Poisson Geometry Related to Atiyah Sequences

We construct and investigate a short exact sequence of Poisson $\mathcal{VB}$-groupoids which is canonically related to the Atiyah sequence of a $G$-principal bundle $P$. Our results include a description of the structure of the symplectic leaves of the Poisson groupoid $\frac{T^*P\times T^*P}{G}\rightrightarrows \frac{T^*P}{G}$. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.


Introduction
For many physical systems the configuration space is the total space of a principal bundle P (M, G) over a base manifold M which parametrizes the external degrees of freedom of the system under consideration. That is, there is a symmetry group G which acts freely on P with a quotient manifold P/G = M and the projection P → M is locally trivial. In this situation the action of G lifts to the phase space T * P with quotient manifold T * P/G. The action of G on T * P is by symplectomorphisms and so the symplectic structure descends to a Poisson structure on T * P/G. This quotient space and its Poisson structure encode the mechanics and the symmetry of the original system.
The typical situation is when the Hamiltonian of the system is invariant with respect to the cotangent lift of the action of G on P . Then the quotient manifold T * P/G becomes the reduced phase space of the system. For example the above happens in rigid body mechanics [8] or when one formulates the equation of motion of a classical particle in a Yang-Mills field: the electromagnetic field is a particular case [12,13].
When the action of G is lifted to T P the quotient manifold T P/G is the Atiyah algebroid of P (M.G), or equivalently the Lie algebroid of the associated gauge groupoid. There is a natural structure of vector bundle on T P/G with base M , and sections of this correspond to vector fields on P which are invariant under the group action; these are closed under the Lie bracket and define a bracket on the sections of T P/G → M .
There are interesting geometric objects which include T P/G and T * P/G as their crucial components. In this paper we will consider the Atiyah sequence (37) and the dual Atiyah sequence (38) as well as the symplectic dual pair (32). From the dual Atiyah sequence we also construct a short exact sequence of VB-groupoids (77) over the gauge groupoid P ×P G ⇒ P/G. The concept of VB-groupoid [10] is recalled in Section 2. The short exact sequences of VB-groupoids which we construct and study (see in particular the diagrams (77) and (86)) relate the various algebraic and Poisson structures inherent in the situation.
Even the case P = G is very productive from the point of view of the applications in Hamiltonian mechanics as well as Lie groupoid theory. Note here that the cotangent groupoid T * G ⇒ T * e G is the symplectic groupoid of the Lie-Poisson structure of the dual T * e G = T * G/G of the Lie algebra T e G of G, see [2]. As we will see, for a general principal bundle P the cotangent groupoid T * G ⇒ T * e G is replaced by the symplectic groupoid T * ( P ×P G ) ⇒ T * P/G which one obtains by the dualization (in the sense of VB-groupoid dualization [10,7]) of the tangent prolongation groupoid T ( P ×P G ) ⇒ T (P/G) of the gauge groupoid.
We now describe the contents of each section in detail. Section 2 is concerned with aspects of VB-groupoids: their duality, principal actions of a Lie group G on a VB-groupoid, and short exact sequences of VB-groupoids. The main results are presented in Theorem 4 and Theorem 10, which show that for VB-groupoids, and for short exact sequences of VB-groupoids, the operations of dualization and quotient, commute.
In Section 3 we recall the definitions of Atiyah sequence and dual symplectic pair and describe the relationship between these notions. Using this relationship we describe in detail the 'double fibration' structure of the symplectic leaves of T * P/G; see Theorem 15. In Section 4, we extend the Atiyah sequence (37) and its dual (38) to short exact sequences of VB-groupoids over the gauge groupoid, see (77) and (86) respectively. From the point of view of Poisson geometry the most important structures are described by the diagram (86), in which all objects are linear Poisson bundles and all arrows are their morphisms. Consequently, using (86), we obtain various relationships between the Poisson and vector bundle structures of T * P/G and T * P ×T * P G , as well as the groupoid structures of T * ( P ×P G ) ⇒ T * P/G and T * P ×T * P G ⇒ T * P/G. The case when P is a group H and G = N is a normal subgroup of H is discussed in Section 5. Some geometric constructions useful in Hamiltonian mechanics of semidirect products, see [5,6,9,11], are also presented in this section.

Short exact sequences of VB-groupoids
The groupoids with which the paper will be chiefly concerned are the tangents and cotangents of gauge groupoids, and certain groupoids related to them. For any groupoid G the tangent bundle T G has both a Lie groupoid and a vector bundle structure and these are related in a natural way. Owing to the relations between these structures, the cotangent bundle T * G also has a natural Lie groupoid structure, and indeed is a symplectic groupoid with respect to the canonical cotangent symplectic structure [3].
The relations between the groupoid and vector bundle structures on T G were abstracted by Pradines [10] to the concept of VB-groupoid, and [10] showed that for a general VB-groupoid Ω, dualizing the vector bundle structure on Ω leads to a VB-groupoid structure on Ω * . We recall this construction in § §2.1, 2.2 and 2.3. For further detail, see [7,Chap. 11].
In §2.4 we show that, under a natural condition, short exact sequences of VB-groupoids may be dualized; see Theorem 4. We then show that when the base of a VB-groupoid is the total space of a principal bundle, the group of which acts on, and preserves the structure of, the VB-groupoid, the quotient spaces form another VB-groupoid (Theorem 5). Finally in §2.5 we show that the two processes -taking the quotient and taking the dualcommute (Theorem 10).

