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Quasi-Hamiltonian Groupoids and Multiplicative Manin Pairs
LI-BLAND, David, SEVERA, Pavol
Abstract
We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.
LI-BLAND, David, SEVERA, Pavol. Quasi-Hamiltonian Groupoids and Multiplicative Manin Pairs.
International Mathematics Research Notices , 2010
DOI : 10.1093/imrn/rnq170
Available at:
http://archive-ouverte.unige.ch/unige:12054
Disclaimer: layout of this document may differ from the published version.
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MANIN PAIRS
DAVID LI-BLAND AND PAVOL ˇSEVERA
Abstract. We reformulate notions from the theory of quasi-Poissong-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids.
We then interpret this result within the theory of Dirac morphisms and multi- plicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.
Contents
Introduction 2
Overview 3
Acknowledgements 3
1. Background and statement of results 3
1.1. Quasi-Poissong-manifolds 3
1.2. Hamiltonian quasi-Poissong-manifolds 6
1.3. Fusion 7
1.4. Hamiltonian quasi-Poissong-groupoids 10
1.5. Main results 10
2. Quasi-Poisson structures and graded Poisson geometry 12
2.1. g-differential algebras 12
2.2. The quadratic graded Lie algebraQ(g) 13
2.3. Quasi-Poissong-manifolds revisited 13
2.4. Hamiltonian quasi-Poissong-manifolds revisited 15 2.5. Hamiltonian quasi-Poissong-groupoids revisited 16
2.6. The Lie bialgebra ˆgis quasi-triangular 17
2.7. Quasi-Poissong-bialgebroids revisited 18
2.8. Proof of Theorem 5 19
2.9. Proof of Theorem 4 20
3. Examples 20
3.1. Quasi-symplectic case 20
3.2. The double 21
3.3. The groupG 21
3.4. Fused double 21
3.5. Actions with coisotropic stabilizers 22
4. Courant algebroids and Manin pairs 23
4.1. Definition of a Manin pair 23
4.2. Morphisms of Manin pairs 24
4.3. Multiplicative Manin pairs 25
5. Manin pairs and MP-manifolds 26
1
arXiv:0911.2179v4 [math.DG] 4 Aug 2010
5.1. Multiplicative Manin pairs and MP-groupoids 28
5.2. MP-algebroids 29
5.3. MP groups 30
6. Manin Pairs and quasi-Poisson structures 31
6.1. Reinterpretation of§2 in terms of MP-manifolds 31
6.2. Alternative proof of Theorem 4 33
7. Hamiltonian quasi-Poissong-groups 34
Appendix A. VB-groupoids 35
References 36
Introduction
LetGdenote a Lie group whose Lie algebragis equipped with an invariant inner product. Quasi-HamiltonianG-manifolds were introduced in [4], where they were shown to be equivalent to the theory of infinite dimensional Hamiltonian loop group spaces. In particular, they were used to simplify the study of the symplectic struc- ture on the moduli space of flat connections by using finite dimensional techniques.
In [2], the more general quasi-Poissong-manifolds were introduced. These notions were further generalized and studied in subsequent papers [1, 3, 10–14, 40, 41, 63] (to cite a few).
The main objective of this paper is to prove that the following three facts hold for an arbitrary quasi-Poissong-manifold,M:
qP-1 T∗M inherits a Lie algebroid structure.
qP-2 Each leaf of the corresponding foliation of M inherits a quasi-symplectic g-structure.
qP-3 If the Lie algebroid T∗M integrates to a Lie groupoid Γ ⇒ M, then Γ inherits a quasi-Hamiltonian g-structure. Moreover, the source map s : Γ →M is a quasi-Poisson morphism, while the target map t : Γ→M is anti-quasi-Poisson.
Besides completing the theory of quasi-Poisson manifolds, our result can provide a new angle towards integrating certain Poisson structures. In addition to this, it has applications towards the integration of certain Courant algebroids. We plan to explore these consequences in forthcoming papers.
