Poisson Geometry Related to Atiyah Sequences
We construct and investigate a short exact sequence of Poisson 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 T*P×T*P/G⇉T*P/G. The semidirect product case,...
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| Cite this: | Poisson Geometry Related to Atiyah Sequences / K. Mackenzie, A. Odzijewicz, A. Sliżewska // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | Poisson Geometry Related to Atiyah Sequences / K. Mackenzie, A. Odzijewicz, A. Sliżewska // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. |
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| description | We construct and investigate a short exact sequence of Poisson 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 T*P×T*P/G⇉T*P/G. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 005, 29 pages
Poisson Geometry Related to Atiyah Sequences
Kirill MACKENZIE †, Anatol ODZIJEWICZ ‡ and Aneta SLIŻEWSKA ‡
† School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK
E-mail: K.Mackenzie@sheffield.ac.uk
‡ Institute of Mathematics, University in Bia lystok, Cio lkowskiego 1M, 15-245 Bia lystok, Poland
E-mail: aodzijew@uwb.edu.pl, anetasl@uwb.edu.pl
Received July 05, 2017, in final form January 06, 2018; Published online January 10, 2018
https://doi.org/10.3842/SIGMA.2018.005
Abstract. We construct and investigate a short exact sequence of Poisson VB-groupoids
which is canonically related to the Atiyah sequence of a G-principal bundle P . Our re-
sults include a description of the structure of the symplectic leaves of the Poisson groupoid
T∗P×T∗P
G ⇒ T∗P
G . The semidirect product case, which is important for applications in
Hamiltonian mechanics, is also discussed.
Key words: Atiyah sequence; VB-groupoid; Poisson groupoid; dualization of VB-groupoid
2010 Mathematics Subject Classification: 58H05; 22A22; 53D17
1 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 [6]
or when one formulates the equation of motion of a classical particle in a Yang–Mills field: the
electromagnetic field is a particular case [10, 11].
When the action of G is lifted to TP the quotient manifold TP/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 TP/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 TP/G→M .
There are interesting geometric objects which include TP/G and T ∗P/G as their crucial
components. In this paper we will consider the Atiyah sequence (3.14) and the dual Atiyah
sequence (3.15) as well as the dual pair of Poisson manifolds (3.11). From the dual Atiyah
sequence we also construct a short exact sequence of VB-groupoids (4.6) over the gauge groupoid
P×P
G ⇒ P/G. The concept of VB-groupoid [8] is recalled in Section 2. The short exact sequences
of VB-groupoids which we construct and study (see in particular the diagrams (4.6) and (4.10))
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
mailto:K.Mackenzie@sheffield.ac.uk
mailto:aodzijew@uwb.edu.pl
mailto:anetasl@uwb.edu.pl
https://doi.org/10.3842/SIGMA.2018.005
2 K. Mackenzie, A. Odzijewicz and A. Sliżewska
T ∗G⇒ T ∗eG is the symplectic groupoid of the Lie–Poisson structure of the dual T ∗eG = T ∗G/G
of the Lie algebra TeG of G, see [1]. As we will see, for a general principal bundle P the cotan-
gent groupoid T ∗G ⇒ T ∗eG 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 [5, 8]) 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 Proposition 2.6 and Proposition 2.12, 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 pair of Poisson manifolds
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 3.5.
In Section 4, we extend the Atiyah sequence (3.14) and its dual (3.15) to short exact sequences
of VB-groupoids over the gauge groupoid, see (4.6) and (4.10) respectively. From the point of
view of Poisson geometry the most important structures are described by the diagram (4.10), in
which all objects are linear Poisson bundles and all arrows are their morphisms. Consequently,
using (4.10), 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 [3, 4, 7, 9], are also presented in this section.
2 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 TG 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 [2].
The relations between the groupoid and vector bundle structures on TG were abstracted by
Pradines [8] to the concept of VB-groupoid, and [8] 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 Sections 2.1, 2.2 and 2.3. For further detail, see [5, Chapter 11].
In Section 2.4 we show that, under a natural condition, short exact sequences of VB-groupoids
may be dualized; see Proposition 2.6. 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 2.7). Finally in
Section 2.5 we show that the two processes – taking the quotient and taking the dual – commute
(Proposition 2.12).
2.1 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
Poisson Geometry Related to Atiyah Sequences 3
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 Fig. 1.
Ω
s̃
��
t̃
��
λ̃ // G
t
��
s
��
B
λ // P
Figure 1.
The crucial equation, as described above, is
(ξ1 + ξ2)(η1 + η2) = ξ1η1 + ξ2η2, (2.1)
where ξi ∈ Ω. In order for ξ1 + ξ2 to be defined we must have λ̃(ξ1) = λ̃(ξ2) where λ̃ : Ω→ G is
the bundle projection. Likewise we must have λ̃(η1) = λ̃(η2). For ξ1η1 to be defined we must
have s̃(ξ1) = t̃(η1) and likewise we must have s̃(ξ2) = t̃(η2). 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 (2.1) is defined. Likewise, if λ̃ is a morphism of groupoids,
then the addition on the right hand side will be defined. We now state the formal definition.
Definition 2.1. 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
ξ 7→ (s̃(ξ), λ̃(ξ)), Ω→ B ×P G = {(b, γ) |λ(b) = s(γ)} (2.2)
formed from the bundle projection λ̃ and the source projection s̃, is a surjective submersion.
The need for the final condition will be clear in Section 2.2. The map in (2.2) may be denoted
(s̃, λ̃). Also, we refer to G ⇒ P as the side groupoid and to B → P as the side vector bundle.
The condition that the identity inclusion be a morphism of vector bundles is the first equation
in (2.3) below. It follows that 1̃0p = 0̃1p ; that is, the groupoid identity over a zero element of B
is the zero element over the corresponding identity element of G . Further, 1̃−b = −1̃b for b ∈ B.
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 (2.3),
1̃b1+b2 = 1̃b1 + 1̃b2 , (ξ + η)−1 = ξ−1 + η−1. (2.3)
Next, the zero sections define a morphism of groupoids. To see this, first consider sources. By
the definition, s̃ ◦ 0̃ = 0 ◦ s, since s̃ and s constitute a morphism of vector bundles. But this
can also be read as stating that 0̃ and 0 commute with the sources. In the same way, 0̃ and 0
commute with the targets and the multiplication, and are therefore morphisms of Lie groupoids.
So we have the first two equations below
0̃γ1γ2 = 0̃γ1 0̃γ2 , 0̃γ−1 = 0̃−1
γ , (−η)(−ξ) = −ηξ.
The third equation follows by a similar argument.
4 K. Mackenzie, A. Odzijewicz and A. Sliżewska
2.2 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 λ̃ : Ω → G
with the kernel of the source s̃ : Ω→ B.
Equivalently, K is the preimage under (s̃, λ̃) of the closed submanifold {(0p, 1p) | p ∈ P}.
Since (s̃, λ̃) is assumed to be a surjective submersion, K is a closed submanifold of Ω.
Equivalently again, K is the pullback of ker(λ̃) along the unit map P → G . This equips K
with a natural structure of vector bundle with base P .
Example 2.2. For any Lie groupoid G ⇒ P , the groupoid TG ⇒ TP obtained by applying
the tangent functor to the structure of G , is a VB-groupoid, (TG ;TP,G ;P ). The core is the Lie
algebroid AG .
When G is a Lie group G, the multiplication, denoted •, in the tangent group TG obtained
by applying the tangent functor to the multiplication in G, is given by the explicit formula
Xg • Yh = TLg(h)Yh + TRh(g)Xg, (2.4)
where Lg(h) := gh and Rh(g) := gh. In particular for Xe, Ye ∈ TeG and zero elements 0g ∈ TgG
and 0h ∈ ThG one has
Xe • Ye = Xe + Ye, 0g • 0h = 0gh, 0g •Xe = TLg(e)Xe. (2.5)
The group TG is isomorphic to the semi-direct product G n TeG with respect to the adjoint
action.
For general groupoids TG ⇒ TP is not a groupoid semi-direct product.
2.3 The dual of a VB-groupoid
Since λ̃ : Ω → G is a vector bundle, it has a dual bundle λ̃∗ : 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 [8].
Take Φ ∈ Ω∗γ where γ = λ̃(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〈
t̃∗(Φ), k
〉
=
〈
Φ, k0̃γ
〉
, k ∈ Kq, 〈s̃∗(Φ), k〉 =
〈
Φ,−0̃γk
−1
〉
, k ∈ Kp.
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 s̃∗(Ψ) = t̃∗(Φ). To define ΨΦ ∈ Ω∗δγ we must define
the pairing of ΨΦ with any element of Ωδγ . Now any element ζ of Ωδγ can be written as
a product ηξ where η ∈ Ωδ and ξ ∈ Ωγ . We claim that
〈ΨΦ, ηξ〉 = 〈Ψ, η〉+ 〈Φ, ξ〉 (2.6)
is well defined. Any other choice of η and ξ must be ητ−1 and τξ for some τ with s̃(τ) = s̃(η) =
t̃(ξ). Also, τ must project to 1q ∈ G where q = s(δ), since ητ−1 must lie in the same fibre
over G as η.
Write b = s̃(τ). Then τ − 1̃b is defined and is a core element, say k. Now we have ητ−1 =
η(1̃b + k)−1 = (η + 0̃δ)(1̃b + k−1) = η + 0̃δk
−1 so〈
Ψ, ητ−1
〉
=
〈
Ψ, η〉+ 〈Ψ, 0̃δk−1
〉
.
Poisson Geometry Related to Atiyah Sequences 5
Proceeding in the same way, we likewise find that
〈Φ, τξ〉 = 〈Φ, ξ〉+
〈
Φ, k0̃γ
〉
.
From s̃∗(Ψ) = t̃∗(Φ) we know that −〈Ψ, 0̃δk−1〉 = 〈Φ, k0̃γ〉, so (2.6) is well-defined.