VB-groupoids
We consider manifolds with both a groupoid structure and a vector bundle structure. The compatibility condition between the groupoid and vector bundle structures can be described succinctly by saying that the groupoid multiplication must be a morphism of vector bundles and that the vector bundle addition must be a morphism of groupoids. In fact these conditions are expressed by a single equation. In order to formulate the equation several conditions on the source, target and bundle projections are necessary.
Consider a manifold Ω with a groupoid structure on base B, and a vector bundle structure on base G . Here B is to be a vector bundle on base P and G is to be a Lie groupoid on the same base P . (At present P is an arbitrary manifold; later it will denote a principal bundle.) These structures are shown in Figure 1.
The crucial equation, as described above, is where ξ i ∈ Ω. In order for ξ 1 + ξ 2 to be defined we must have λ(ξ 1 ) = λ(ξ 2 ) where λ : Ω → G is the bundle projection. Likewise we must have λ(η 1 ) = λ(η 2 ). For ξ 1 η 1 to be defined we must have s(ξ 1 ) = t(η 1 ) and likewise we must have s(ξ 2 ) = t(η 2 ). If we impose the conditions that the source and target projections Ω → B are morphisms of vector bundles, then it will follow that the product on the left hand side of (1) is defined. Likewise, if λ is a morphism of groupoids, then the addition on the right hand side will be defined. We now state the formal definition.

Definition 1.
A VB-groupoid consists of four manifolds Ω, B, G , P together with Lie groupoid structures Ω ⇒ B and G ⇒ P and vector bundle structures Ω → G and B → P , such that the maps defining the groupoid structure (source, target, identity inclusion, multiplication) are morphisms of vector bundles, and such that the map formed from the bundle projection λ and the source projection s, is a surjective submersion.
The need for the final condition will be clear in §2.2. The map in (2) may be denoted ( s, λ). Also, we refer to G ⇒ P as the side groupoid and to B → P as the side vector bundle.

4
The condition that the identity inclusion be a morphism of vector bundles is the first equation in (3) below. It follows that 1 0p = 0 1p ; that is, the groupoid identity over a zero element of B is the zero element over the corresponding identity element of G . Further, Since multiplication and identity are morphisms of vector bundles, it follows in the usual way that inversion is also a morphism of vector bundles, and this is the second equation in (3).
Next, the zero sections define a morphism of groupoids. To see this, first consider sources. By the definition, s • 0 = 0 • s, since s and s constitute a morphism of vector bundles. But this can also be read as stating that 0 and 0 commute with the sources. In the same way, 0 and 0 commute with the targets and the multiplication, and are therefore a morphism of Lie groupoids. So we have the first two equations below.
The third equation follows by a similar argument.

The core of a VB-groupoid
A necessary ingredient in the duality of VB-groupoids is the concept of core. The core K of a VB-groupoid (Ω; B; G ; P ) is the intersection of the kernel of the bundle projection λ : Ω → G with the kernel of the source s : Ω → B.
Equivalently, K is the preimage under ( s, λ) of the closed submanifold {(0 p , 1 p ) | p ∈ P }. Since ( s, λ) is assumed to be a surjective submersion, K is a closed submanifold of Ω.
The core inherits a vector bundle structure, with base P . Define λ K : K → P by k → s( λ(k)). For k 1 , k 2 ∈ K with λ K (k 1 ) = λ K (k 2 ), the sum k 1 + k 1 ∈ Ω is defined and is in K. The vector bundle conditions are easily verified.
Example 2.1. For any Lie groupoid G ⇒ P , the groupoid T G ⇒ T P obtained by applying the tangent functor to the structure of G , is a VB-groupoid, (T G ; T P, G ; P ). The core is the Lie algebroid AG .
When G is a Lie group G, the multiplication in the tangent group T G is given by (8) and T G is isomorphic to the semi-direct product G ⋉ T e G. For general groupoids T G ⇒ T P is not a groupoid semi-direct product.

The dual of a VB-groupoid
Since λ : Ω → G is a vector bundle, it has a dual bundle λ * : Ω * → G . Somewhat surprisingly, this has a natural Lie groupoid structure with base K * and these structures make (Ω * ; K * , G ; P ) a VB-groupoid [10].
Take Φ ∈ Ω * γ where γ = λ(Φ) ∈ G has source p = s(γ) and target q = t(γ). Then the target and source of Φ in K * q and K * p respectively are defined to be 5 The lack of symmetry is unavoidable: in defining the core one must take either the kernel of s or of t. For the composition, take Ψ ∈ Ω * δ with s * (Ψ) = t * (Φ). To define ΨΦ ∈ Ω * δγ we must define the pairing of ΨΦ with any element of Ω δγ . Now any element ζ of Ω δγ can be written as a product ηξ where η ∈ Ω δ and ξ ∈ Ω γ . We claim that is well defined. Any other choice of η and ξ must be ητ −1 and τ ξ for some τ with s(τ ) = s(η) = t(ξ). Also, τ must project to 1 q ∈ G where q = s(δ), since ητ −1 must lie in the same fibre over G as η.
Write b = s(τ ). Then τ − 1 b is defined and is a core element, say k. Now we have Proceeding in the same way, we likewise find that We now define the identity element of Ω * at χ ∈ K * p . To do this we need to pair 1 χ with elements of Ω which project to 1 p ∈ G . Take such an element ξ and write b = λ(ξ). Then ξ − 1 b is a core element k and we define It is now straightforward to check the proof of the following result.
Given ω ∈ B * p , we identify it with the element ω ∈ Ω * defined by Example 2.2. Given a Lie groupoid G ⇒ P , the dual of the VB-groupoid (T G ; G , T P ; P ) is (T * G ; G , A * G ; P ) where AG is the Lie algebroid of G . The groupoid T * G ⇒ A * G is a symplectic groupoid with respect to the canonical symplectic structure on T * G [3, 15].