The methods we use to prove the results (qP-1,2, and 3) are of independent inter- est. We reformulate notions from the theory of quasi-Poissong-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups. Then we prove the re- sults (qP-1,2, and 3) using well known theorems established for Poisson manifolds and Poisson Lie groups. In particular, we prove (qP-3) by interpreting structures in terms of Lie algebroid/groupoid morphisms, as was done in [9, 33, 39, 40]. As a result, we are able to avoid infinite dimensional path spaces. We hope our ap- proach will provide the reader with a fresh and insightful perspective on the theory of quasi-Poissong-manifolds.
The methods we use to derive the results (qP 1,2 and 3) are an application of the more general theory of MP-groupoids. We use the latter half of our paper to describe the theory of MP-groupoids. MP-groupoids are a reinterpretation of multiplicative Manin pairs [39] in terms of graded Poisson geometry [14, 47]. In
this portion of the paper, we also describe the link between our approach to quasi- Poissong-manifolds and the standard approach in terms of Dirac morphisms and multiplicative Manin pairs [1, 10, 13, 14, 39].
Overview. Our paper is organized as follows. In§1, we briefly summarize some background material and introduce our main result, the integration of quasi-Poisson g-manifolds. In§2, we provide a new perspective on the theory of quasi-Poissong- manifolds using the theory graded Poisson geometry and graded Poisson-Lie groups.
We then use this new perspective to prove the results from §1. Next, in §3, we provide some detailed examples of the integration of quasi-Poissong-manifolds.
The remainder of the paper is spent relating our approach to the theory of Manin pairs. In§4 we review the definitions of Courant algebroids, Manin pairs, their morphisms and the category of multiplicative Manin pairs. Following this, in §5 we recall the relationship between the categories of Manin pairs and graded Poisson manifolds. Using this relationship, we introduce the infinitesimal notion corresponding to a multiplicative Manin pair. We apply these concepts in §6 to relate the content of§2 to the theory of Manin pairs.
Acknowledgements. We would like to thank Eckhard Meinrenken for his advice and suggestions. D.L.-B. was supported by an NSERC CGS-D Grant, and thanks the Universit´e de Gen`eve, and Anton Alekseev in particular, for their hospitality during his visit. P. ˇS. was supported by the Swiss National Science Foundation (grant 200020-120042 and 200020-126817). Finally, we would like to thank the referees for their helpful comments.
1. Background and statement of results
In this section we want to recall the theory of quasi-Poisson g-manifolds. To provide some intuition and motivate the definitions, we will develop both the theory of quasi-Poisson g-manifolds and the theory of Poisson manifolds in parallel. We will describe the Poisson case in a series of remarks.
In§ 2 we will use a graded version of the theory of Poisson manifolds and Lie bialgebra actions to prove some results about quasi-Poissong-manifolds. In partic- ular, for§2 we will require some understanding of quasi-triangular Lie bialgebras.
We intend to summarize this background material in this section.
1.1. Quasi-Poissong-manifolds. Letgbe a Lie algebra with a chosen ad-invariant element s ∈ S2g. Define s] : g∗ → g by β(s](α)) = s(α, β) for α, β ∈ g∗. Let φ∈ ∧3gbe given by
φ(α, β, γ) =1
2α([s]β, s]γ]) (α, β, γ∈g∗).
Definition 1. [2, 3] A quasi-Poissong-manifold is triple (M, ρ, π), where
• M is a g-manifold,
• ρ:M ×g→T M is the anchor map for the action Lie algebroid, and
• π∈Γ(∧2T M)g is ag-invariant bivector field satisfying
(1) [π, π] =ρ(φ), and [π, ρ(ξ)] = 0,
for anyξ∈g. Here we viewφandξas constant sections ofM × ∧gand extendρ to a morphismρ:M× ∧g→ ∧T M. The bracket in (1) is the Schouten bracket on multivector fields.
Let (Mi, ρi, πi) be two quasi-Poissong-manifolds (fori= 1,2). A mapf :M1→ M2is a quasi-Poisson morphism ifπ2=f∗π1andρ2=f∗◦ρ1.
Example 1. Suppose that ρis an action of gon M. Forx∈ M let gx = ker(ρx) be the stabilizer at the point x, andg⊥x ⊂g∗ be its annihilator. Thenρ(φ) = 0 if s](g⊥x)⊂gx, in which case we say the stabilizers are coisotropic (with respect to s). In this case, the triple (M, ρ,0) is a quasi-Poissong-manifold.