We now define the identity element of Ω∗ at χ ∈ K∗p . To do this we need to pair 1̃χ with
elements of Ω which project to 1p ∈ G . Take such an element ξ and write b = λ̃(ξ). Then ξ− 1̃b
is a core element k and we define〈
1̃χ, 1̃b + k
〉
= 〈χ, k〉.
It is now straightforward to check the proof of the following result.
Proposition 2.3. Given a VB-groupoid (Ω, B,G ;P ), the construction above yields a VB-groupoid
(Ω∗;K∗,G ;P ) with core B∗.
Given ω ∈ B∗p , we identify it with the element ω ∈ Ω∗ defined by〈
ω, 1̃b + k
〉
=
〈
ω, b+ t̃(k)
〉
.
Example 2.4. Given a Lie groupoid G ⇒ P , the dual of the VB-groupoid (TG ; G , TP ;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 [2, 12].
2.4 Short exact sequences of VB-groupoids and their duals
The main constructions of the paper rely on dualizing short exact sequences of VB-groupoids
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 F : Ω→
Ω′, fB : B → B′, fG : G → G ′ and f : P → P ′ such that (F, fG ) and (fB, f) are morphisms of
vector bundles, and (F, fB) and (fG , f) are morphisms of Lie groupoids.
It follows that F restricts to a vector bundle morphism of the cores fK : K → K ′.
For the purposes of this paper we only need to consider the dualization of morphisms which
preserve G ⇒ P .
Proposition 2.5. Let F : Ω1 → Ω2 and fB : B1 → B2, together with the identity maps on G
and P , be a morphism of VB-groupoids (Ω1;B1,G ;P ) → (Ω2;B2,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 Fig. 2, such that the three VB-groupoids have the same side groupoid G ⇒ P and
the morphisms (F, fB) and (H,hB) preserve G and P , and such that Ω1
F→ Ω2
H→ Ω3 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 Ω3, as well as exactness at Ω2.
Proposition 2.6. Consider a short exact sequence of VB-groupoids, as just defined and as shown
in Fig. 2. Then:
6 K. Mackenzie, A. Odzijewicz and A. Sliżewska
Ω1
//
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OOO
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OOO
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OOO
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OOO
O
OOO
OOO
OOO
OOO
OOO
O
B1
//
fB ''OO
OOO
OOO
OOO
OOO
O P
OOO
OOO
OOO
OOO
OOO
OO
OOO
OOO
OOO
OOO
OOO
OO Ω2
//
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O
OOO
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B2
//
hB ''OO
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OOO
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OOO
OOO
OOO
OOO
OOO
OO
OOO
OOO
OOO
OOO
OOO
OO Ω3
//
�� ��
G
�� ��
B3
// P
Figure 2.
(i) B1
fB→ B2
hB→ B3 is a short exact sequence of vector bundles over P ;
(ii) K1
fK→ K2
hK→ K3 is a short exact sequence of vector bundles over P , where Ki is the core
of Ωi, and fK , hK 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 Fig. 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 fB : B1 → B2 is injective. Take b1 ∈ B1 and suppose that fB(b1) = 0.
Then F (1b1) = 1fB(b1) = 10. It is also true that F (10) = 1fB(0) = 10 so since F is injective we
have 1b1 = 10 and therefore b1 = 0.
Secondly we prove that hK : K2 → K3 is surjective. Take k3 ∈ K3. Since K3 ⊆ Ω3 and
H : Ω2 → Ω3 is surjective, there is ξ ∈ Ω2 such that H(ξ) = k3. Now if ξ projects to g ∈ G it
follows that k3 also projects to g; since k3 is a core element we have g = 1m for some m ∈ P .
Now write X ∈ B2 for the source of ξ. We have hB(X) = 0.
The subtraction ξ − 1̃X is defined since both project to 1m in G . And ξ − 1̃X has source
X −X = 0 so is a core element. Now H(ξ − 1̃X) = k3 − 1̃0 and 1̃0 = 0̃1, the zero element in the
fibre over 1m. So ξ − 1̃X is a core element which is mapped by H to k3.
Thirdly we prove exactness at K2. Since fK and hK are restrictions of F and H it follows
from H ◦ F = 0 that hK ◦ fK = 0. Now suppose that hK(k2) = 0 where k2 ∈ K2. By exactness
at Ω2, there exists ξ1 ∈ Ω1 such that F (ξ1) = k2. Write b1 = s̃(ξ1). Then fB(b1) = 0 and
since fB is injective, it follows that b1 = 0. Now λ̃(ξ1) = λ̃(k2) = 1p for some p ∈ P , so ξ1 ∈ K1.
So the sequence of cores is exact at K2.
The other conditions are proved in the same way.
Now apply Proposition 2.5 to F and H as morphisms of VB-groupoids. It follows that F ∗
and H∗ are morphisms of VB-groupoids as shown in Fig. 3.
Since Ω1
F→ Ω2
H→ Ω3 is a short exact sequence of vector bundles over G , the duals form a
short exact sequence Ω∗3
H∗
→ Ω∗2
F ∗
→ Ω∗1, again of vector bundles over G . This completes the proof
of (iii). �
2.5 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 with
projection µ : P → P/G and we need to quotient over the action of G. In this situation the
required quotient manifolds exist, and the constructions are straightfoward. Denote the group
Poisson Geometry Related to Atiyah Sequences 7
Ω∗3
//
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K∗3
//
h∗K
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OOO
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OOO
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//
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K∗2
//
f∗K
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OOO
OOO
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OOO
OOO
OO
OOO
OOO
OOO
OOO
OOO
OO Ω∗1
//
�� ��
G
�� ��
K∗1
// P
Figure 3.
action by κ : P ×G→ P . Recall that the action in a principal bundle is free and that the orbits
are equal to the fibres of µ. We always assume that the bundle is locally trivial.
First consider vector bundles over the total space P of a principal bundle. A PBG-vector
bundle over P is a vector bundle λ : 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〉 and the orbit of p ∈ P by 〈p〉. The projection λ : E/G→ P/G
is defined by λ(〈e〉) = 〈λ(e)〉. Take 〈e〉, 〈e′〉 ∈ E/G such that λ(〈e〉) = λ(〈e′〉). 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 [5, Section 3.1].
Proposition 2.7. Let λ : E → P be a PBG-vector bundle over a principal bundle P with group G
and projection µ : P → P/G. Then the quotient manifold E/G exists, and has a vector bundle
structure with base P/G, such that the natural projection E → E/G is a vector bundle morphism
over µ.
The case E = TP arises in constructing the Atiyah sequence of a principal bundle.
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 [5, Definition 2.5.4].
Denote the orbit of γ ∈ G by 〈γ〉. The source and target maps s : G /G→ P/G and t : G /G→
P/G are defined by s(〈γ〉) = µ(s(γ)) and t(〈γ〉) = µ(t(γ)). Given 〈γ〉 and 〈γ′〉 with s(〈γ〉) =
t(〈γ′〉) there exists a unique g ∈ G such that s(γ)g = t(γ′). We define
〈γ〉 〈γ′〉 = 〈(γg)γ′〉.
That G /G ⇒ P/G is a groupoid is straightforward to check. The remaining details of the
following proposition can be found in [5, Section 3.1].
Proposition 2.8. Let G be a PBG-groupoid over a principal bundle P with group G and pro-
jection µ : P → P/G. Then the quotient manifold G /G exists, and has a Lie groupoid structure
with base P/G, such that the natural projection G → G /G is a morphism of Lie groupoids
over µ.
See [5, Proposition 2.5.5] for the proof. These two constructions can be combined so as to
apply to VB-groupoids.
Definition 2.9. Let P be a principal bundle with group G and projection µ : P → P/G.
A PBG-VB-groupoid over P is a VB-groupoid (Ω;B,G ;P ) together with right actions of G on
8 K. Mackenzie, A. Odzijewicz and A. Sliżewska
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.
Proposition 2.10. Let (Ω;B,G ;P ) be a PBG-VB-groupoid over P , described above, with co-
re K. 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, constitute a morphism of VB-groupoids.
The proof only requires assembling the results of Propositions 2.7 and 2.8.
We now need to establish that this quotienting process commutes with dualization. Con-
sider 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,
〈Φg, ξ〉 =
〈
Φ, ξg−1
〉
, 〈φg, k〉 =
〈
φ, kg−1
〉
,
for Φ ∈ Ωγ , γ ∈ G , ξ ∈ Ωγg, g ∈ G, φ ∈ K∗ and k ∈ K.
Proposition 2.11. With the structures just defined, (Ω∗;K∗,G ;P ) is a PBG-VB-groupoid and
the canonical maps
Ω∗/G→ (Ω/G)∗, K∗/G→ (K/G)∗,
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.
Proposition 2.12. In the short exact sequence shown in Fig. 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 Fig. 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 Ω∗i /G → (Ωi/G)∗ and K∗i /G → (Ki/G)∗ together
with the identities on G /G and P/G, constitute an isomorphism of VB-groupoids over P/G.
3 The dual Atiyah sequence
The Lie algebroid of a gauge groupoid is the Atiyah algebroid of the corresponding principal
bundle P , 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 pair together with
that of Atiyah sequence. The main results are collected in Theorem 3.5. Throughout the section,
P denotes a principal bundle unless otherwise specified.
3.1 Principal bundles
We consider a principal G-bundle P with the notation of Section 2.5. One may take the tangent
of the right action κ : P ×G→ P and obtain the action
Tκ : TP × TG→ TP
of the tangent group TG on the tangent bundle TP .
Poisson Geometry Related to Atiyah Sequences 9
Applying the tangent functor to the multiplication in G we obtain the group structure on TG
given in (2.4), and denoted by •. We see from (2.5) that the zero section 0: G → TG of the
tangent bundle TG is a group monomorphism and one has the decomposition
TG = G • TeG ∼= Gn TeG
of TG as a semi-direct product of G and the normal subgroup TeG ⊆ TG.