Short exact sequences of VB-groupoids and their duals
The main constructions of the paper rely on dualizing short exact sequences of VBgroupoids and in this subsection we give the basic results for this process. First we need the concept of morphism.
A morphism of VB-groupoids (Ω; B, G ; P ) → (Ω ′ ; B ′ , G ′ ; P ′ ) is a quadruple of maps It follows that F restricts to a vector bundle morphism of the cores f K : K → K ′ . For the purposes of this paper we only need consider the dualization of morphisms which preserve G ⇒ P . Proposition 3. Let F : Ω 1 → Ω 2 and f B : B 1 → B 2 , together with the identity maps on G and P , be a morphism of VB-groupoids (Ω 1 ; B 1 , G ; P ) → (Ω 2 ; B 2 , G ; P ). Then the dual maps F * : Ω * 2 → Ω * 1 and f * K : K * 2 → K * 1 together with the identity maps on G and P are a morphism of VB-groupoids (Ω * 2 ; K * 2 , G ; P ) → (Ω * 1 ; K * 1 , G ; P ). The proof is a straightforward verification. We will apply the duality of VB-groupoids to structures in which the side groupoid is a gauge groupoid, or is closely related to a gauge groupoid. At the moment we continue to allow the base manifold P to be arbitrary; later in the section it will denote a principal bundle.
A short exact sequence of VB-groupoids consists of three VB-groupoids and two morphisms, as shown in Figure 2, such that the three VB-groupoids have the same side groupoid G ⇒ P and the morphisms (F, f B ) and (H, h B ) preserve G and P , and such → Ω 3 is a short exact sequence of vector bundles over G . This includes the conditions that the kernel of F is the zero bundle over G and that the image of H is Ω 3 , as well as exactness at Ω 2 .
Theorem 4. Consider a short exact sequence of VB-groupoids, as just defined and as shown in Figure 2. Then: → B 3 is a short exact sequence of vector bundles over P ; is a short exact sequence of vector bundles over P , where K i is the core of Ω i , and f K , h K are the induced morphisms of the cores; (iii) the dualization of the short exact sequence of VB-groupoids over G ⇒ P results in the situation shown in Figure 3, which is therefore also a short exact sequence of VB-groupoids.
Proof. For (i) and (ii) there are six conditions to establish. We give three representative proofs.
First we prove that f B : B 1 → B 2 is injective. Take b 1 ∈ B 1 and suppose that Secondly we prove that h K : K 2 → K 3 is surjective. Take k 3 ∈ K 3 . Since K 3 ⊆ Ω 3 and H : Ω 2 → Ω 3 is surjective, there is ξ ∈ Ω 2 such that H(ξ) = k 3 . Now if ξ projects to g ∈ G it follows that k 3 also projects to g; since k 3 is a core element we have g = 1 m for some m ∈ P . Now write X ∈ B 2 for the source of ξ. We have h B (X) = 0.
The subtraction ξ − 1 X is defined since both project to 1 m in G . And ξ − 1 X has source X − X = 0 so is a core element. Now H(ξ − 1 X ) = k 3 − 1 0 and 1 0 = 0 1 , the zero element in the fibre over 1 m . So ξ − 1 X is a core element which is mapped by H to k 3 .
Thirdly we prove exactness at K 2 . Since f K and h K are restrictions of F and H it follows from The other conditions are proved in the same way. Now apply Proposition 3 to F and H as morphisms of VB-groupoids. It follows that F * and H * are morphisms of VB-groupoids as shown in Figure 3. Since → Ω 3 is a short exact sequence of vector bundles over G , the duals form a short exact sequence Ω * , again of vector bundles over G . This completes the proof of (iii).

Quotients of VB-groupoids over group actions
We now need to establish that this dualization process commutes with quotienting over a group action. In the cases which we consider, the base manifold P is a principal G-bundle and we need to quotient over the action of G. In this situation the required quotient manifolds exist, and the constructions are straightfoward.
First consider vector bundles over the total space P of a principal bundle. A PBGvector bundle over P is a vector bundle E → P together with an action E × G → E by vector bundle automorphisms over the principal action κ : P × G → P .
Denote the orbit of e ∈ E by e . The projection λ : . Then there exists a unique g ∈ G such that λ(e ′ ) = λ(e)g. We define e + e ′ = eg + e ′ and t e = te for t ∈ R. It remains to prove that E/G → P/G is locally trivial; for this and the rest of the proof of the following proposition see [7, §3.1].
Proposition 5. Let E be a PBG-vector bundle over P (M, G, µ). Then the quotient manifold E/G exists, and has a vector bundle structure with base P/G ∼ = M , such that the natural projection E → E/G is a vector bundle morphism over µ.
The case E = T P arises in constructing the Atiyah sequence of a Lie algebroid. There is a corresponding notion for Lie groupoids. A PBG-groupoid over P (M, G) is a Lie groupoid G ⇒ P together with an action G × G → G by groupoid automorphisms over the principal action P × G → P ([7, 2.5.4]).
Denote the orbit of γ ∈ G by γ . The source and target maps s : That G /G ⇒ P/G is a groupoid is straightforward to check. The remaining details of the following proposition can be found in [7, §3.1].
Proposition 6. Let G be a PBG-groupoid over P (M, G, µ). Then the quotient manifold G /G exists, and has a Lie groupoid structure with base P/G ∼ = M , such that the natural projection G → G /G is a morphism of Lie groupoids over µ.
See [7, 2.5.5] for the proof. These two constructions can be combined so as to apply to VB-groupoids. Definition 7. Let P (M, G, µ) be a principal bundle. A PBG-VB-groupoid over P (M, G) is a VB-groupoid (Ω; B, G ; P ) together with right actions of G on each of the manifolds Ω, B and G such that Ω ⇒ B and G ⇒ P are PBG-groupoids and B → P is a PBG-vector bundle.
Then the quotient manifolds Ω/G, B/G, G /G and K/G exist, and form a VB-groupoid (Ω/G; B/G, G /G; P/G) with core K/G such that the natural maps Ω → Ω/G, B → B/G, G → G /G and P → P/G = M , constitute a morphism of VB-groupoids.
The proof only requires assembling the results of Propositions 5 and 6.
We now need to establish that this quotienting process commutes with dualization. Consider a PBG-VB-groupoid (Ω; B, G ; P ) and its dual (Ω * ; K * , G ; P ). Equip Ω * and K * with the contragredient actions of G; that is, Proposition 9. With the structures just defined, (Ω * ; K * , G ; P ) is a PBG-VB-groupoid and the canonical maps together with the identities on G and P , constitute an isomorphism of VB-groupoids.
The proof is a lengthy but straightforward verification. Finally in this section we need to consider the preservation of exact sequences of PBG-VB-groupoids under dualization and quotient. The proof is a straightforward application of the techniques used above.
Theorem 10. In the short exact sequence shown in Figure 2 assume that each VB-groupoid has the structure of a PBG-VB-groupoid with respect to a principal G-bundle structure on P , and that F and H are G-equivariant. Equip the dual sequence, shown in Figure 3, with the contragredient G-actions. Then the dual short exact sequence is also a short exact sequence of PBG-VB-groupoids and the canonical maps together with the identities on G /G and P/G, constitute an isomorphism of VBgroupoids over P/G.