Remark 1 (Poisson Parallel). We now recall the related notion in Poisson geometry.
Lethbe a Lie bialgebra with cobracketδ:h→ ∧2h. A Poissonh-manifold is a triple (N, ρ, π), where
• N is ah-manifold,
• ρ:M ×h→T N is the anchor map for the action Lie algebroid, and
• π∈Γ(∧2T N) is a bivector field, satisfying
(2) [π, π] = 0, and [π, ρ(ξ)] =ρ(δ(ξ)) for everyξ∈h, which we view as a constant section ofM×h.
IfH denotes the Poisson-Lie group corresponding to the Lie bialgebrah, and if ρ can be integrated to an action ofH on N (or we work with local actions), the actionH×N →N becomes a Poisson map [30, 31].
When h = 0, then we may refer to a Poisson h-manifold (N, ρ = 0, π) as a Poisson manifold (N, π).
Remark 2. Recall that the Lie bracket ongextends to a Gerstenhaber bracket on Vg. Suppose there is an elementu∈V2
gsuch that [u, u] =−φ. Thengbecomes a Lie bialgebra with cobracket δ:ξ→[u, ξ]. Lie bialgebras of this type are called quasi-triangular [7, 24] and the combinationr=s+u∈g⊗gis called aclassical r-matrix.
In this case, quasi-Poissong-manifolds and Poisson g-manifolds are equivalent via a “twist” by u, as shown in [2]. To recall the details, if (M, ρ, π) is a quasi- Poisson g-manifold then π0 = π+ρ(u) satisfies [π0, π0] = 0, i.e. π0 is a Poisson structure. Moreover the action ofgviaρbecomes an action of the Lie bialgebrag on the Poisson manifold (M, π0).
Note that in§2 we will use a graded version of this equivalence.
For simplicity, we shallrestrict from now onto the case wheresis non-degenerate.
We shall identifygwithg∗vias, and leth·,·idenote the correspondingad-invariant inner-product on g. Such a Lie algebra is called quadratic. Let ei and ei denote two bases ofgdual with respect toh·,·i.
Remark 3. Suppose (M, ρ,0) is as in Example 1,sis non-degenerate, andg=f⊕h wherefandhare Lagrangian subalgebras. We can choose bases{fi}offand{hi} of hsuch thathfi, hii= 1 andhfi, hji= 0 fori 6=j. Then with u:=P
ifi∧hi, r=s+u∈g⊗gis a classical r-matrix. As in Remark 2,π0 :=π+ρ(u) defines a Poisson structure onM.
The Poisson structureπ0 was constructed in [28] via a Courant algebroid struc- ture on M ×g. The approach via quasi-Poissong-manifolds appears to be more direct.
Recall that Lie algebroid structure on a vector bundleA→M is equivalent to a differential on the algebraΓ(∧∗A∗) [15, 52].
Theorem 1. If (M, ρ, π)is a quasi-Poisson g-manifold, thenT∗M becomes a Lie algebroid, where the Lie algebroid differentialdT∗M on Γ(∧∗T M) is
(3) dT∗M = [π,·] +1
2 X
i
ρ(ei)∧[ρ(ei),·]
Furthermore, the induced action ofgonT∗M preserves the Lie algebroid structure.
Let prg : M ×g→ g be the projection to the second factor, and ρ∗ : T∗M → M×g∗∼=M×gthe transpose ofρ, thenµρ:= prg◦ρ∗:T∗M →gis a Lie algebroid morphism.
This can be proven by a direct calculation: the two parts ofdT∗M commute with each other and their squares cancel each other. We give a conceptual proof of the theorem in§2.3.
The corresponding Lie bracket on 1-formsα, β∈Ω1(M) is [α, β] = [α, β]π+1
2 X
i
α ρ(ei)
Lρ(ei)β−β ρ(ei) Lρ(ei)α
,
where
(4) [α, β]π=dπ(α, β) +ιπ](α)dβ−ιπ](β)dα
is the Koszul bracket (hereβ(π](α)) =π(α, β)). The anchor map,a:T∗M →T M, is
(5) a=π]+1
2ρ◦ρ∗.