We will use the following notations:
κ(p, g) = κp(g) = κg(p) = pg
where κp : G→ P and κg : P → P . Thus we have
Tκp(g) : TgG→ TpgP and Tκg(p) : TpP → TpgP
for g ∈ G. The action Tκ : TP ×TG→ TP of the tangent group TG on the tangent bundle TP
can be expressed by
Tκ(p, g)(vp, Xg) = Tκg(p)vp + Tκp(g)Xg. (3.1)
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 3.1. For g, h ∈ G and p ∈ P ,
κg ◦ κp = κp ◦Rg,
κg ◦ κp = κpg ◦ Ig−1 ,
κpg = κp ◦ Lg,
Tκpg(e) = Tκg(p) ◦ Tκp(e) ◦Adg,
Tκg(p)
−1 = Tκg−1(pg),
Tκgh(p) = Tκh(pg) ◦ Tκg(p),
T ∗κgh(p) = T ∗κh(pg) ◦ T ∗κg(p),
where Ig = Lg ◦Rg−1, Adg = TIg(e) and T ∗κg(p) : T ∗pP → T ∗pgP is defined by
T ∗κg(p) :=
(
Tκg(p)
−1
)∗
.
Write T V P for the vertical subbundle of TP ; that is, T V P = kerTµ, and T V ∗P for its dual.
Proposition 3.2. There are the following isomorphisms of vector bundles:
TP/TeG ∼= TP/T V P, (3.2a)
TP/TG ∼= (TP/TeG)/G, (3.2b)
T (P/G) ∼= TP/TG, (3.2c)
T V P/G ∼= P ×AdG TeG, (3.2d)
T V ∗P/G ∼= P ×Ad∗G T
∗
eG, (3.2e)
where 〈p,X〉 ∈ P ×AdG TeG is defined by 〈p,X〉 := {(pg,Adg−1 X) : g ∈ G}.
Proof. Using the vector space isomorphisms
Tκp(e) : TeG→ T Vp P
10 K. Mackenzie, A. Odzijewicz and A. Sliżewska
and the action of the subgroup TeG ⊆ TG on TP defined by
TpP × TeG 3 (vp, Xe) 7→ vp + Tκp(e)Xe ∈ TpP
we obtain (3.2a).
In order to prove (3.2b) we observe that
Tκg(p)(vp + Tκp(e)Xe) = Tκg(p)vp + (Tκg(p) ◦ Tκp(e))Xe
= Tκg(p)vp + T (κg ◦ κp)(e)Xe = Tκg(p)vp + T (κp ◦Rg)(e)Xe
= Tκg(p)vp + Tκp(g)(TRg(e)Xe). (3.3)
Assuming Xg=TRg(e)Xe we find from (3.3) that the double quotient vector bundle (TP/TeG)/G
is isomorphic to the quotient bundle TP/TG.
Since the fibres of Tµ are orbits of the action (3.1) we obtain the vector bundle isomorphism
mentioned in (3.2c).
Let us define actions φg : TP → TP and φ∗g : T ∗P → T ∗P of g ∈ G as follows
φg(v)(pg) := Tκg(p)v and φ∗g(ϕ)(pg) := T ∗κg(p)ϕ (3.4)
for v ∈ TpP , ϕ ∈ T ∗pP . For the vector bundle trivialization I : P × TeG
∼−→ T V P defined by
I(p,Xe) := Tκp(e)Xe (3.5)
one has
φg ◦ I = I ◦ (κg ×Adg−1), (3.6)
where g ∈ G. Thus the bundle isomorphism I : P × TeG
∼−→ T V P defines the isomorphism
[I] : P ×AdG TeG
∼= (P × TeG)/G
∼−→ T V P/G
of quotient vector bundles presented in (3.2d). Dualizing (3.6) we obtain the isomorphism
(3.2e). �
3.2 The dual pair of Poisson manifolds
We recall that the canonical 1-form γ on T ∗P is defined by
〈γϕ, ξϕ〉 := 〈ϕ, Tπ∗(ϕ)ξϕ〉, (3.7)
where ϕ ∈ T ∗P and ξϕ ∈ Tϕ(T ∗P ). Here π∗ : T ∗P → P is the projection of T ∗P on the base P .
We note that
Tπ∗ ◦ Tφ∗g = φg ◦ Tπ∗ (3.8)
for any g ∈ G. From the definition (3.7) and (3.8) one has
〈γφ∗g(ϕ), Tφ
∗
g(ϕ)ξϕ〉 = 〈φ∗g(ϕ), Tπ∗(φ∗g(ϕ))Tφ∗g(ϕ)ξϕ〉 = 〈φ∗g(ϕ), (φg ◦ Tπ∗(ϕ))ξϕ〉
= 〈T ∗κg(p)ϕ, Tκg(p) ◦ Tπ∗(ϕ)ξϕ〉 = 〈ϕ, Tκg(p)−1◦ Tκg(p) ◦ Tπ∗(ϕ)ξϕ〉
= 〈ϕ, Tπ∗(ϕ)(ξϕ)〉 = 〈γϕ, ξϕ〉,
which means that γ is invariant with respect to the action φ∗g : T ∗P → T ∗P defined in (3.4).
This action generates two maps:
π∗G : T ∗P → T ∗P/G, and J : T ∗P → T ∗eG,
Poisson Geometry Related to Atiyah Sequences 11
where π∗G is the projection to the quotient manifold and J is the G-equivariant
J(φ∗gϕ) = Ad∗g−1 J(ϕ)
momentum map defined by
J(ϕ) := ϕ ◦ Tκp(e), (3.9)
for ϕ ∈ T ∗pP .
One has the canonical Lie–Poisson structure on T ∗eG defined by
πL-P(Tf, Tg)(χ) := 〈χ, [Tf(χ), T g(χ)]〉, (3.10)
where f, g ∈ C∞(T ∗eG) and [·, ·] is the Lie bracket of the Lie algebra TeG.
Since the symplectic form dγ is invariant with respect to the action φ∗g : T ∗P → T ∗P of the
group G, defined in (3.4), 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 ∗pP → T ∗eG is surjective. Thus we see that the momentum map J : T ∗P → T ∗eG is a surjec-
tive submersion. So, one has two surjective Poisson submersions:
T ∗P
T ∗P/G T ∗eG
�
�
�
�
�
�
@
@
@
@
@
@R
π∗G J (3.11)
from the symplectic manifold T ∗P , such that the Poisson subalgebras (π∗G)∗(C∞(T ∗P/G)) and
J∗(C∞(T ∗eG)) commute, see [1]. Therefore, the diagram (3.11) gives a dual pair in the sense of
the definition in [1, Section 9.3].
Let us mention that π∗G is a complete Poisson map; that is, the pullback of every function
on T ∗P/G that has a complete Hamiltonian vector field, also has a complete Hamiltonian vector
field (see [1, Proposition 6.6]). 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 pair (3.11) mentioned above one finds (see [1]) that there is
a one-to-one correspondence between the symplectic leaves
S = π∗G
(
J−1(O)
)
(3.12)
of T ∗P/G and the coadjoint orbits O = J((π∗G)−1(S)) which are the symplectic leaves of T ∗eG.
3.3 The dual Atiyah sequence
Using the vector bundle monomorphism I : P × TeG→ T V P ⊂ TP , defined in (3.5), we obtain
the following exact sequence
P × TeG
I−→ TP
A−→ TP/TeG (3.13)
of vector bundles over P . The map I is given by (3.5) and A is a quotient map defined as
A(vp) := [vp] = vp + Tκp(e)TeG,
where vp ∈ TP .
12 K. Mackenzie, A. Odzijewicz and A. Sliżewska
Quotienting (3.13) over the G-action and using the isomorphisms (3.2b), (3.2c) and (3.2d),
we obtain the Atiyah sequence
P ×AdG TeG TP/G T (P/G)
P/G P/G P/G
?? ?
-
--
-
[pr1] π̃[π]
a
idid
ι
(3.14)
where ι =: [I] and a := [A] is the anchor map, e.g., see [5].
It follows from µ ◦ κg = µ that Tµ : TP → T (P/G) is constant on the orbits of the action
φg : TP → TP , 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 [5, Chapter 3]) that the short exact sequence
T ∗(P/G) T ∗P/G P ×Ad∗
G
T ∗eG
P/G P/G P/G
?? ?
-
--
-
π̃∗ [π∗] [pr1]
ι∗
idid
a∗
(3.15)
dual to the Atiyah sequence (3.14), 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 ∗eG→ P/G is defined as follows. Let π0
be the zero Poisson tensor on P and let πL−P be the Lie Poisson tensor on T ∗eG defined by (3.10).
The product Poisson structure π0 × πL−P defined on P × T ∗eG is invariant with respect to the
action φg × Ad∗g : P × T ∗eG→ P × T ∗eG, g ∈ G. Hence one has on the associated vector bundle
P ×Ad∗
G
T ∗eG → 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 (3.10).
To conclude we mention that a∗ and ι∗ are Poisson maps and preserve the vector bundle
structures in the short exact sequence (3.15).
3.4 Relationship between the 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
diagrams (3.11) and (3.15), respectively. We now discuss the relationship between them. For
this reason, using the isomorphism I : P × TeG → T V P defined in (3.5), we consider the short
exact sequence of vector bundles
0→ T V 0P
A∗
↪→ T ∗P
I∗−→ P × T ∗eG→ 0,
Poisson Geometry Related to Atiyah Sequences 13
dual to (3.13), where the subbundle T V 0P ⊆ T ∗P is the annihilator of T V P in TP ; that is,
T V 0
p P consists of those ϕ ∈ T ∗pP which satisfy
ϕ ◦ Tκp(e) = 0. (3.16)
The vector bundle epimorphism I∗ dual to I is related to the momentum map (3.9) by
I∗(ϕ) = (π∗(ϕ), J(ϕ));
that is, I∗ = π∗ × J and the map ι∗ : T ∗P/G→ P ×Ad∗
G
T ∗eG given by
ι∗ = [I∗],
is the quotient of I∗ over the group G. We see from (3.9) and (3.16) that T V 0P = J−1(0) which
implies that T V 0P is a G-invariant vector subbundle of T ∗P .