The dual Atiyah sequence
The Lie algebroid of a gauge groupoid is the Atiyah algebroid of the corresponding principal bundle, and this Lie algebroid is the central term of a short exact sequence of Lie algebroids, the Atiyah sequence. Accordingly, the dual of this Lie algebroid is the central term of a short exact sequence of vector bundles with Poisson structures. In this section we set up the notation needed for the study of this dual sequence. We also investigate the fibre structure of the symplectic leaves of the Poisson manifold T * P/G using the notion of dual symplectic pair together with that of Atiyah sequence. The main results are collected in Theorem 15.

Principal bundles
We consider a principal bundle P (M, G, µ) where κ : P × G → P is a free right action of a Lie group G on a smooth manifold P , and µ : P → M is a surjective submersion for which the fibres equal the orbits of G; thus M = P/G. As in the previous sections, we always assume that the bundle is locally trivial.
One may take the tangent right action T κ : T P × T G → T P of the tangent group T G on the tangent bundle T P .
Applying the tangent functor to the multiplication in G we obtain a group structure on T G, denoted here by where L g (h) := gh and R h (g) := gh. In particular for X e , Y e ∈ T e G and zero elements We see from (9) that the zero section 0 : G → T G of the tangent bundle T G is a group monomorphism and one has the decomposition of T G as a semi-direct product of G and the normal subgroup T e G ⊆ T G.
We will use the following notations: where κ p : G → P and κ g : P → P . Thus we have for g ∈ G. The action T κ : T P × T G → T P of the tangent group T G on the tangent bundle T P can be expressed by In the following two propositions we collect various equalities and bundle isomorphisms which will be useful in what follows. The proof of the first proposition is a direct calculation.
Proposition 11. For g, h ∈ G and p ∈ P , Write T V P for the vertical subbundle of T P ; that is, T V P = ker T µ, and T V * P for its dual.
Proposition 12. There are the following isomorphisms of vector bundles: Proof. Using the vector space isomorphisms and the action of the subgroup T e G ⊆ T G on T P defined by we obtain (16a). In order to prove (16b) we observe that Assuming X g = T R g (e)X e we find from (19) that the double quotient vector bundle (T P/T e G)/G is isomorphic to the quotient bundle T P/T G.
Since the fibres of T µ are orbits of the action (13) we obtain the vector bundle isomorphism mentioned in (16c).
Let us define actions φ g : T P → T P and φ * g : for v ∈ T p P , ϕ ∈ T * p P . For the vector bundle trivialization I : one has where g ∈ G. Thus the bundle isomorphism I : of quotient vector bundles presented in (16d). Dualizing (22) we obtain the isomorphism (16e).

The symplectic dual pair
We recall that the canonical 1-form γ on T * P is defined by where ϕ ∈ T * P and ξ ϕ ∈ T ϕ (T * P ). Here π * : T * P → P is the projection of T * P on the base P . We note that for any g ∈ G. From the definition (23) and (24) one has which means that γ is invariant with respect to the action φ * g : T * P → T * P defined in (20). This action generates two maps: where π * G is the projection to the quotient manifold and J is the momentum map defined by for φ ∈ T * p P . In order to see that (26) is the momentum map for the symplectic manifold (T * P, dγ) let us take X ∈ T e G and consider the corresponding vector field ξ X ∈ Γ ∞ (T (T * P )), i.e. the field defined by where f ∈ C ∞ (T * P ). For ξ X one has γ(ϕ), ξ X (ϕ) = ϕ, T π * (ϕ)ξ X (ϕ) = ϕ, T κ p (e)X = J(ϕ), X .
Hence we obtain The first equality in (29) follows from L ξ X dγ = 0 and the second one from (28). Next, for g ∈ G one has which shows that J : T * P → T * e G is an equivariant map. So, from (29) and (30) we conclude that J : T * P → T * e G is the momentum map for the symplectic manifold (T * P, dγ).

13
One has the canonical Lie-Poisson structure on T * e G defined by where f, g ∈ C ∞ (T * e G) and [·, ·] is the Lie bracket of the Lie algebra T e G. Since the symplectic form dγ is invariant with respect to the action φ * g : T * P → T * P of the group G, defined in (20), the Poisson bracket {f, g} of G-invariant functions f, g ∈ C ∞ (T * P ) is also an G-invariant function. So, the quotient manifold T * P/G is a Poisson manifold, the Poisson structure of which is defined by the quotienting of the structure on T * P .
Remembering that the action of G on P is free, we conclude that for any p ∈ P the map J : T * p P → T * e G is a submersion onto. Thus we see that the momentum map J : T * P → T * e G is a surjective submersion. So, one has two surjective Poisson submersions: from the symplectic manifold T * P , such that the Poisson subalgebras (π * G ) * (C ∞ (T * P/G)) and J * (C ∞ (T * e G)) are polar one to another, see [2]. Therefore, the diagram (32) gives a full symplectic dual pair in the sense of the definition in Subsection 9.3 in [2].
Let us mention that π * G is a complete Poisson map, see Proposition 6.6 in [2]. The completness of the momentum map J follows from its G-equivariance. In the rest of the paper we will assume that G and P are connected manifolds. This implies the connectness of the fibres of π * G and J. Taking into consideration this assumption and the properties of the dual symplectic pair (32) mentioned above one finds (see [2]) that there is a one-to-one correspondence between the symplectic leaves of T * P/G and the coadjoint orbits which are the symplectic leaves of T * e G.