We call a quasi-Poisson g-manifold integrable if the Lie algebroid structure on the cotangent bundle is integrable to a Lie groupoid.
Remark 4 (Poisson Parallel). If (N, π) is a Poisson manifold, there is a Lie al- gebroid structure on T∗N whose corresponding Lie algebroid differential dT∗N : Γ(∧nT N)→Γ(∧n+1T N) is
dT∗N = [π,·].
In this case, the anchor mapa:T∗M →T M, is given by
(6) a=π],
and the bracket is given by (4).
If a Lie bialgebrahacts onN, thenµρ:= prh∗◦ρ∗:T∗N →h∗is a Lie algebroid morphism, where prh∗ : M ×h∗ → h∗ is the projection to the second factor and ρ∗:T∗M →M ×h∗ is the transpose ofρ.
Remark 5. It was shown in [10] that whenever (M, ρ, π) is a quasi-Poisson g- manifold, there is a Lie algebroid structure on A =T∗M ⊕g. The Lie algebroid T∗M described in Theorem 1 is embedded as a subalgebroid of T∗M ⊕g by the mapα→α+ρ∗(α).
The following was pointed out by a referee: Suppose that (M, ρ, π) also possesses a moment map Φ :M →G(see Definition 3). Letθdenote the (left) Maurer-Cartan form on G, and let η = 12h[θ, θ], θi. In [10], H. Bursztyn and M. Crainic describe
a Φ∗η-twisted Dirac structure L ⊂ T M ⊕T∗M associated to the quasi-Poisson g-structure onM. In [10, Proposition 3.19] they describe a map
A→L⊂T M⊕T∗M,
which restricts to T∗M ∼= Grρ∗ ⊂ A to define an isomorphism of Lie algebroids T∗M ∼=L.
In particular, [10] shows that in the presence of a moment map the Lie algebroid described in Theorem 1 can be viewed as a Φ∗η-twisted Dirac structure.
Remark 6. The foliation of the quasi-Poissong-manifold (M, ρ, π) given by the Lie algebroidT∗M isdifferent from the foliation given in [2, 3, 10]. The latter foliation is tangent toρx(g) +πx]Tx∗M at any point x∈M; in particular, the leaves contain theg-orbits. This is not the case for the foliation given by the Lie algebroidT∗M; for instance, if as in Example 1, (M, ρ,0) is a quasi-Poissong-manifold, the anchor mapa:T∗M →T M is trivial (ρx◦ρ∗x= 0 for any pointx∈M, since the stabilizers ofρare coisotropic), while theg-orbits may not be.
On the other hand, as we shall see below, for a Hamiltonian quasi-Poisson g- manifold these two foliations coincide.
Definition 2. Aquasi-symplecticg-manifold is a quasi-Poissong-manifold (M, π, ρ) such that the anchor map (5) is bijective.
Remark 7 (Poisson Parallel). A Poisson manifold (N, π) is calledsymplectic if the corresponding anchor map (6) is bijective.
1.2. Hamiltonian quasi-Poisson g-manifolds. There is a concept of a group valued moment map for quasi-Poissong-manifolds [2, 3]. LetGbe a Lie group with Lie algebrag. For anyξ∈gletξL, ξR∈Γ(T G) denote the corresponding left and right invariant vector fields. LetθL, θR∈Ω1(G,g) denote the left and right Maurer Cartan forms onGdefined by
θL(ξL) =ξ, θR(ξR) =ξ.
Definition 3. [2, 3] A map Φ : M →G is called a moment map for the quasi- Poissong-manifold (M, ρ, π) if
• Φ isg-equivariant, and
• π](Φ∗(α)) =ρ(Φ∗(b∗α)) for anyα∈Ω1(G), where the vector bundle mapb:G×g→T Gis given by
(7) b: (g, ξ)→ 1
2 ξL(g) +ξR(g) .
Under these conditions, we call the quadruple (M, ρ, π,Φ) aHamiltonian quasi- Poissong-manifold, or a Hamiltonian quasi-Poisson g-structure onM.