It follows from the Marsden–Weinstein symplectic reduction procedure [6], that T V 0P/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 ∗eG. Proposition 3.3 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 [9, Lemma 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 0P/G. However it is interesting to prove this result explicitly.
Proposition 3.3. One has the vector bundle isomorphism
T ∗(P/G) T V 0P/G
P/G P/G
? ?
-
-
π̃ [π∗]
a∗
id
which is also a symplectomorphism.
Proof. In order to describe a∗ in an explicit way we recall that µ ◦ κg = µ and thus
Tµ(pg) ◦ Tκg(p) = Tµ(p). (3.17)
Dualizing (3.17) and noting that Tµ(p)∗ : T ∗µ(p)(P/G)→ T V 0
p P we obtain
T ∗µ(p)(P/G)
T V 0
p P
T V 0
pg P
��
�
��
�*
6
H
HHH
HHj
Tκ∗g(p)
Tµ(p)∗
Tµ(pg)∗
(3.18)
where Tµ(p)∗, Tµ(pg)∗ are monomorphisms of vector spaces and T ∗κg(p) is a vector space
isomorphism, for any p ∈ P and g ∈ G.
From (3.18) we see that a∗ : T ∗(P/G)→ T V 0P/G is given by
T ∗(P/G) 3 ρ 7→ a∗(ρ) = {Tµ(pg)∗ρ : g ∈ G} ∈ T V 0P/G.
14 K. Mackenzie, A. Odzijewicz and A. Sliżewska
Now we will show that the canonical 1-form γ̃ of the cotangent bundle T ∗(P/G) is obtained
as the reduction by a∗ of the canonical form γ defined in (3.7). For this reason we choose a local
trivialization
ψα : µ−1(Ωα)→ Ωα ×G,
where ∪α∈IΩα = M := P/G, of the principal bundle P (M,G). Using local sections sα : Ωα →
µ−1(Ωα), where sα(m) := ψ−1
α (m, e), we define
a∗α(ρ) := Tµ(sα(m))∗ρ
the maps a∗α : (ν∗)−1(Ωα)→ (µ◦π∗)−1(Ωα) which “trivialize ” the vector bundle isomorphism a∗.
Here ν : TM →M and ν∗ : T ∗M →M are the bundle projections.
Note that for p = sα(m) one has
a∗α(ρ) ◦ Tκp(e) = (Tµ(sα(m))∗ρ) ◦ Tκp(e)
= ρ ◦ (Tµ(sα(m)) ◦ Tκp(e)) = ρ ◦ T (µ ◦ κp)(e) = 0. (3.19)
The last equality in (3.19) follows from the fact that the map µ ◦ κp : G→M
(µ ◦ κp)(g) = µ(pg) = µ(p)
is constant on G. Thus we find that a∗α(ρ) ∈ T V 0P .
Now for ρ ∈ (ν∗)−1(Ωα) and ϕ = a∗α(ρ) ∈ (π∗ ◦ µ)−1(Ωα) we have
〈((a∗α)∗γρ, ξρ〉 = 〈γϕ, T a∗α(ρ)ξρ〉 = 〈ϕ, Tπ∗(ϕ)T a∗α(ρ)ξρ〉 = 〈ϕ, T (π∗ ◦ a∗α)(ρ)ξρ〉
= 〈Tµ(sα(ν∗(ρ))∗ρ, T (ν∗ ◦ a∗α)(ρ)ξρ〉 = 〈ρ, Tµ(sα(ν∗(ρ))T (ν∗ ◦ a∗α)(ρ)ξρ〉
= 〈ρ, T (µ ◦ ν∗ ◦ a∗α)(ρ)ξρ〉 = 〈ρ, T (id)(ρ)ξρ〉 = 〈ρ, ξρ〉 = 〈γ̃ρ, ξρ〉, (3.20)
where we used the following equalities
µ ◦ π∗ ◦ a∗α = idΩα and (sα ◦ ν∗)(ρ) = (π∗ ◦ a∗α)(ρ).
From (3.20) we conclude that (a∗α)∗γ = γ̃ on (ν∗)−1(Ωα). The above shows that (a∗α)∗γ =
(a∗t)
∗γ on ν̃−1(Ωα ∩ Ωt). As a consequence we obtain (a∗α)∗γ = γ̃. �
As we have seen, the dual pair (3.11) gives a one-to-one correspondence between the coadjoint
orbits O ⊂ T ∗eG of G and the symplectic leaves S ⊂ T ∗P/G of T ∗P/G given by (3.12). Using
the Atiyah dual sequence (3.15) we can investigate this correspondence in more detail. At first
let us make the following observation:
Proposition 3.4. For any coadjoint orbit O ⊂ T ∗eG, we have
J−1(O)/G = ι∗−1(P ×Ad∗
G
O). (3.21)
Proof. At first we note that
J−1(O) =
⊔
p∈P
(
T ∗pP ∩ J−1(O)
)
= I∗−1(P ×O). (3.22)
Since J and I∗ are G-equivariant maps and ι∗ = [I∗] we obtain (3.21) by quotienting (3.22)
over G. �
Poisson Geometry Related to Atiyah Sequences 15
We observe that the vertical arrows in the diagram
T ∗P/G P ×Ad∗
G
T ∗eG
P/G P/G
? ?
-
-
[π∗] [pr1]
ι∗
id
(3.23)
which is a part of the dual Atiyah sequence (3.15), are the projections on the base of the corre-
sponding vector bundle, whereas the map ι∗ : T ∗P/G → P ×Ad∗
G
T ∗eG is the bundle projection
of an affine bundle over P ×Ad∗
G
T ∗eG.
In order to see that the fibre ι∗−1(〈p, χ〉), where 〈p, χ〉 ∈ [pr1]−1(〈p〉) and (p, χ) ∈ P × T ∗eG,
is an affine space over the vector space T ∗〈p〉(P/G), let us take 〈ϕ1〉, 〈ϕ2〉 ∈ ι∗−1(〈p, χ〉). From
im a∗ = ker ι∗ it follows that there exists a unique ρ ∈ T ∗〈p〉(P/G) such that
〈ϕ1〉 = 〈ϕ2〉+ a∗(ρ). (3.24)
So, the vector space T ∗〈p〉(P/G) acts freely and transitively on ι∗−1(〈p, χ〉). Note that
dim ι∗−1(〈p, χ〉) = dimT ∗〈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 (3.24). The orbits of this action are the fibres of ι∗ : T ∗P/G → P ×Ad∗
G
T ∗G. From
the above and from Proposition 3.4 it follows that all concerning (3.23) is also valid for the
fibration
J−1(O)/G P ×Ad∗
G
O
P/G P/G
? ?
-
-
[π∗] [pr1]
ι∗
id
(3.25)
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 ∗eG→ T ∗P/G of the affine bundle ι∗ : T ∗P/G→ P ×Ad∗
G
T ∗eG we obtain
the vector bundle epimorphism σ̃ : T ∗P/G→ T ∗(P/G) defined by the action (3.24) as follows
a∗(σ̃(〈ϕ〉) = 〈ϕ〉 − σ(〈p, χ〉), (3.26)
where ι∗(〈ϕ〉) = 〈p, χ〉. Note here that a∗ is a monomorphism of vector bundles, thus the
equality (3.26) defines σ̃(〈ϕ〉) ∈ T ∗(P/G) uniquely.
The following theorem summarizes the observations mentioned above .
Theorem 3.5.
(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 ∗eG over the total space of the vector bundle [pr1] : P ×Ad∗
G
T ∗eG→
P/G, i.e., the fibre ι∗−1(〈p, χ〉) of 〈p, χ〉 ∈ P ×Ad∗
G
T ∗eG is an affine space over the vector
space T ∗〈p〉(P/G).
16 K. Mackenzie, A. Odzijewicz and A. Sliżewska
(ii) The symplectic leaf S = J−1(O)/G has the structure of an affine fibre bundle over the total
space of the bundle [pr1] : P ×Ad∗
G
O → P/G with the orbit O ⊂ T ∗eG as a typical fibre.
(iii) Fixing a section σ : P ×Ad∗
G
O → T ∗P/G we could consider the symplectic leaf J−1(O)/G
as a fibre bundle
πσ : J−1(O)/G→ T ∗(P/G) (3.27)
over the cotangent bundle T ∗(P/G) with O as the typical fibre. The total space and the
base of (3.27) are symplectic manifolds. However, the bundle projection πσ is not a Poisson
map in general.
(iv) In the case when O is a one-element orbit O = {χ} (such an orbit corresponds to a char-
acter of G) the bundle map πσ : J−1(χ)/G→ T ∗(P/G) defines a diffeomorphism of man-
ifolds, but it is not a symplectomorphism. The difference ωχ − π∗σdγ̃ of symplectic forms,
where ωχ is the symplectic form of the symplectic leaf J−1(χ)/G and γ̃ is the canonical
symplectic form on T ∗(P/G), and is called the magnetic term, see [10, 11].
If G is a commutative Lie group all the coadjoint orbits are one element sets O = {χ}, where
χ ∈ T ∗eG. Hence we have
J−1(χ)/G = ι∗−1(P ×Ad∗
G
{χ}) = ι∗−1(P/G× {χ})
= {〈ϕp〉 ∈ T ∗P/G : ϕp ◦ Tκp(e) = χ},
where ϕp ∈ T ∗pP and 〈ϕp〉 ∈ [π∗]−1(〈p〉). If ϕp, ϕ̃p ∈ T ∗pP satisfy
ϕp ◦ Tκp(e) = ϕ̃p ◦ Tκp(e) = χ
then 〈ϕp − ϕ̃p〉 ∈ J−1{O}/G ∼= T ∗(P/G). Taking the above facts into account and identifying
P/G× {χ} with P/G we conclude the following
Remark 3.6. If G is a commutative group then the symplectic leaves J−1{χ}/G, χ ∈ T ∗eG,
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 [11]), i.e., such α ∈ Γ∞(T ∗P, TeG) that
αp ◦ Tκp(e) = idTeG (3.28)
and
αpg ◦ Tκg(p) = Adg−1 ◦αp. (3.29)
Conditions (3.28) and (3.29) imply that the bundle epimorphism α̃ : TP → P × TeG defined
on TP by
α̃(vp) := (p, αp(vp)),
where vp ∈ TpP , after quotienting by G gives the map
[α̃] : TP/G→ P ×AdG TeG. (3.30)
The dual of (3.30)
σ := [α̃]∗ : P ×Ad∗
G
T ∗eG→ T ∗P/G
is the section of ι∗ : T ∗P/G→ P ×Ad∗
G
T ∗eG mentioned above.