The dual Atiyah sequence
Using the vector bundle monomorphism I : P × T e G → T V P ⊂ T P , defined in (21), we obtain the following exact sequence of vector bundles over P . The map I is given by (21) and A is a quotient map defined as where v p ∈ T P . Quotienting (35) over the G-action and using the isomorphisms (16b), (16c) and (16d), we obtain the Atiyah sequence where ι =: [I] and a := [A] is the anchor map, e.g. see [7].
It follows from µ • κ g = µ that T µ : T P → T (P/G) is constant on the orbits of the action φ g : T P → T P , g ∈ G. Note that all terms of the above short exact sequence have a Lie algebroid structure over P/G and the central term is the Atiyah algebroid. So, it follows from Lie algebroid theory (e.g. see [7,Chap. 3]) that the short exact sequence dual to the Atiyah sequence (37), is a short exact sequence of Poisson maps of linear Poisson bundles. Let us explain more precisely the above statement.
The action φ * g : T * P → T * P, g ∈ G, preserves the vector bundle structure π * : T * P → P of T * P as well as its Poisson structure. Hence the quotient manifold T * P/G is a vector bundle [π * ] : T * P/G → P/G over P/G. The linearity of the Poisson bracket {·, ·} on C ∞ (T * P/G) means that if f, g ∈ C ∞ (T * P/G) are linear on the fibres of [π * ] : T * P/G → P/G then the Poisson bracket {f, g} has the same property. So, we have consistency between vector bundle and Poisson manifold structures of T * P/G. The above is also valid for the Poisson structure of T * (P/G).
The linear Poisson structure of the bundle P × Ad * G T * e G → P/G is defined as follows. Let π 0 be the zero Poisson tensor on P and let π L−P be the Lie Poisson tensor on T * e G defined by (31). The product Poisson structure π 0 × π L−P defined on P × T * e G is invariant with respect to the action φ g × Ad * g : P × T * e G → P × T * e G, g ∈ G. Hence one has on the associated vector bundle P × Ad * G T * e G → P/G the quotient linear Poisson structure. This property follows from the linearity and the Ad * G -invariance of the Lie-Poisson bracket defined in (31).
To conclude we mention that a * and ι * are Poisson maps and preserve the vector bundle structures in the short exact sequence (38).