Theorem 2. If (M, ρ, π,Φ) is a Hamiltonian quasi-Poissong-manifold, then the map
(8) i:g→Ω1(M), i(ξ) = Φ∗hξ, θLi is a morphism of Lie algebras such that a◦i=ρ.
Again this can be proved by a direct calculation, and we give a conceptual proof in§2.4.
Remark 8 (Poisson Parallel). In§2, we will need the corresponding notion of group valued moment maps for Poisson geometry [30], summarized as follows.
Let again hbe a Lie bialgebra, and let H∗ denote the 1-connected Poisson Lie group integratingh∗. Recall [30] that a Poisson map
Φ :N →H∗ gives rise to a morphism of Lie algebras
i:h→Ω1(N), i(ξ) = Φ∗(ξL)
where ξL is the left-invariant 1-form on H∗ equal to ξ at the group unit. The morphism i then in turn produces an actionρ of hon N via the anchor map on T∗N, i.e.
ρ(ξ) =π](i(ξ)).
The map Φ is automaticallyh-equivariant, where the so-called dressing action ofh onH∗comes from the identity moment mapH∗ →H∗. Φ is called a moment map for the actionρ.
Remark 9. Whens∈(S2g)g is not assumed to be non-degenerate, one can proceed as follows. The elementsis equivalent to a triple (d,g,g0), where
• dis a quadratic Lie algebra,
• g⊂d is a Lagrangian subalgebra,
• g0⊂dan ideal such thatd=g⊕g0 as vector spaces, and
the restriction tog0 of the inner product ondiss(where we identifyg0 withg∗via the inner product ind).
Moment maps then have value in a groupG0 integratingg0.
Theorem 3. If (M, ρ, π,Φ)is a Hamiltonian quasi-Poissong-manifold then ρx(g) +πx](Tx∗M) =a(Tx∗M).
Proof. Recall thata =π]+12ρ◦ρ∗, hence ρx(g) +πx](Tx∗M)⊇a(Tx∗M). On the other hand, by Theorem 2 we havea◦i=ρ. Hence,
π]=a−1
2ρ◦ρ∗=a◦(id−1 2i◦ρ∗), andπ]x(Tx∗M)⊆ax(Tx∗M).
Remark10. A Hamiltonian quasi-Poissong-manifold (M, ρ, π,Φ) is aquasi-Hamiltonian g-manifold [1–4, 10], if for every pointx∈M,
(9) ρx(g) +πx](Tx∗M) =TxM.
It follows from Theorem 3 that a quasi-Hamiltoniang-manifold is equivalent to a Hamiltonian quasi-symplecticg-manifold. We will use the latter term in this paper.
1.3. Fusion. The category of quasi-Poissong-manifolds, has a braided monoidal structure given by fusion [3]. Let (M, ρ, π) be a quasi-Poissong⊕g-manifold,
(10) ψ=1
2 X
i
(ei,0)∧(0, ei)∈ ∧2(g⊕g),
and let diag(g) ∼= g denote the diagonal subalgebra of g⊕g. The quasi-Poisson g-manifold
(11) (M, ρ|diag(g), πfusion), withπfusion=π+ρ(ψ),
is called the fusion ofM [3]. Moreover, if (Φ1,Φ2) :M →G×Gis a moment map, then the pointwise product Φ1Φ2:M →Gis a moment map for the fusion [3].
If (Mi, ρi, πi,Φi) (for i = 1,2) are two Hamiltonian quasi-Poisson g-manifolds, then (M1×M2, ρ1×ρ2, π1+π2,Φ1×Φ2) is a quasi-Poisson g⊕g-manifold. The fusion
(M1×M2,(ρ1×ρ2)|diag(g),(π1+π2)fusion,Φ1Φ2) is called the fusion product, and denoted
(M1, ρ1, π1)~(M2, ρ2, π2), or justM1~M2.
The fusion product of two quasi-Poissong-manifolds is defined similarly, one just ignores the moment maps.
The monoidal category of Hamiltonian quasi-PoissonG-manifolds is braided [3]:
if (Mi, ρi, πi,Φi) are two Hamiltonian quasi-PoissonG-manifolds, the corresponding isomorphism between fusion products
M1~M2→M2~M1
is given by (x1, x2)7→(Φ1(x1)·x2, x1).
