (3.6), ands| M = Id, we see that ds| M is onto. It follows from (5.1) and J A = dJ| A that , and a dimension count shows that (5.3) rank(J −1 A (0)) = dim(J −1 (0)) − dim(M ). For g ∈ G, let r g : s −1 (t(g)) → s −1 (s(g)) be the associated right-translation map. Condition (3.6) implies that (dJ) g (dr g ) t(g) a = (dJ) t(g) (a) for a ∈ A t(g) , which, along with (5.1), shows that, for g ∈ J −1 (0), ker(ds) g = ker(ds) g ∩ ker(dJ) g = (dr g ) t(g) A ∩ ker(dJ) g = (dr g ) t(g) (J −1 A (0)). It now follows from a dimension count using (5.3) that rank(ds) g = dim(J −1 (0)) − rank(J −1 A (0)) = dim(M ), sos is a submersion. A direct consequence of (3.6) is that J −1 (0) is closed under the multiplication on G, which endows J −1 (0) with a Lie groupoid structure with source and target maps given bys andt. As a result of (5.2), its Lie algebroid agrees with J −1 A (0). As in Section 4, we henceforth assume that the G-action on M is free and proper, which implies that the lifted G-action on G is also free and proper [11,Prop. 4.4]. If 0 is a clean value for J, so that J −1 (0) is a submanifold (necessarily G-invariant by the equivariance of J), then the quotient G ′ red := J −1 (0)/G is a smooth manifold that naturally inherits a closed 2-form ω ′ red ; indeed, just as in usual Marsden-Weinstein reduction, condition (3.9) guarantees that the pull-back of ω to J −1 (0) is basic with respect to the G-action. Let A red and µ red be as in Prop. 4.3.
Proposition 5.2. Suppose 0 ∈ g * is a clean value for J. Then (1) G ′ red is a Lie groupoid over M/G such that the quotient map J −1 (0) → G ′ red is a groupoid homomorphism, and whose Lie algebroid is A red . (2) The closed 2-form ω ′ red on G ′ red is multiplicative and integrates the closed IM 2-form µ red on A red .
Proof. The fact that G ′ red inherits a groupoid structure, uniquely determined by the property that the quotient map is a homomorphism, can be directly checked using the freeness of the G-action (note that the restriction of this map to identity section M ⊆ J −1 (0) agrees with the quotient map (4.1), hence the abuse of notation). Let A(G ′ red ) be the Lie algebroid of G ′ red . The map p induces a surjective morphism of Lie algebroids , covering the quotient map M → M/G. We observe that ker(dp) ∩ ker(ds) = {0} since X ∈ ker(dp) if and only if X = u G for some u ∈ g, and ds(u G ) = u M = 0 if and only if u = 0 by freeness. It follows that the kernel of (5.4) is trivial, so that (5.4) induces an identification between J −1 A (0) and the pull-back bundle p * A(G ′ red ), from where we obtain a natural identification between A(G ′ red ) and J −1 A (0)/G = A red . For the first assertion in (2), note that the pull-back ι * ω with respect to the inclusion ι : J −1 (0) → G ′ is a closed, multiplicative 2-form on J −1 (0), and one can check that ω ′ red is multiplicative from the equality p * ω ′ red = ι * ω and the fact that p is a groupoid homomorphism.
To compare IM 2-forms, recall that µ red : A red → T * (M/G) is defined as the G-quotient of the restriction (see Prop. 4.3) Consider a ∈ A red and X ∈ T (M/G) at a point p(x) ∈ M/G, and let a ∈ J −1 A (0)| x , X ∈ T x M be such that dp(X) = X and dp(a) = a. Then ω ′ red (a, X) = p * ω red (a, X) = ι * ω(a, X) = µ(a)(X) = µ red (a)(X), which concludes the proof.
When the 2-form ω on G is nondegenerate, then condition (3.9) and the freeness of the G-action on G guarantee that 0 is a regular value for J. In general, however, ω may be degenerate and (3.9) may give no information about the regularity of J −1 (0). There is, nevertheless, an alternative route for constructing a "reduced" Lie groupoid over M/G that always works.