Relationship between the symplectic dual pair and the dual Atiyah sequence
As we have shown in the two previous subsections the principal G-bundle structure µ : P → P/G of P leads to two crucial, from the point of view of Poisson geometry, structures described in the diagrams (32) and (38), respectively. We now discuss the relationship between them. For this reason, using the isomorphism I : P × T e G → T V P defined in (21), we consider the short exact sequence of vector bundles dual to (35), where the subbundle T V 0 P ⊆ T * P is the annihilator of T V P in T P ; that is, The vector bundle epimorphism I * dual to I is related to the momentum map (26) by I * (ϕ) = (π * (ϕ), J(ϕ)); that is, I * = π * × J and the map ι * : T * P/G → P × Ad * G T * e G given by is the quotient of I * over the group G. We see from (26) and (40) that It follows from the Marsden-Weinstein symplectic reduction procedure [8], that T V 0 P/G = J −1 (0)/G is a symplectic leaf of T * P/G corresponding to the one-element Ad * G -orbit consisting of the zero element of T * e G. Proposition 13 below shows that the dual anchor map a * : T * (P/G) → T * P/G defines a fibrewise linear symplectic diffeomorphism between T * (P/G) and J −1 (0)/G. The result can also be found in [11, 5.4]. Note that this fact follows also from the general theory of Lie algebroids, where the dual a * of the anchor map a is a Poisson map from the symplectic manifold T * (P/G) to the symplectic leaf J −1 (0)/G ⊂ T * P/G which in this case is equal to (ι * ) −1 (P × Ad * G {0}) = T V 0 P/G. However it is interesting to prove this result explicitly.
Proposition 13. One has the vector bundle isomorphism which is also a natural isomorphism of the symplectic manifolds.
Proof. In order to describe a * in an explicit way we recall that µ • κ g = µ and thus T µ(pg) • T κ g (p) = T µ(p).
As we have seen, the dual symplectic pair (32) gives a one-to-one correspondence between the coadjoint orbits Ø ⊂ T * e G of G and the symplectic leaves S ⊂ T * P/G of T * P/G given by (33). Using the Atiyah dual sequence (38) we can investigate this correspondence in more detail. At first let us make the following observation: Proposition 14. For any coadjoint orbit Ø ⊂ T * e G, we have: Proof. At first we note that Since J and I * are G-equivariant maps and ι * = [I * ] we obtain (53) by quotienting (54) over G.
We observe that the vertical arrows in the diagram which is a part of the dual Atiyah sequence (38), are the projections on the base of the corresponding vector bundle, whereas the map ι * : T * P/G → P × Ad * G T * e G is the bundle projection of an affine bundle over P × Ad * G T * e G. In order to see that the fibre ι * −1 ( p, χ ), where p, χ ∈ [pr 1 ] −1 ( p ) and (p, χ) ∈ P × T * e G, is an affine space over the vector space T * p (P/G), let us take ϕ 1 , ϕ 2 ∈ ι * −1 ( p, χ ). From im a * = ker ι * it follows that there exists a unique ρ ∈ T * p (P/G) such that So, the vector space T * p (P/G) acts freely and transitively on ι * −1 ( p, χ ). Note that dim ι * −1 ( p, χ ) = dim T * p (P/G). The affine fibre bundle ι * : T * P/G → P × Ad * G T * G can be described in the groupoid language. Namely, let us consider the cotangent bundle T * (P/G) as a groupoid T * (P/G) ⇒ P/G in which the source and target maps are equal to ν * : T * (P/G) → P/G. Then the dual anchor map a * : T * (P/G) → T * P/G is the momentum map for the action of T * (P/G) ⇒ P/G on T * P/G defined in (56). The orbits of this action are the fibres of ι * : T * P/G → P × Ad * G T * G. From the above and from Proposition 14 it follows that all concerning (55) is also valid for the fibration Let us mention that T * P/G does not have a fibre structure over T * (P/G). However, choosing a section σ : P × Ad * G T * e G → T * P/G of the affine bundle ι * : T * P/G → P × Ad * G T * e G we obtain the vector bundle epimorphismσ : T * P/G → T * (P/G) defined by the action (56) as follows where ι * ( ϕ ) = p, χ . Note here that a * is a monomorphism of vector bundles, thus the equality (58) definesσ( ϕ ) ∈ T * (P/G) uniquely.
The following theorem summarizes observations mentioned above .
Theorem 15. (i) The Poisson vector bundle [π * ] : T * P/G → P/G has the structure of an affine bundle ι * : T * P/G → P × Ad * G T * e G over the total space of the vector bundle [pr 1 ] : P × Ad * G T * e G → P/G, i.e the fibre ι * −1 ( p, χ ) of p, χ ∈ P × Ad * G T * e G is an affine space over the vector space T * p (P/G).
(ii) The symplectic leaf S = J −1 (Ø)/G has the structure of an affine fibre bundle over over the total space of the bundle [pr 1 ] : P × Ad * G Ø → P/G with the orbit Ø ⊂ T * e G as a typical fibre.
(iii) Fixing a section σ : P × Ad * G Ø → T * P/G we could consider the symplectic leaf J −1 (Ø)/G as a fibre bundle over the cotangent bundle T * (P/G) with Ø as the typical fibre. The total space and the base of (59) are symplectic manifolds. However, the bundle projection π σ is not a Poisson map in general.
(iv) In the case when Ø is a one-element orbit Ø = {χ} (such an orbit corresponds to a character of G) the bundle map π σ : J −1 (χ)/G → T * (P/G) defines a diffeomorphism of manifolds, but it is not a symplectomorphism. The difference ω χ − π * σ dγ of symplectic forms, where ω χ is the symplectic form of the symplectic leaf J −1 (χ)/G andγ is the canonical symplectic form on T * (P/G), and is called the magnetic term, see [12], [13].
If G is a commutative Lie group all the coadjoint orbits are one element sets Ø = {χ}, where χ ∈ T * e G. Hence we have Taking the above facts into account and identifying P/G × {χ} with P/G we conclude the following Remark 16. If G is a commutative group then the symplectic leaves J −1 {χ}/G, χ ∈ T * e G, are affine bundles over P/G modeled over J −1 (0)/G ∼ = T * (P/G), i.e. for p ∈ P/G the fibres ι * −1 ( p ) are affine spaces over the vector spaces T * p (P/G). A nice geometrical way to define a section σ is given by the choice of a connection form α on the principal bundle µ : P → P/G (see [13]), i.e. such α ∈ Γ ∞ (T * P, T e G) that and The conditions (61) and (62) imply that the bundle epimorphismα : where v p ∈ T p P , after quotienting by G gives the map The dual of (64) is the section of ι * : T * P/G → P × Ad * G T * e G mentioned above.

20
Applying the constructions investigated in Section 2, we will construct two short exact sequences, (77) and (86), of VB-groupoids over the gauge groupoid P ×P G ⇒ P/G. These are related to each other by the dualization procedure described in §2.4. Applying the results obtained in Section 3 we investigate the groupoid and Poisson structures of the objects involved (86).

The tangent VB-groupoid of a gauge groupoid
Let us consider the tangent VB-groupoid of the pair groupoid P × P ⇒ P . Since T pr 1 × T pr 2 : T (P × P )→T P × T P is a vector bundle isomorphism we will identify the tangent VB-groupoid T (P × P ) ⇒ T P with the pair VB-groupoid T P × T P ⇒ T P . By T V (P × P ) we denote the vertical subbundle of the tangent bundle T (P × P ) defined by the action of G on the product P × P . In consequence X ∈ T e G acts on (v p , w q ) ∈ T P × T P as follows (v p , w q ) → (v p + T κ p (e)X, w q + T κ q (e)X).
It follows from the properties of morphisms of VB-groupoids which preserve the side groupoids, that T V (P × P ) ⇒ T V P is a subgroupoid of T (P × P ) ⇒ T P . The groupoids mentioned above form a short exact sequence of VB-groupoids as defined in §2. 4.
Here each side groupoid is the pair groupoid P × P ⇒ P , and T P ×T P TeG ⇒ T P TeG is the quotient of the pair groupoid T P × T P ⇒ T P by T e G.
Now we define a VB-groupoid over the pair groupoid P × P ⇒ P as follows: s(p, X, q) := (q, X),t(p, X, q) := (p, X),ε(p, X) := (p, X, p), ι(p, X, q) := (q, X, p), (p, X, q)(q, X, r) := (p, X, r), where p, q, r ∈ P and X ∈ T e G, and the horizontal arrows in (71) are projections on the suitable terms of the product manifolds. Let us define a map I 2 : P × T e G × P → T V (P × P ) by Proposition 17. The map I given by (21), and the map I 2 define an isomorphism of VB-groupoids over P × P ⇒ P .
We conclude from the above that I and I 2 trivialize the vertical subbundles T V P → P and T V (P ×P ) → P ×P , respectively. Let us also note that the VB-groupoid isomorphism given in (73) is equivariant with respect to the action of the group G defined on P ×T e G×P by (p, X, q) → (pg, Ad g −1 X, qg) and on T V (P × P ) by (T κ p (e)X, T κ q (e)X) → ((T κ g (p) • T κ p (e))X, (T κ g (q) • T κ q (e))X) where g ∈ G. From (70) and the isomorphism (73) we obtain the following short exact sequence of VB-groupoids over P ×P ⇒ P in which all arrows commute with the action of G. Thus, after quotienting by G, we obtain the short exact sequence of VB-groupoids over the gauge groupoid P ×P G ⇒ P/G. The oblique upper and lower arrows in (77) give the Atiyah sequences for the principal bundles P × P → P ×P G and P → P/G, respectively. Note that the last groupoid in the short exact sequence (77) is the tangent prolongation groupoid of the gauge groupoid P ×P G ⇒ P/G.