To provide an alternate explanation for this monoidal structure in§ 2, we will need to understand the story for Poisson manifolds.
Remark 11 (Poisson Parallel). If (M, πM) and (N, πN) are two Poisson manifolds, then (M ×N, πM +πN) is also a Poisson manifold. If M → H∗, N → H∗ are Poisson (moment) maps, we can compose
M×N →H∗×H∗→H∗
(the latter arrow is the product inH∗) to get a Poisson map M ×N →H∗. The category of those Poisson manifolds with a Poisson map toH∗is thus monoidal (but not necessarily braided). Notice, that unlike the case of quasi-Poisson manifolds, the resulting action of hon M ×N is not just the diagonal action – it is twisted by the moment map onM. On the other hand, the Poisson bivector is simply the sumπM1+πM2.
If (M1, π1) and (M2, π2) are Poisson manifolds, then the Lie algebroid structure onT∗(M1×M2) is the direct sum of the Lie algebroidsT∗M1 andT∗M2.
For quasi-Poissong-manifolds, one may ask how the Lie algebroidsT∗(M1~M2) andT∗M1⊕T∗M2 are related. A direct computation shows
dT∗(M1~M2)=dT∗M1+dT∗M2+X
i
ρ1(ei)∧[ρ2(ei),·],
so that the obvious isomorphism of vector spacesT∗(M1~M2)∼=T∗M1⊕T∗M2
is not an isomorphism of Lie algebroids. However, for Hamiltonian quasi-Poisson manifolds, there is a non-standard isomorphism T∗(M1×M2) ∼= T∗M1⊕T∗M2
which is an isomorphism of Lie algebroids.
By acomorphism [32] from a Lie algebroidA→M to a Lie algebroidA0 →M0 we mean a morphism of Gerstenhaber algebrasΓ(V
A0)→Γ(V
A). The assignment M 7→T∗M is a functor from the category of quasi-Poissong-manifolds and quasi- Poisson morphisms to the category of Lie algebroids and comorphisms.
Proposition 1. Let(Mi, ρi, πi,Φi) (i= 1,2) be two Hamiltonian quasi-Poissong- manifolds and let the isomorphism J:T∗M1⊕T∗M2→T∗(M1~M2)be given by (α, β)7→(α, β−i2(ρ∗1(α))). ThenJ is a isomorphism of Lie algebroids. Moreover,J is a natural transformation which makes the functorM 7→T∗M (from the category of Hamiltonian quasi-Poisson g-manifolds to the category of Lie algebroids and comorphisms) strongly monoidal.
The proof is in § 2.6, but it requires a deeper understanding of the relation- ship between the monoidal structures for the categories of quasi-triangular Poisson manifolds and quasi-Poissong-manifolds, which we shall now recall.
Let h be a quasi-triangular Lie-bialgebra. As described in Remark 1, h corre- sponds to a Manin triple (d,h,h∗). Leth0 ⊂d=h⊕h∗be the graph of ther-matrix.
Thenh0is an ideal, so that (d,h,h0) is as in Remark 9. LetH, H∗, H0⊂Dbe groups with Lie algebras h,h∗,h0 ⊂d. Suppose further that the maps H∗ → D/H and H0→D/H are bijections.
Suppose (M, ρ, π,Φ) is a quasi-Poisson H-manifold with H-action ρ : H × M → M, bivector π, and with moment map Φ : M → D/G ∼= H0. Define Fh(M, ρ, π,Φ) := (M, ρ, π0,Φ) to be the PoissonH-manifold with bivectorπ0 given by twisting π as in in Remark 2, the sameH-action, and the same moment map Φ :M →D/G∼=H∗. Then as shown in [2, 11, 14, 41], the functor
Fh: Ham-qPoish→Ham-Poish
describes an equivalence between the category Ham-qPoish of quasi-Poisson H- manifolds with H0-valued moment maps and the category Ham-Poish of Poisson H-manifolds withH∗-valued moment maps.