Let G 0 be the source-simply-connected Lie groupoid integrating the Lie algebroid J −1 A (0) (since A is integrable, so is any Lie subalgebroid of A). Since J −1 A (0) is a Lie subalgebroid of A, G 0 comes equipped with a groupoid homomorphism ι 0 : G 0 → G, (which is an immersion but may fail to be injective, see e.g. [18]). The G-action on J −1 A (0) integrates to a G-action on G 0 for which ι 0 is G-equivariant; moreover, since the G-action on M is free and proper, so is the action on G 0 . The map J • ι 0 : G 0 → g * is a multiplicative function whose infinitesimal counterpart d(J •ι 0 )| J −1 As a result, one may use (3.9) to directly verify that ι * 0 ω is a closed multiplicative 2-form on G 0 which is basic with respect to the G-action. Hence G red := G 0 /G is a smooth manifold, and it inherits a closed 2-form ω red from ι * 0 ω. Proposition 5.3. The following holds: (1) G red is a Lie groupoid over M/G for which the quotient map G 0 → G red is a groupoid homomorphism, and whose Lie algebroid is A red . (2) The closed 2-form ω red on G red is multiplicative, and it integrates the closed IM 2-form µ red on A red . The proof of Proposition 5.3 is similar to that of Proposition 5.2, but with no regularity assumptions on J.
As discussed in [11], the reduced groupoids G red or G ′ red generally do not agree with the source-simply-connected integration of A red ; [11,Thm. 4.11] shows how one can approach the problem of finding obstructions.

The case of Dirac structures
Let M be a manifold equipped with a Dirac structure L and a free and proper action of a Lie group G by Dirac diffeomorphisms. As explained in Section 2.3, this fits into the set-up in (i), (ii), (iii) as follows: A is the natural Lie algebroid defined on the vector-bundle L → M by (2.14) and µ = µ L is the closed IM 2-form defined in (2.15). As in Section 4, let K ⊆ T M be the distribution tangent to the orbits of G on M , and let K ⊥ = T M ⊕ K • denote its orthogonal in T M ⊕ T * M relative to the canonical pairing (2.12). Note that . Assuming that L ∩ K ⊥ has constant rank, following Prop. 4.3, one may reduce A and µ to obtain A red and µ red , where (6.2) A red = J −1 A (0)/G = L ∩ K ⊥ /G is a Lie algebroid over M/G and µ red is a closed IM 2-form on A red . On the other hand, one may consider the reduction of L in the sense of Dirac structures, recalled below, which produces a new Dirac structure on M/G. We will now compare these two possible reduction procedures. 6.1. Comparing reductions. The lifted G-action on TM (see (2.16)) preserves the bundles K, K ⊥ , and L, and we have a natural identification Assuming that the intersection L ∩ K ⊥ has constant rank 2 , the quotient defines a Dirac structure on M/G, see [4,Sec. 4]. Equivalently, the Dirac structure (6.4) is the result of a push-forward by the quotient map p : M → M/G: , (X, dp * β) ∈ L x }. Being a Dirac structure, L quot carries a Lie-algebroid structure over M/G and is naturally equipped with the closed IM 2-form (c.f (2.15)) (6.6) µ quot := pr T * | Lquot : L quot → T * (M/G).
A direct comparison of (6.2) and (6.4) indicates that A red and L quot generally disagree. The following simple example illustrates this fact. We assume henceforth that L∩K ⊥ has constant rank, so that both A red and L quot are well-defined. Let us consider the map (6.7) r : µ red (a)).
The following result relates A red , µ red and L quot , µ quot in general.
Lemma 6.2. The image of the map r is L quot and its kernel is (K ∩L)/G. Moreover, r : A red → L quot is a Lie-algebroid morphism and µ quot • r = µ red .
The fact that r : A red → L quot preserves Lie-algebroid structures follows from ρ red preserving Lie brackets as well as conditions (2.4), (2.5) for µ red . The compatibility µ quot • r = µ red is immediate from (6.6).
The next result follows directly from Lemma 6.2.