The cotangent VB-groupoid of a gauge groupoid
Now let us dualize (77) in the sense of §2.4. Since dualization commutes with the action of G we start from the dualization of (76).
Proposition 18. There are the following vector bundle isomorphisms: Proof. (i) From the definition we see that (v p , w q ) ∈ core (T P × T P ) if and only if p = q and w q = 0. Thus core (T P × T P ) = T P × ({0} × P ) ∼ = T P.
(iii) An element v p , w q ∈ T P ×T P TeG is defined by v p , w q := {(v p + T κ p (e)X, w q + T κ q (e)X); X ∈ T e G}.
So, v p , w q ∈ core T P ×T P TeG if and only if p = q and w q = T κ q (e)Y for some Y ∈ T e G. Choosing in (78) X = −Y we find that core Applying the dualization procedure presented in Subsection 2.4 and Proposition 18 we see that the short VB-groupoid sequence (over P × P ⇒ P ) dual to (70) is where the dual A * 2 of A 2 is the inclusion map and I * 2 is given by I * 2 (ϕ, ψ) := (p, ϕ p • T κ p (e) + ψ q • T κ q (e), q) for p = π * (ϕ) and q = π * (ψ). The map I * 2 : T * P × T * P → P × T * e G × P is a groupoid morphism over the bundle projection π * : T * P → P .
We note here that for T * P × T * P ⇒ T * P we have as the source map, target map, identity section, inverse map and groupoid product, respectively.
Remark 19. It is easy to see that the involution δ : gives an isomorphism of symplectic groupoids between the pair groupoid T * P × T * P ⇒ T * P with the symplectic form d(pr * 1 γ − pr * 2 γ) on T * P × T * P and the symplectic groupoid T * P × T * P ⇒ T * P for which the groupoid structure is given in (81) and the symplectic form is d(pr * 1 γ + pr * 2 γ).
The second component of the bundle epimorphism I * 2 defines the momentum map for the cotangent bundle T * (P × P ) = T * P × T * P whose symplectic structure is given by the canonical symplectic form dγ 2 = d(pr * 1 γ + pr * 2 γ) and the action of the group G on P × P is defined in (67).
In order to see that J 2 : T * P × T * P → T * e G is a momentum map we note that (P × P, µ 2 : P × P → P ×P G , G) is a G-principal bundle and apply the same consideration as for J : T * P → T * e G in (26).
We note also that the vector bundle T P ×T P TeG * → P × P is isomorphic to the vector Hence we can identify the dual VB-groupoid T P ×T P TeG * ⇒ T * P with the subgroupoid T V 0 (P × P ) ⇒ T * P of the dual VB-groupoid T * P × T * P ⇒ T * P .
So, according to Proposition 10, quotienting (79) by G we obtain the short exact sequence of VB-groupoids over the gauge groupoid P ×P G ⇒ P/G 25 which is the dual of the short exact sequence of (77) in sense of the dualization procedure described in Section 2. In particular the first groupoid of (86) is the VB-groupoid dual to the tangent prolongation groupoid of the gauge groupoid P ×P G ⇒ P/G.