There is a canonical choice for the inverse functor,Fh−1(M, ρ, π0,Φ) := (M, ρ, π,Φ), whereπis constructed fromπ0 by reversing the procedure in Remark 2.
Remark 12. This equivalence can be understood more intrinsically. What is signif- icant is that there is a natural Dirac structure living overD/G,
(d×D/G,g×D/G)
(see Example 7 for more details). It was shown in [11, 14, 41] that a morphism of Manin Pairs
(Φ, K) : (TM, T M)99K(d×D/G,g×D/G)
is the intrinsic data underlying both (M, ρ, π,Φ) and (M, ρ, π0,Φ). Indeed, the actionρ is specified by the morphism of Manin pairs, and the bivectorsπ and π0 arise from choosing two different Lagrangian complements togind[2, 41].
Remark 13. As shown by A. Weinstein and P. Xu [61], when the Lie-bialgebra h is quasi-triangular, the category of Poisson H-manifolds withH∗-valued moment maps is braided monoidal. In fact,Fhis a strong monoidal functor with the natural transformation given by
J :Fh(M1~M2)→Fh(M1)×Fh(M2), (x1, x2)7→(x1, j(Φ1(x1))·x2), where Φ1: M1 →D/G∼=H∗ is the moment map forM1 and j :H∗ →H is the map specified by the condition
g j(g)∈H0 for every g∈H∗.
(To definej:H∗→H, we used the fact that the mapsH∗→D/HandH0→D/H are bijections.)
We shall need a graded version of this fact in the proof of Proposition 1 in§2.6.
1.4. Hamiltonian quasi-Poisson g-groupoids. Let Γ⇒M be a groupoid, and let GrmultΓ = {(g, h, g·h)} ⊂ Γ×Γ×Γ denote the graph of the multiplication map. A bivector fieldπΓ∈Γ(∧2TΓ) is said to bemultiplicative[40] if GrmultΓ is a coisotropic submanifold of (Γ, πΓ)×(Γ, πΓ)×(Γ,−πΓ).
Definition 4. Suppose that Γ⇒M is a groupoid, and (Γ, ρ, πΓ,Φ) is a Hamilton- ian quasi-Poisson g-manifold. It is called aHamiltonian quasi-Poisson g-groupoid if
• Φ : Γ→Gis a morphism of groupoids,
• gacts on Γ by (infinitesimal) groupoid automorphisms, and
• GrmultΓ is coisotropic with respect to the bivector field ((πΓ+πΓ)fusion)1,2−(πΓ)3,
where ((πΓ+πΓ)fusion)1,2 appears on the first two factors of Γ×Γ×Γ and (πΓ)3 appears on the third.
We refer to the last condition asπbeingfusion multiplicative.
A Hamiltonian quasi-symplectic g-groupoid is a Hamiltonian quasi-Poisson g- groupoid such that the anchor map (5) is bijective.
A Hamiltonian quasi-Poisson g-groupoid is called source 1-connected if Γ is source 1-connected andGis 1-connected.
Remark 14 (Poisson Parallel). If (N, π) is a Poisson manifold, and the Lie algebroid T∗N integrates to a (possibly local) Lie groupoid Γ⇒N, then [59]
• there is a bivector fieldπΓ∈Γ(∧2TΓ) such that (Γ, πΓ) is a Poisson mani- fold,
• πΓ is non-degenerate (so that Γ is in fact symplectic).
• πΓ is multiplicative, so that (Γ, πΓ) is a Poisson groupoid [60] (in fact, a symplectic groupoid).
Suppose, in addition, that a Lie bialgebrahacts onN. We can interpret the action as a Lie bialgebroid morphism T∗N → h∗, which then integrates to a Poisson groupoid morphism Γ→H∗ (see [62]).
1.5. Main results. We may now state the first of our main results
Theorem 4. There is a one-to-one correspondence between source 1-connected Hamiltonian quasi-symplecticg-groupoids(Γ, ρΓ, πΓ,Φ)and integrable quasi-Poisson g-manifolds (M, ρ, π). Under this correspondence, the Lie algebroid of Γ is T∗M and Φ integrates the Lie algebroid morphism µρ : T∗M → g. Furthermore, the source maps: Γ→M is a quasi-Poisson morphism, while the target mapt: Γ→M is anti-quasi-Poisson.