Theorem 6.3. The map r defines a Lie-algebroid isomorphism from A red to L quot , such that µ quot • r = µ red , if and only if K ∩ L = {0}.
We have the following consequences of Thm. 6.3: (1) Whenever A red and L quot do not coincide, i.e., r fails to be an isomorphism, the Lie groupoid (G red , ω red ) (resp. (G ′ red , ω ′ red ), provided J −1 (0) is a smooth submanifold) constructed in Section 5 is not a presymplectic groupoid for L quot ; by Thm. 6.3, this happens if and only if K ∩ L = {0}. In general, (G red , ω red ) (resp. (G ′ red , ω ′ red )) is an over-presymplectic groupoid in the sense of [5,Def. 4.5]; in particular, the target map t : (G red , ω red ) → (M/G, L quot ) is always a forward Dirac map.
(2) If L is the graph of a Poisson structure π on M , then it immediately follows that K ∩L = {0} and L∩K ⊥ has constant rank. In this case, L quot is just the graph of the natural Poisson structure π quot on M/G, uniquely determined by p : M → M/G being a Poisson map. By Thm. 6.3, A red is identified with L quot via r, and (G red , ω red ) is a symplectic groupoid for π quot ; since 0 is automatically a regular value for J in this case, one may also consider the (possibly different) symplectic groupoid (G ′ red , ω ′ red ). This situation is fully treated and further developed in [11].
(3) Let us suppose that the Dirac structure L on M is such that L quot is given by the graph of a Poisson structure π quot . In this case, we note that r defines an isomorphism As a consequence, the reduced groupoid (G red , ω red ) (resp. (G ′ red , ω ′ red )) is not a symplectic groupoid for π quot unless L is the graph of a Poisson structure to begin with. This observation indicates that the reduction in [11,Thm. 4.21] for non-free Poisson actions may not generally yield symplectic groupoids on each strata (just over -symplectic [5,Def. 4.5]).

6.2.
A non-integrable quotient. Let us keep considering a manifold M equipped with a Dirac structure L and carrying a free and proper G-action by Dirac diffeomorphisms. We saw in Thm. 6.3 that A red and L quot do not coincide in general. We now illustrate how different these Lie algebroids can be concerning integrability.
It is a direct consequence of Prop. 5.3 that, if L is integrable as a Lie algebroid, then so is A red . It may happen, however, that the quotient Dirac structure L quot is not integrable as a Lie algebroid. Note that this feature is not present when L is defined by a Poisson structure, as in this case L quot necessarily coincides with A red ; i.e, the quotient of an integrable Poisson structure by a free and proper G-action by Poisson diffeomorphisms is always integrable (see [11,Prop. 4.6]). We will verify how this picture changes in the realm of Dirac structures through a concrete example.
Recall that P = S 2 × R, with coordinates (x, t), may be equipped with a Poisson structure π P that does not admit an integration to a symplectic groupoid [20] (see also [10]). This nonintegrable Poisson structure is characterized by the fact that its symplectic leaves are the spheres obtained by constant values of t endowed with (1 + t 2 ) times the usual symplectic structure of S 2 . Proposition 6.4. There exists a presymplectic manifold (M, ω) carrying a free and proper G-action preserving ω for which P = M/G and the quotient map M → P is a forward Dirac map; i.e., if L is the graph of ω, then L quot is the graph of π P .
Observe that whenever L is the graph of a presymplectic structure, it is automatically integrable as a Lie algebroid, as the projection pr T | L : L → T M is a Lie-algebroid isomorphism, so L is integrated by the fundamental groupoid of M , see e.g. [7]. As already mentioned, since π P is nonintegrable, it is impossible to replace ω in Prop. 6.4 by a symplectic form (or any other integrable Poisson structure), so ω must be degenerate.
The proof of Prop. 6.4 is a direct consequence of the construction in [5, Example 6.8], that we recall for completeness.