Symplectic leaves of duals of Atiyah sequences
The results of Section 3, in particular Theorem 15, can be applied to the short exact sequence of linear Poisson bundles; this is the dual of the Atiyah sequence of the G-principal bundle (P × P, µ 2 : P × P → P ×P G , G). Note that (87) is part of the VB-groupoid short exact sequence (86). In (86) both ι * 2 and a * 2 are Poisson maps, and are VB-groupoid morphisms. Summarizing the above, we can say that the structures of groupoid, vector bundle and Poisson manifold are consistently involved in the diagram (86).
From the considerations in Subsection 4.2, especially of (86), we see that the core of the VB-groupoid consists of all ϕ p , 0 p ∈ T * P ×T * P G such that ϕ p • T κ p (e) = 0, where p ∈ P . Thus, using (38), we have the following isomorphisms of vector bundles over P/G. In particular, the core of T * ( P ×P G ) is isomorphic to T * (P/G). The vector bundle T * (P/G) → P/G can be regarded as a totally intransitive groupoid on base P/G and as such acts on T * P/G by Recall here that a * (ρ p ), ϕ p ∈ [π * ] −1 ( p ). The moment map for the action (90) is [π * ] : T * P/G → P/G, i.e. [π * ]( ϕ p ) = p ∈ P/G and i.e. (ρ, ϕ ) ∈ T * (P/G) * T * P/G if and only if q = p . The action groupoid of (90) is T * (P/G) < [π * ] T * P/G ⇒ T * P/G and according to Proposition 11.2.3 in [7], valid for a general VB-groupoid, one has the short exact sequence of Lie groupoids There are other crucial statements which we collect in the following proposition: The symplectic groupoid T * ( P ×P G ) ⇒ T * P G is a subgroupoid as well as a symplectic leaf of the Poisson groupoid T * P ×T * P G ⇒ T * P G . (iii) Therefore the symplectic leaves of T * P G , defined as J −1 (Ø)/G are orbits of the standard action of the groupoid T * ( P ×P G ) ⇒ T * P G on its base T * P G . (iv) As a consequence of (92), any symplectic leaf J −1 (Ø)/G is foliated by the orbits of the action of T * (P/G) ⇒ P/G on T * P/G; these orbits are the fibres of the affine bundle ι * : J −1 (Ø)/G → P × Ad * G Ø. Here (i) is a particular case of the theory introduced in [15]. The symplectic leaves J −1 2 (Ø)/G of T * P ×T * P G ⇒ T * P G may be included in a sequence of morphisms of fibre bundles, shown in (93), similarly to (57).
So, Proposition 15 is valid in this case as well as in the case of symplectic leaves J −1 (Ø)/G, where Ø ∈ T * e G. From (86) we have we find that inverse of T Σ is given by It factorizes T * H into the product of cotangent bundles T * K × T * N and allows us to simplify the form of the momentum map where ϕ h ∈ T * h H. Namely, for (J • T * Σ) : For reasons which will be clear later we take the pullback of the momentum map (106) and the pullback of the action (105) on 114) In order to obtain the explicit form of the action T * R (l,w) := (T * Σ) −1 • T * R g • T * Σ : T * K × T * N → T * K × T * N we take g = Σ(l, w), where (l, w) ∈ K × N , and use the product formula (k, u)(l, w) := Σ −1 (hg) = (kl, m(k, l)̺(l)(u)w) , where the cocycle m : K × K → N and the map ̺ : K → Aut N are defined by We simplify the problem assuming triviality of the cocycle m : K×K → N , i.e. we consider the case when m : K × K → {e}. Then ̺ : K → Aut N is a group anti-homomorphism and σ : K → H is a group homomorphism. The group product (115) in this case assumes the following form (k, u)(l, w) = (kl, ̺(l)(u)w) = (kl, (R w • ̺(l))(u)), and H is the semidirect product K ⋉ ρ N . The product (118) leads to the formula for the action T * R (l,w) : Taking in (108) k = e and u = e we obtain the isomorphism T Σ e (ξ e , ν e ) := T σ(e)ξ e + T ι(e)ν e (120) of T e K × T e N with T e H and, thus the decomposition Superposing the dual (T Σ e ) * : T * e H → T * e K ×T * e N of (120) with J H •T * Σ : T * K ×T * N → T * e H we find that the factorized momentum map is given by where J K : T * K → T * e K and J N : T * N → T * e N are the momentum maps for K and N , respectively. Combining (119) and (122) we obtain the following equivariance property: for J Σ , where is an anti-homomorphism of K ⋉ ̺ N ∼ = H in Aut(T * e K × T * e N ). Note that J Σ is the momentum map related to the symplectic form d((T * Σ) * γ H ) where (T * Σ) * γ H is the pullback of the canonical one-form γ H of T * H by T * Σ : T * K × T * N → T * H. The subsequent proposition expresses (T * Σ) * γ H in terms of the canonical form γ K of T * K and the connection form α defined in (103).
Proposition 23. One has the following equality where π * K : T * K → K and π * N : T * N → N are the projections on the bundle bases, and pr * K : T * K × T * N → T * K and pr * N : T * K × T * N → T * N are projections on the corresponding components of the manifolds products.
The first term in (125) is the pullback of γ K and the second one, called the "magnetic term", is the pullback of the connection form α (superposed with χ u • T L ι(u) (e) ∈ T * e N ) on the product T * K × T * N . Let us note that (pr * K ) * γ K as a section of T * (T * K × T * N ) is linear on the fibres of T * K and constant on the second factor of T * K × T * N . Similarly, the magnetic term is linear on the fibres of T * N , constant on the fibres of T * K, but not constant on the base K × N of T * K × T * N .
Since J Σ : T * K×T * N → T * e K×T * e N ∼ = T * e H is a Poisson map of (T * K×T * N, d(T * Σ) * γ H ) into T * e K × T * e N with the usual Lie-Poisson bracket {·, ·} T * e H of T * e H, any function f ∈ C ∞ (T * e K × T * e N ) defines a Hamiltonian f • J Σ on T * K × T * N . According to (123), if f is invariant with respect to the action Then f • J Σ ∈ C ∞ (T * K × T * N/N ) ∼ = C ∞ (T * K × T * e N ) defines a Hamiltonian system on the Poisson manifold T * K × T * e N . This is always so if N is an abelian group. In this case the coadjoint representation Ad * : N → Aut T * e N of N is trivial, i. e. Ad * = id. So, its orbits are one-element subsets {a} ⊂ T * e N of T * e N . Thus and from The- The symplectic form ω a of (J • T * Σ) −1 ((0, a)) is obtained as the reduction of d(T * Σ) * γ H . So, from (123) and the diffeomorphism J −1 (0)/N ∼ = T * K we find that where A is a one-form on K defined by the connection one-form α and a ∈ T * e N , and (π * K ) * A is the pullback of A on T * K by π * K : T * K → K. As an example one can present the case of the heavy top, where K = SO(3), N = R 3 and ρ : SO(3) → Aut R 3 is the usual action of SO(3) on R 3 . For detailed description of this case see for example [9]. For the description of the n-dimensional tops in the framework of semi-direct product Hamiltonian mechanics we refer the reader to [11]. One can find many others examples of application of semidirect product Hamiltonian mechanics in [5,6,9] which concern also infinite dimensional physical systems.
The short exact sequence of VB-groupoids (86) in the case when P = H ∼ = K ⋉ ̺ N and G = N assumes the following form: where we used the following diffeomorphisms Let us note that we can consider (134) as a product of the short exact sequence of VB-groupoids (86), taken for P = K and G = {e}, with the short exact sequence of VB-groupoids (86) taken for P = N and G = N . But we have to stress here that the Poisson structures in (134) are not the products of the respective Poisson structures and they depend on the anti-homomorphism ̺ : K → Aut N .
To end, we mention that one can consider the Poisson manifolds which are included in the upper dual Atiyah sequence of (134) as the phase spaces of some composed systems related to the ones presented in the lower dual Atiyah sequence of (134).