Poisson Geometry Related to Atiyah Sequences 17
4 Short exact sequences of VB-groupoids over gauge groupoids
Applying the constructions investigated in Section 2, we will construct two short exact sequences,
(4.6) and (4.10), of VB-groupoids over the gauge groupoid P×P
G ⇒ P/G. These are related to
each other by the dualization procedure described in Section 2.4. Applying the results obtained
in Section 3 we investigate the groupoid and Poisson structures of the objects involved (4.10).
4.1 The tangent VB-groupoid of a gauge groupoid
Let us consider the tangent VB-groupoid
T (P × P ) P × P
TP P
?? ??
-
-
T pr2
pr2T pr1
pr1
of the pair groupoid P × P ⇒ P . Since
T pr1×T pr2 : T (P × P )→̃TP × TP
is a vector bundle isomorphism we will identify the tangent VB-groupoid T (P × P ) ⇒ TP with
the pair VB-groupoid TP × TP ⇒ TP . By T V (P × P ) we denote the vertical subbundle of the
tangent bundle T (P × P ) defined by the action
κ2 : (P × P )×G→ (P × P ), ((p, q), g) 7→ (pg, qg), (4.1)
of G on the product P × P . In consequence X ∈ TeG acts on (vp, wq) ∈ TP × TP as follows
(vp, wq) 7→ (vp + Tκp(e)X,wq + Tκq(e)X).
Note that (vp, wq) = (T pr1(p, q)× T pr2(p, q))v(p,q), where v(p,q) ∈ T(p,q)(P × P ), and thus
T V(p,q)(P × P ) := {(Tκp(e)X,Tκq(e)X) ∈ TpP × TqP ; X ∈ TeG}.
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 ) ⇒ TP .
The groupoids mentioned above form a short exact sequence of VB-groupoids as defined in
Section 2.4
T V (P × P ) TP × TP TP×TP
TeG
T V P TP TP/TeG
???? ??
-
--
- A2
A
(4.2)
Here each side groupoid is the pair groupoid P × P ⇒ P , and TP×TP
TeG
⇒ TP
TeG
is the quotient of
the pair groupoid TP × TP ⇒ TP by TeG.
Now we define a VB-groupoid
P × TeG× P P × P
P × TeG P
?? ??
-
-
s̃ pr2t̃ pr1
(4.3)
18 K. Mackenzie, A. Odzijewicz and A. Sliżewska
over the pair groupoid P × P ⇒ P as follows
s̃(p,X, q) := (q,X), t̃(p,X, q) := (p,X), ε̃(p,X) := (p,X, p),
ι̃(p,X, q) := (q,X, p), (p,X, q)(q,X, r) := (p,X, r),
where p, q, r ∈ P and X ∈ TeG, and the horizontal arrows in (4.3) are projections on the suitable
terms of the product manifolds.
Let us define a map I2 : P × TeG× P → T V (P × P ) by
I2(p,X, q) := (Tκp(e)X,Tκq(e)X).
Proposition 4.1. The map I given by (3.5), and the map I2 define an isomorphism
P × TeG× P T V (P × P )
P × TeG T V P
?? ??
-
-
s̃ st̃ t
I2
I
(4.4)
of VB-groupoids over P × P ⇒ P .
We conclude from the above that I and I2 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 (4.4) is equivariant with respect to the action of the group G defined on P × TeG× P by
(p,X, q) 7→ (pg,Adg−1 X, qg)
and on T V (P × P ) by
(Tκp(e)X,Tκq(e)X) 7→ ((Tκg(p) ◦ Tκp(e))X, (Tκg(q) ◦ Tκq(e))X)
= ((Tκpg(e) ◦Adg−1)X, (Tκqg(e) ◦Adg−1)X),
where g ∈ G. From (4.2) and the isomorphism (4.4) we obtain the following short exact sequence
of VB-groupoids
P × TeG× P TP × TP TP×TP
TeG
P × TeG TP TP/TeG
???? ??
-
--
- A2
AI
I2
(4.5)
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
P×TeG×P
G
P×TeG
G
P×P
G
P
G
?? ??
-
-
XXXXXXXz
XXXXXXXXz
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
TP×TP
G
TP/G
P×P
G
P
G
?? ??
-
-
XXXXXXXz
XXXXXXXz
XXXXXXXXX
XXXXXXXXX
XXXXXXXX
XXXXXXXX
T
(
P×P
G
)
T (P/G)
P×P
G
P
G
?? ??
-
-
(4.6)
Poisson Geometry Related to Atiyah Sequences 19
over the gauge groupoid P×P
G ⇒ P/G. The oblique upper and lower arrows in (4.6) 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 (4.6) is the tangent prolongation groupoid of
the gauge groupoid P×P
G ⇒ P/G.
4.2 The cotangent VB-groupoid of a gauge groupoid
Now let us dualize (4.6) in the sense of Section 2.4. Since dualization commutes with the action
of G we start from the dualization of (4.5).
Proposition 4.2. There are the following vector bundle isomorphisms:
(i) core (TP × TP ) ∼= TP ,
(ii) core (T V (P × P )) ∼= core (P × TeG× P ) ∼= P × {0},
(iii) core
(
TP×TP
TeG
) ∼= TP .
Proof. (i) From the definition we see that (vp, wq) ∈ core (TP × TP ) if and only if p = q and
wq = 0. Thus core (TP × TP ) = TP × ({0} × P ) ∼= TP .
(ii) The element (p,X, q) ∈ core (P × TeG × P ) if and only if p = q and X = 0. Thus
core ((P × TeG× P )) = {0} × P .
(iii) An element 〈vp, wq〉 ∈ TP×TP
TeG
is defined by
〈vp, wq〉 := {(vp + Tκp(e)X, wq + Tκq(e)X); X ∈ TeG}. (4.7)
So, 〈vp, wq〉 ∈ core
(
TP×TP
TeG
)
if and only if p = q and wq = Tκq(e)Y for some Y ∈ TeG. Choosing
in (4.7) X = −Y we find that
core
(
TP×TP
TeG
)
= {〈vp, 0p〉; vp ∈ TP} ∼= TP. �
Applying the dualization procedure presented in Section 2.4 and Proposition 4.2 we see that
the short exact VB-groupoid sequence (over P × P ⇒ P ) dual to (4.2) is
T V 0(P × P ) T ∗P × T ∗P P × T ∗eG× P
T ∗P T ∗P P × {0}∗ ∼= P
???? ??
-
--
-A
∗
2 I∗2
π∗id
(4.8)
where the dual A∗2 of A2 is the inclusion map and I∗2 is given by
I∗2 (ϕ,ψ) := (p, ϕp ◦ Tκp(e) + ψq ◦ Tκq(e), q)
for p = π∗(ϕ) and q = π∗(ψ). The map I∗2 : T ∗P ×T ∗P → P ×T ∗eG×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
s(ϕp, ψq) = −ψq, t(ϕp, ψq) = ϕp, ε(ϕp) = (ϕp,−ϕp),
ι(ϕp, ψq) = (−ψq,−ϕp), (ϕp, ψq)(−ψq, λr) = (ϕp, λr), (4.9)
as the source map, target map, identity section, inverse map and groupoid product, respectively.
20 K. Mackenzie, A. Odzijewicz and A. Sliżewska
Remark 4.3. It is easy to see that the involution δ : T ∗P × T ∗P → T ∗P × T ∗P defined by
δ(ϕp, ψq) = (ϕp,−ψq)
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 (4.9) and the symplectic form is d(pr∗1 γ+pr∗2 γ).
The groupoid maps for the groupoid P × T ∗eG× P ⇒ P are defined in the following way
s(p,X , q) := q, t(p,X , q) := p, ε(p) := (p, 0, p), ι(p,X , q) := (q,−X , p),
and the groupoid product of (p,X , q), (q,Y, r) ∈ P × T ∗eG× P is defined as
(p,X , q)(q,Y, r) := (p,X + Y, r).
The second component of the bundle epimorphism I∗2 defines the momentum map
J2(ϕp, ψq) = ϕp ◦ Tκp(e) + ψq ◦ Tκq(e)
for the cotangent bundle T ∗(P × P ) = T ∗P × T ∗P whose symplectic structure is given by the
canonical symplectic form dγ2 = d(pr∗1 γ + pr∗2 γ) and the action of the group G on P × P is
defined in (4.1).
In order to see that J2 : T ∗P ×T ∗P → T ∗eG is a momentum map we note that (P ×P, µ2 : P ×
P → P×P
G , G) is a G-principal bundle and apply the same consideration as for J : T ∗P → T ∗eG
in (3.9).
We note also that the vector bundle
(
TP×TP
TeG
)∗ → P×P is isomorphic to the vector subbundle
T V 0(P × P )→ P × P which contains such (ϕp, ψq) ∈ T ∗pP × T ∗q P that
ϕp ◦ Tκp(e) + ψq ◦ Tκq(e) = 0.
Hence we can identify the dual VB-groupoid
(
TP×TP
TeG
)∗
⇒ T ∗P with the subgroupoid T V 0(P ×
P ) ⇒ T ∗P of the dual VB-groupoid T ∗P × T ∗P ⇒ T ∗P .