Let g be a Lie algebra endowed with an Ad-invariant, symmetric and nondegenerate bilinear form B : g × g → R. We use B to identify g * ∼ = g, in such a way that the adjoint and co-adjoint actions are intertwined. We consider the function This function is a Casimir for the linear Poisson structure π g * on g * , as a result of the Ad-invariance of B. It follows that C −1 (λ) is a Poisson submanifold of g * for any λ ∈ R, i.e., π g * restricts to a Poisson structure π λ on C −1 (λ) for which the inclusion Let G be a connected Lie group whose Lie algebra is g, and let G × g * be endowed with the symplectic structure Ω coming from the identification G × g * ∼ = T * G via right translations, so that the projection on the second factor, is a Poisson map. The submanifold M λ = (C • p) −1 (λ) = G × C −1 (λ) of G × g * is equipped with a closed 2-form ω λ , given by the pull-back of Ω by the inclusion ι λ : M λ ֒→ G × g * , and carries a free and proper G-action (by right multiplication on the first factor) so that M λ /G = C −1 (λ). We denote the quotient map by p λ : M λ → C −1 (λ), so that Let L be the Dirac structure on M λ given by the graph of ω λ . Consider the distribution K = ker(p λ ) = T G ⊆ T M λ . Since M λ ⊂ G × g * , we may view K in T (G × g * ) and consider its symplectic orthogonal K Ω ⊆ T (G × g * ).
Proof. The distribution K Ω is spanned by hamiltonian vector fields X p * f of functions of the form p * f , for f ∈ C ∞ (g * ). Since p is a Poisson map, we have dp(X p * f ) = X f . It follows that dC(dp(X p * f )) = dC(X f ) = 0, for C is a Casimir. Hence K Ω ⊆ ker(d(C • p)) = T M λ .
Since K ⊥ = T M λ ⊕ K • , L ∩ K ⊥ = {(X, i X ω λ ) | X ∈ T M λ , i X ω λ ∈ K • }. But one can directly check that i X ω λ ∈ K • if and only if X ∈ T M λ ∩ K Ω = K Ω . Let L quot be the reduction of L as in (6.4), (6.5). Lemma 6.6. L quot is the graph of π λ .
Letf ∈ C ∞ (g * ) be any extension of f , so that f =f • j λ . Then ι * λ (i X Ω) = i X ι * λ Ω = i X ω λ = dp * λ j * λf = ι * λ dp * f , which implies that (6.10) i X Ω = dp * f + kd(p * C) for some k ∈ R. Denoting by π Ω the Poisson structure defined by Ω, the fact that p : G × g * → g * is a Poisson map implies that dp(π ♯ Ω dp * C) = π ♯ g * (dC) = 0, since C is a Casimir. It follows from (6.10) that X = π ♯ Ω (dp * f + kd(p * C)), so dp(X) = dp(π ♯ Ω (dp * f + kd(p * C))) = dp(π ♯ Ω (dp * f )) = π ♯ g * (df ). We conclude that (6.9) holds as a direct consequence of j λ being a Poisson map and X being tangent to M λ . Remark 6.7. It follows from Lemma 6.5 that L ∩ K ⊥ is isomorphic to the distribution K Ω ⊆ T M λ , which can be identified with the action Lie algebroid g ⋉ M λ arising from the diagonal action of G on M λ by left multiplication on the first factor and the coadjoint action on the second. By (6.2), one can check that A red = K Ω /G is identified with the action Lie algebroid g ⋉ C −1 (λ) (relative to the coadjoint action). If G is simply connected, the reduced groupoid G red agrees with the action Lie groupoid G ⋉ C −1 (λ), and ω red coincides with ω λ .
For the proof of Proposition 6.4, consider the Lie algebras su(2) and R, equipped with their canonical bilinear forms B su (2) and B R . With the usual identification su(2) ∼ = R 3 , B su(2) agrees with the euclidean inner product. We let g be the Liealgebra direct sum su(2) ⊕ R, equipped with the bilinear form B = B su(2) − B R . Then C −1 (1/2) = {(ξ, t) ∈ R 3 × R | ξ, ξ = 1 + t 2 } can be identified with P = S 2 ×R in such a way that π 1/2 agrees with π P . Lemma 6.6 shows that M = SU(2) × R × C −1 (1/2) can be equipped with a presymplectic form that pushes forward to π P . This concludes the proof of Proposition 6.4.