For any g ∈ G one has
I∗2 (T ∗κg(p)ϕp, T
∗κg(q)ψq) = (pg,Ad∗g−1(ϕp ◦ Tκg(e) + ψq ◦ Tκg(e)), qg),
where (ϕp, ψq) ∈ T ∗pP × T ∗q P . Thus all horizontal arrows in (4.8) define G-equivariant VB-
groupoid morphism. All groupoid maps and products for the groupoids in (4.8) are also G-
equivariant.
So, according to Proposition 2.12, quotienting (4.8) by G we obtain the short exact sequence
of VB-groupoids over the gauge groupoid P×P
G ⇒ P/G
T ∗
(
P×P
G
)
T ∗P/G
P×P
G
P
G
?? ??
-
-
XXXXXXXz
XXXXXXXXz
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
a∗2
id
T ∗P×T ∗P
G
T ∗P/G
P×P
G
P
G
?? ??
-
-
XXXXXXXz
XXXXXXXz
XXXXXXXXX
XXXXXXXXX
XXXXXXXX
XXXXXXXXι∗2
[π∗]
P×T ∗
eG×P
G
P/G
P×P
G
P
G
?? ??
-
-
(4.10)
which is the dual of the short exact sequence of (4.6) in sense of the dualization procedure
described in Section 2. In particular the first groupoid of (4.10) is the VB-groupoid dual to the
tangent prolongation groupoid of the gauge groupoid P×P
G ⇒ P/G.
Poisson Geometry Related to Atiyah Sequences 21
4.3 Symplectic leaves of duals of Atiyah sequences
The results of Section 3, in particular Theorem 3.5, can be applied to the short exact sequence
T ∗
(
P×P
G
)
T ∗P×T ∗P
G
(P×T ∗
eG×P )
G
P×P
G
P×P
G
P×P
G
?? ?
-
--
- ι∗2
idid
a∗2
(4.11)
of linear Poisson bundles; this is the dual of the Atiyah sequence of the G-principal bundle (P×P ,
µ2 : P ×P → P×P
G , G). Note that (4.11) is part of the VB-groupoid short exact sequence (4.10).
In (4.10) 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 (4.10).
From the considerations in Section 4.2, especially of (4.10), we see that the core of the VB-
groupoid
T ∗
(
P×P
G
)
P×P
G
T ∗P/G P/G
?? ??
-
-
〈s〉 〈pr2〉〈t〉 〈pr1〉
[π∗2]
[π∗] ,
consists of all 〈ϕp, 0p〉 ∈ T ∗P×T ∗P
G such that ϕp ◦ Tκp(e) = 0, where p ∈ P . Thus, using (3.15),
we have the following isomorphisms
core T ∗
(
P×P
G
) 〈t〉−→ J−1(0)/G and T ∗(P/G)
a∗
−→ J−1(0)/G
of vector bundles over P/G. In particular, the core of T ∗(P×PG ) 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
T ∗(P/G) ∗ T ∗P/G 3 (ρ〈p〉, 〈ϕp〉)→ a∗(ρ〈p〉) + 〈ϕp〉 ∈ T ∗P/G. (4.12)
Recall here that a∗(ρ〈p〉), 〈ϕp〉 ∈ [π∗]−1(〈p〉). The moment map for the action (4.12) is [π∗] :
T ∗P/G→ P/G, i.e., [π∗](〈ϕp〉) = 〈p〉 ∈ P/G and
T ∗(P/G) ∗ T ∗P/G = {(ρ, 〈ϕ〉) ∈ T ∗(P/G)× T ∗P/G; ν∗(ρ) = [π∗](〈ϕ〉)},
i.e., (ρ, 〈ϕ〉) ∈ T ∗(P/G) ∗ T ∗P/G if and only if 〈q〉 = 〈p〉.
The action groupoid of (4.12) is T ∗(P/G)<7 [π∗]T
∗P/G⇒ T ∗P/G and according to [5, Propo-
sition 11.2.3], valid for a general VB-groupoid, one has the short exact sequence of Lie groupoids
T ∗(P/G)<7 [π∗] T
∗P/G T ∗
(
P×P
G
)
P×P
G
T ∗P/G T ∗P/G P/G
???? ??
-
--
-
id
(4.13)
There are other crucial statements which we collect in the following proposition:
22 K. Mackenzie, A. Odzijewicz and A. Sliżewska
Proposition 4.4.
(i) The groupoid T ∗
(
P×P
G
)
⇒ T ∗P
G is the symplectic groupoid of the Poisson manifold T ∗P
G .
(ii) 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(O)/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 (4.13), any symplectic leaf J−1(O)/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(O)/G→ P ×Ad∗
G
O.
Here (i) is a particular case of the theory introduced in [12].
The symplectic leaves J−1
2 (O)/G of T ∗P×T ∗P
G ⇒ T ∗P
G may be included in a sequence of
morphisms of fibre bundles, shown in (4.14), similarly to (3.25)
T ∗
(
P×P
G
)
J−1
2 (O)/G (P×O×P )
G
P×P
G
P×P
G
P×P
G
?? ?
-
--
- ι∗2
idid
a∗2
(4.14)
So, Proposition 3.5 is valid in this case as well as in the case of symplectic leaves J−1(O)/G,
where O ∈ T ∗eG. From (4.10) we have
T ∗
(
P×P
G
)
J−1
2 (O)/G (P×O×P )
G
T ∗P/G T ∗P/G P/G
???? ??
-
--
- ι∗2
[π∗]id
a∗2
(4.15)
Note that in diagram (4.15) only T ∗
(
P×P
G
)
⇒ T ∗P
G is a groupoid. The target and source maps
of (T ∗P × T ∗P )/G restricted to J−1
2 (O)/G define symplectic and anti-symplectic realizations
of T ∗P/G, respectively. (In general these realizations are not full.) Thereby, we have the family
of symplectic (anti-symplectic) realizations of T ∗P/G, parametrized by the coadjoint orbits
O ⊂ T ∗eG.
Proposition 4.5. The action of the symplectic groupoid T ∗
(
P×P
G
)
⇒ T ∗P
G on the symplectic
realization t : J−1
2 (O)/G→ T ∗P
G is a symplectic action; that is, the graph of the action
T ∗
(
P×P
G
)
×T ∗P/G J
−1
2 (O)/G→ J−1
2 (O)/G (4.16)
is Lagrangian and t : J−1
2 (O)/G → T ∗P
G is a Poisson map which is equivariant with respect to
this action and the natural action of T ∗
(
P×P
G
)
on its base. Corresponding results with signs
reversed hold for the action of T ∗
(
P×P
G
)
⇒ T ∗P
G on s : J−1
2 (O)/G→ T ∗P
G .
Proof. First consider the multiplication
T ∗P×T ∗P
G ×T ∗P/G
T ∗P×T ∗P
G → T ∗P×T ∗P
G (4.17)
Poisson Geometry Related to Atiyah Sequences 23
in the groupoid T ∗P×T ∗P
G ⇒ T ∗P/G. This is the quotient over G of the groupoid T ∗P ×
T ∗P ⇒ T ∗P (with the structure from (4.9)), which is a symplectic groupoid with respect
to the form d(pr∗1 γ + pr∗2 γ), where γ is the canonical 1-form on T ∗P ; since G preserves the
symplectic structure, T ∗P×T ∗P
G ⇒ T ∗P/G is a Poisson groupoid. In particular, the graph of the
multiplication (4.17) is coisotropic in G × G × G where we write G = T ∗P×T ∗P
G for brevity.
The map (4.16) is a restriction of the multiplication (4.17). By (4.11), T ∗
(
P×P
G
)
is (ι∗2)−1(0)
and J−1
2 (O)/G is (ι∗2)−1((P ×O × P )/G). Since ι∗2 is a morphism, it follows that the groupoid
multiplication of an element of T ∗
(
P×P
G
)
by an element of J−1
2 (O)/G is again an element
of J−1
2 (O)/G.
Further, since J−1
2 (O)/G is a symplectic leaf in T ∗P×T ∗P
G , and T ∗
(
P×P
G
)
is a symplectic sub-
manifold of G , the restriction of (4.17) again has coisotropic graph, and by counting dimensions,
is Lagrangian. Thus (4.16) is a symplectic action. �
5 Applications in Hamiltonian mechanics
of semidirect products
In this section we investigate a few special cases of the results obtained in the two previous
sections. We also indicate applications in the theory of Hamiltonian systems.
Let us begin with the simplest case when P = G. Then the dual Atiyah sequence (3.15)
reduces to the well-known isomorphism of Poisson manifolds T ∗G/G ∼= T ∗eG. The interest
of the situation becomes clear if one considers the reciprocally dual short exact sequences of
VB-groupoid presented in diagrams (4.6) and (4.10).
Setting P = G in (4.6) we obtain the following short exact sequence of VB-groupoids over
the group(oid) G⇒ {e}:
TG
TeG
G
{e}
?? ??
-
-
XXXXXXXz
XXXXXXXXz
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
ι2
id
TG×TG
G
TeG
G
{e}
?? ??
-
-
XXXXXXXz
XXXXXXXz
XXXXXXXXX
XXXXXXXXX
XXXXXXXX
XXXXXXXXa2
[π]
TG
{e}
G
{e}
?? ??
-
-
(5.1)
The central VB-groupoid in (5.1) is the gauge groupoid of the principal bundle
(TG,G, µ : TG→ TG/G ∼= TeG).
Note that the group G acts on the tangent bundle TG by the tangent lift of the right action
of G on itself. The final VB-groupoid of the short exact sequence (5.1) is the tangent group
TG ⇒ {e} of the group G. The upper short exact sequence in (5.1) is the Atiyah sequence of
the principal G-bundle(
G×G,G, µ : G×G→ G×G
G
∼= G
)
.
Remark 5.1. The initial VB-groupoid in (5.1) has core zero and is a special case of the initial
VB-groupoid in (4.6). Note that there is no natural groupoid structure TG ⇒ AG for a general
Lie groupoid G .
24 K. Mackenzie, A. Odzijewicz and A. Sliżewska
The dualization of (5.1), according to (4.10), leads to the dual sequence of VB-groupoids
T ∗G
T ∗eG
G
{e}
?? ??
-
-
XXXXXXXz
XXXXXXXXz
XXXXXXXX
XXXXXXXX
XXXXXXXX
XXXXXXXX
a∗2
id
T ∗G×T ∗G
G
T ∗eG
G
{e}
?? ??
-
-
XXXXXXXz
XXXXXXXz
XXXXXXXXX
XXXXXXXXX
XXXXXXXX
XXXXXXXXι∗2
[π∗]
T ∗G
{e}
G
{e}
?? ??
-
-
in which ι∗2 and a∗2 are morphisms of Poisson manifolds.
We now consider the case in which the total space P is a Lie group H and G = N ⊂ H is
a normal subgroup of H. This includes the case just treated. So, K := H/N is a group and one
has the short exact sequence of Lie groups
{e} → N
ι
↪→ H
µ→ K → {e}.
We further assume that there is a global section σ : K → H of the principal bundle µ : H → K;
we do not require σ to be a morphism of groups. Without loss of generality we can assume
that σ(eK) = eH , where eK and eH are the neutral elements of K and H, respectively. Since
σ : K → H is transversal to the fibres of µ : H → K, we define an isomorphism of principal
bundles Σ: K ×N → H (trivialization of µ : H → K) as follows
Σ(k, u) := σ(k)ι(u). (5.2)
From (5.2) one obtains the decomposition
vh = T (Rι(u) ◦ σ)(k)ξk + T (Lσ(k) ◦ ι)(u)νu (5.3)
of vh ∈ ThH on (νu, ξk) ∈ TkK × TuN which defines the connection Γ: TK → TH by
Γ(k)(ξk) := T (Rι(u) ◦ σ)(k)ξk
and the connection form α ∈ Γ∞(TH, TeN) by
α(vh) = T (ι−1 ◦ Lh−1)(h) ◦
(
idThH − T (Rι(u) ◦ σ ◦ µ)
)
(h)vh, (5.4)
where u = σ(µ(h))−1h.
Let us note here that T (Rι(u)◦σ◦µ)(h) : ThH → ThH is a projection of ThH on the horizontal
subspace T (Rι(u) ◦ σ)(k)(TkK) and (idThH − T (Rι(u) ◦ σ ◦ µ)(h)) : ThH → ThH is a projection
of ThH on the vertical subspace T (Lσ(k) ◦ ι)(TuN) of the connection Γ: TK → TH.
Dualizing Γ: TK → TH we obtain Γ∗ : T ∗H → T ∗K which for ϕh ∈ T ∗hH is given by
Γ∗(ϕh) = ϕh ◦ T (Rh ◦ σ)(k), (5.5)
where h = σ(k)ι(u).
Note that the cotangent lift T ∗Rg : T ∗hH → T ∗hgH of the right action Rg : H → H is given by
T ∗Rg(ϕh) = ϕh ◦ (TRg(h))−1 = ϕh ◦ TRg−1(hg) (5.6)
Poisson Geometry Related to Atiyah Sequences 25
and the corresponding momentum map JH : T ∗H → T ∗eH is given by
JH(ϕh) = ϕh ◦ TLh(e) (5.7)
and satisfies JH ◦ T ∗Rg = Ad∗g−1 ◦JH for g ∈ H.
Since for u ∈ N one has Γ∗◦T ∗Rι(u) = Γ∗ the map (5.5) defines the map [Γ∗] : T ∗H/N → T ∗K
which is not necessarily a Poisson map.
Using (5.3) we obtain the explicit formula of the map TΣ: TK × TN → TH
TΣ(ξk, νu) := T (Rι(u) ◦ σ)(k)ξk + T (Lσ(k) ◦ ι)(u)νu, (5.8)
where ξk ∈ TkK and νu ∈ TuN , tangent to the trivialization map Σ: K × N → H defined
in (5.2). Since T (Lσ(k) ◦ ι)(u)νu ∈ kerTµ(h) and µ ◦ Rι(u) ◦ σ = idK we find that the inverse
of TΣ is given by
(TΣ)−1(vh) =
(
Tµ(h)vh, T
(
ι−1 ◦ Lh−1
)
(h) ◦ (idThH − T (Rι(u) ◦ σ ◦ µ))(h)vh
)
,
where k = µ(h), ι(u) = σ(µ(h))−1h.
The diffeomorphism T ∗Σ := ((TΣ)−1)∗ : T ∗K ×T ∗N → T ∗H dual to (TΣ)−1 takes the form
T ∗Σ(θk, χu) = θk ◦ Tµ(h) + χu ◦ T
(
ι−1 ◦ Lh−1
)
(h) ◦
(
idThH − T (Rι(u) ◦ σ ◦ µ)(h)
)
on (θk, χu) ∈ T ∗kK×T ∗uN . 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
J(ϕh) = ϕh ◦ TLh(e) ◦ Tι(e),
(J ◦ T ∗Σ)(θk, χu) = χu ◦ T (Lι(u) ◦ ι)(e).
We take the pullback of the momentum map (5.7) and the pullback of the action (5.6) on
T ∗K × T ∗N . For (JH ◦ T ∗Σ): T ∗K × T ∗N → T ∗eH, where h = σ(k)ι(u), one has
(JH ◦ T ∗Σ)(θk, χu) = θk ◦ T (µ ◦ Lh)(e)
+ χu ◦ T (ι−1 ◦ Lh−1)(h) ◦
(
idThH − T (Rι(u) ◦ σ ◦ µ)
)
(h) ◦ TLh(e).
In order to obtain the explicit form of the action
T ∗R(l,w) := (T ∗Σ)−1 ◦ T ∗Rg ◦ 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), (5.9)
where the cocycle m : K ×K → N and the map % : K → AutN are defined by
m(k, l) := σ(kl)−1σ(k)σ(l) and %(l)(u) := σ(l)−1uσ(l).
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 → AutN is a group anti-homomorphism and
σ : K → H is a group homomorphism. The group product (5.9) in this case assumes the
following form
(k, u)(l, w) = (kl, %(l)(u)w) = (kl, (Rw ◦ %(l))(u)), (5.10)
26 K. Mackenzie, A. Odzijewicz and A. Sliżewska
and H is the semidirect product K nρ N . The product (5.10) leads to the formula
T ∗R(l,w)(θk, χu) =
(
θk ◦ TRl−1(kl), χu ◦ T
[
(Rι(w) ◦ %(l) ◦ ι)−1
]
(u)
)
(5.11)
for the action T ∗R(l,w) : T ∗kK × T ∗uN → T ∗klK × T ∗(Rι(w)◦%(l)◦ι)(u)N .
Taking in (5.8) k = e and u = e we obtain the isomorphism
TΣe(ξe, νe) := Tσ(e)ξe + Tι(e)νe (5.12)
of TeK × TeN with TeH and, thus the decomposition
TeH = Tσ(e)(TeK)⊕ Tι(e)(TeN).
Superposing the dual (TΣe)
∗ : T ∗eH → T ∗eK×T ∗eN of (5.12) with JH ◦T ∗Σ: T ∗K×T ∗N → T ∗eH
we find that the factorized momentum map
JΣ := (TΣe)
∗ ◦ JH ◦ T ∗Σ: T ∗K × T ∗N → T ∗eK × T ∗eN
is given by
JΣ(θk, νu) = (θk ◦ TLk(e), χu ◦ T (Lι(u) ◦ ι)(e))
= (JK(θk), (J ◦ T ∗Σ)(θk, χu)) = (JK(θk), JN (χu)), (5.13)
where JK : T ∗K → T ∗eK and JN : T ∗N → T ∗eN are the momentum maps for K and N , respec-
tively. Combining (5.11) and (5.13) we obtain the following equivariance property
JΣ ◦ T ∗R(l,w) =
(
Ad∗l−1 ×Ad∗w−1 ◦
(
T%
(
l−1
)
(e)
)∗) ◦ JΣ, (5.14)
for JΣ, where
K n% N 3 (l, w) 7→
(
Ad∗l−1 ×Ad∗w−1 ◦
(
T%
(
l−1
)
(e)
)∗) ∈ Aut(T ∗eK × T ∗eN)
is an anti-homomorphism of K n% N ∼= H in Aut(T ∗eK × T ∗eN).
Note that JΣ is the momentum map related to the symplectic form d((T ∗Σ)∗γH) where
(T ∗Σ)∗γH is the pullback of the canonical 1-form γH of T ∗H by T ∗Σ: T ∗K×T ∗N → T ∗H. The
subsequent proposition expresses (T ∗Σ)∗γH in terms of the canonical form γK of T ∗K and the
connection form α defined in (5.4).
Proposition 5.2. One has the following equality
((T ∗Σ)∗γH)(θk, χu) = (pr∗K)∗γK(θk)
+ χu ◦ TLι(u)(e) ◦ (Σ ◦ [(π∗K ◦ pr∗K)× (π∗N ◦ pr∗N )])∗α, (5.15)
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 the canonical projections.
Proof. For X(θk,χu) ∈ T(θk,χu)(T
∗K × T ∗N) we have
〈((T ∗Σ)∗γH)(θK , χu), X(θk,χu)〉 = 〈(γH((T ∗Σ)(θk, χu)), T (T ∗Σ)(θk, χu)X(θk,χu)〉
= 〈((T ∗Σ)(θk, χu), Tπ∗((T ∗Σ)(θk, χu))T (T ∗Σ)(θk, χu)X(θk,χu)〉
= 〈((T ∗Σ)(θk, χu), T (π∗ ◦ T ∗Σ)(θk, χu))X(θk,χu)〉
= 〈θk ◦ Tµ(h) + χu ◦ TLσ(k)−1(h)
◦ (idThH − T (Rι(u) ◦ σ ◦ µ)(h)), T (π∗ ◦ T ∗Σ)(θk, χu))X(θk,χu)〉
Poisson Geometry Related to Atiyah Sequences 27
= 〈θk, T (µ ◦ π∗ ◦ T ∗Σ)(θk, χu)X(θk,χu)〉
+ 〈χu, TLι(u)(e) ◦ α ◦ T (π∗ ◦ T ∗Σ)(θk, χu)X(θk,χu)〉, (5.16)
where for the last equality we used (5.4). Now we observe that
π∗ ◦ T ∗Σ = Σ ◦ ((π∗K ◦ pr∗K)× (π∗N ◦ pr∗N )) (5.17)
and
ν ◦ π∗ ◦ T ∗Σ = π∗K ◦ pr∗K . (5.18)
Substituting (5.17) and (5.18) into (5.16) we obtain
〈((T ∗Σ)∗γH)(θK , χu), X(θk,χu)〉 = 〈θk, T (π∗K ◦ pr∗K)(θk, χu)X(θk,χu)〉
+ 〈χu, TLι(u)(e) ◦ α ◦ T (Σ ◦ ((π∗K ◦ pr∗K)× (π∗N ◦ pr∗N )))(θk, χu)X(θk,χu)〉
= 〈(pr∗K)∗γK , X(θk,χu)〉+ 〈χu ◦ TLι(u)(e) ◦ α ◦ T (Σ ◦ ((π∗K ◦ pr∗K)
× (π∗N ◦ pr∗N )))∗(θk, χu), X(θk,χu)〉,
which gives (5.15). �
The first term in (5.15) is the pullback of γK and the second one, called the “magnetic term”,
is the pullback of the connection form α (superposed with χu ◦TLι(u)(e) ∈ T ∗eN) on the product
T ∗K × T ∗N . Let us note that (pr∗K)∗γK 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 .
Using the diffeomorphism T ∗Σ: T ∗K × T ∗N
∼→ T ∗H we can write the dual Atiyah se-
quence (3.15) for P = H = K nρ N and G = N as follows
0 −→ T ∗K
a∗
−→ T ∗K × T ∗eN
ι∗−→ K × T ∗eN −→ 0,
where a∗(θk) = (θk, 0) and ι∗(θk, χl) = (k, χl). Hence the symplectic leaves of T ∗K × T ∗eN are
T ∗K ×O ∼= (J ◦ T ∗Σ)−1(O)/N = ι∗−1(K ×O)/N,
where O ⊂ T ∗eN are coadjoint orbits of N . One obtains the symplectic structure of the leaf
T ∗K ×O by the reduction to ι∗−1(K ×O) ⊂ T ∗K × T ∗N of d((T ∗Σ)∗γH) presented in (5.15).
Since JΣ : T ∗K×T ∗N → T ∗eK×T ∗eN ∼= T ∗eH is a Poisson map, where T ∗eH is equipped with
the linear Poisson structure, any function f ∈ C∞(T ∗eK × T ∗eN) defines a Hamiltonian f ◦ JΣ
on T ∗K × T ∗N . According to (5.14), if f is invariant with respect to the action
{e} ×N 3 (e, w) 7→
(
id×Ad−1
w−1
)
∈ Aut(T ∗K × T ∗eN).
Then f ◦ JΣ ∈ C∞(T ∗K × T ∗N/N) ∼= C∞(T ∗K × T ∗eN) defines a Hamiltonian system on the
Poisson manifold T ∗K × T ∗eN . This is always so if N is an abelian group.
In this case the coadjoint representation Ad∗ : N → AutT ∗eN of N is trivial, i.e., Ad∗ = id.
So, its orbits are one-element subsets {a} ⊂ T ∗eN of T ∗eN . It follows from Theorem 3.5 that
[Γ∗] : T ∗H/N → T ∗K defines diffeomorphisms [Γ∗] : J−1(a)/N → T ∗K, see also (3.27), of the
symplectic leaves J−1(a)/N , a ∈ T ∗eN , with the cotangent bundle T ∗K.
From point (ii) of Theorem 3.5 we see that J−1(a)/N → H×Ad∗
N
{a} ∼= K is an affine bundle
over K. The diffeomorphism [Γ∗] : J−1(a)/N
∼→ T ∗K preserves the fibre bundle structures, i.e.,
it covers the identity map of the base K. Let us note here that
J−1(a)/N ∼= (J ◦ T ∗Σ)−1((0, a))/N ∼= T ∗K × J−1
N (a)/N ∼= T ∗K × {0} ∼= T ∗K.
28 K. Mackenzie, A. Odzijewicz and A. Sliżewska
The symplectic form ωa of (J ◦ T ∗Σ)−1((0, a)) is obtained as the reduction of d(T ∗Σ)∗γH . So,
from (5.14) and the diffeomorphism J−1(0)/N ∼= T ∗K we find that
ωa = d(γK + (π∗K)∗A),
where A is a 1-form on K defined by the connection 1-form α and a ∈ T ∗eN , 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 = R3 and
ρ : SO(3) → AutR3 is the usual action of SO(3) on R3. For detailed description of this case
see for example [7]. For the description of n-dimensional tops in the framework of semi-direct
product Hamiltonian mechanics we refer the reader to [9]. One can find many other examples of
application of semidirect product Hamiltonian mechanics in [3, 4, 7] which concern also infinite
dimensional physical systems.
The short exact sequence of VB-groupoids (4.10) in the case when P = H ∼= K n% N and
G = N assumes the following form
T ∗(K ×N ×K)
T ∗K × T ∗
eN
K ×N ×K
K
?? ??
-
-
XXXXXXXz
XXXXXXXXz
XXXXXXX
XXXXXXX
XXXXXXXX
XXXXXXXX
a∗2
id
T ∗K× T∗N×T∗N
N
×T ∗K
T ∗K × T ∗
eN
K ×N ×K
K
?? ??
-
-
XXXXXXXz
XXXXXXXz
XXXXXXXXX
XXXXXXXXX
XXXXXXXX
XXXXXXXXι∗2 K×T ∗N×K
K
K ×N ×K
K
?? ??
-
-
(5.19)
where we used the following diffeomorphisms
K n% N/N ∼= K,
(K n% N)× (K n% N)
N
∼= K ×N ×K,
T ∗(K n% N)/N ∼= T ∗K × T ∗eN,
T ∗((K n% N)× (K n% N))
N
∼= T ∗K × T ∗N × T ∗N
N
× T ∗K,
(K n% N)× T ∗eN × (K n% N))
N
∼= K × T ∗N ×K.
Let us note that we can consider (5.19) as a product of the short exact sequence of VB-
groupoids (4.10), taken for P = K and G = {e}, with the short exact sequence of VB-groupoids
(4.10) taken for P = N and G = N . But we have to stress here that the Poisson structures
in (5.19) are not the products of the respective Poisson structures and they depend on the
anti-homomorphism % : K → AutN .
To end, we mention that one can consider the Poisson manifolds which are included in the
upper dual Atiyah sequence of (5.19) as the phase spaces of some composed systems related to
the ones presented in the lower dual Atiyah sequence of (5.19).
Acknowledgements
We extend our best thanks to the referees, who caught many slips and obscurities.
Poisson Geometry Related to Atiyah Sequences 29
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https://doi.org/10.1016/j.aam.2008.06.002
https://arxiv.org/abs/0903.4294
https://doi.org/10.1006/aima.1998.1721
https://arxiv.org/abs/chao-dyn/9801015
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.1017/CBO9781107325883
https://doi.org/10.1016/0034-4877(74)90021-4
https://doi.org/10.2307/1999527
https://doi.org/10.2307/1999527
https://doi.org/10.1007/978-3-662-06796-3_7
https://doi.org/10.1007/BF00400169
https://doi.org/10.1090/S0273-0979-1987-15473-5
1 Introduction
2 Short exact sequences of VB-groupoids
2.1 VB-groupoids
2.2 The core of a VB-groupoid
2.3 The dual of a VB-groupoid
2.4 Short exact sequences of VB-groupoids and their duals
2.5 Quotients of VB-groupoids over group actions
3 The dual Atiyah sequence
3.1 Principal bundles
3.2 The dual pair of Poisson manifolds
3.3 The dual Atiyah sequence
3.4 Relationship between the dual pair and the dual Atiyah sequence
4 Short exact sequences of VB-groupoids over gauge groupoids
4.1 The tangent VB-groupoid of a gauge groupoid
4.2 The cotangent VB-groupoid of a gauge groupoid
4.3 Symplectic leaves of duals of Atiyah sequences
5 Applications in Hamiltonian mechanics of semidirect products
References
|
| id | nasplib_isofts_kiev_ua-123456789-209459 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:14:19Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mackenzie, K. Odzijewicz, A. Sliżewska, A. 2025-11-21T19:14:21Z 2018 Poisson Geometry Related to Atiyah Sequences / K. Mackenzie, A. Odzijewicz, A. Sliżewska // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 12 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58H05; 22A22; 53D17 arXiv: 1506.03216 https://nasplib.isofts.kiev.ua/handle/123456789/209459 https://doi.org/10.3842/SIGMA.2018.005 We construct and investigate a short exact sequence of Poisson 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 T*P×T*P/G⇉T*P/G. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed. We extend our best thanks to the referees, who caught many slips and obscurities. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Poisson Geometry Related to Atiyah Sequences Article published earlier |
| spellingShingle | Poisson Geometry Related to Atiyah Sequences Mackenzie, K. Odzijewicz, A. Sliżewska, A. |
| title | Poisson Geometry Related to Atiyah Sequences |
| title_full | Poisson Geometry Related to Atiyah Sequences |
| title_fullStr | Poisson Geometry Related to Atiyah Sequences |
| title_full_unstemmed | Poisson Geometry Related to Atiyah Sequences |
| title_short | Poisson Geometry Related to Atiyah Sequences |
| title_sort | poisson geometry related to atiyah sequences |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209459 |
| work_keys_str_mv | AT mackenziek poissongeometryrelatedtoatiyahsequences AT odzijewicza poissongeometryrelatedtoatiyahsequences AT slizewskaa poissongeometryrelatedtoatiyahsequences |