Remark 15.Theorem 4 was already established for the case where the quasi-Poisson g-manifold (M, ρ, π) possess a moment map Φ :M →G. This fact was pointed out to us by a referee, and we explain it in Remark 32 after recalling some background.
Theorem 4 prompts one to ask what a general Hamiltonian quasi-Poisson g- groupoid corresponds to infinitesimally. To answer this, we extend the notion of a quasi-Poissong-manifold, by replacing the tangent bundle with an arbitrary Lie algebroid.
Definition 5 (quasi-Poisson g-bialgebroid). A quasi-Poisson g-bialgebroid over a g-manifoldM is a triple (A, ρ,D) consisting of
• a Lie algebroidA→M,
• a Lie algebroid morphism ρ:M ×g→A, where M ×gis the action Lie algebroid, and
• a degree +1 derivationDof the Gerstenhaber algebraΓ(VA), such that
• Dρ(ξ) = 0 for any constant sectionξΓ(M×g), and
• D2=12[ρ(φ),·] , where we view φas a constant section ofM×g.
Let prg :M ×g→gdenote the projection to the second factor, andρ∗:A∗→ M×g∗∼=M×gbe the transpose ofρ. Define µρ:A∗→gbyµρ:= prg◦ρ∗. Remark 16. Quasi-Poisson g-bialgebroids are examples of Lie quasi-bialgebroids, a notion introduced in [42, 44] by D. Roytenberg (see also [25]). A Lie quasi- bialgebroid is a triple (A,D, χ), where A is a Lie algebroid, D is a degree +1 derivation of the Gerstenhaber algebraΓ(∧A), andχ∈Γ(∧3A). They must satisfy the equationsD2= 12[χ,·] andDχ= 0.
Therefore, if (A, ρ,D) is a quasi-Poissong-bialgebroid, then (A,D, ρ(φ)) is a Lie quasi-bialgebroid.
Example 2 (quasi-Poissong-manifolds). Suppose that (M, ρ, π) is a quasi-Poisson g-manifold. Let
Dπ= [π,·]Schouten
be the derivation of Γ(∧∗T M) given by the Schouten bracket. Then (T M, ρ,Dπ) is a quasi-Poissong-bialgebroid.
Proposition 2. LetA→M be a Lie algebroid with anchor mapaA:A→T M. A compatible quasi-Poissong-bialgebroid structure(A, ρ,D)defines a canonical quasi- Poissong-structure(M,aA◦ρ, πD)on M via
πD](df) =aA◦ Df,
where we view the function f ∈C∞(M)as an element of Γ(∧0A).
The result is just a special case of [40, Proposition 4.8], proven for general Lie quasi-bialgebroids.
We refer to (M,aA◦ρ, πD) as theinduced quasi-Poissong-structure.
Remark 17. As a converse to Example 2, suppose that (T M, ρ,D) is a quasi-Poisson g-bialgebroid. Then by [43, Lemma 2.2.], D = [π,·] for a unique bivector field π∈Γ(∧2T M). SinceπD](df) =aA◦ Df, it follows thatπ=πD.
Consequently (T M, ρ,D) is of the form given in Example 2 for the quasi-Poisson g-structure (M, ρ, πD).
Proposition 3. If (A, ρ,D) is a quasi-Poisson g-bialgebroid, then A∗ becomes a Lie algebroid, where the Lie algebroid differentialdA∗ on Γ(∧A)is
(12) dA∗=D+1
2 X
i
ρ(ei)∧[ρ(ei),·]A.
Furthermore, the action ofgon A∗ preserves the Lie algebroid structure, andµρ : A∗→gis a Lie algebroid morphism, where µρ:= prg◦ρ∗.
A proof of this is given in§2.7.
A quasi-Poissong-bialgebroid will be called integrable ifA∗ is an integrable Lie algebroid. In particular, (A, ρ,D) may be integrable even ifAis not.
We can now state our second theorem.
Theorem 5. There is a one-to-one correspondence between source 1-connected Hamiltonian quasi-Poissong-groupoids,(Γ, ρΓ, πΓ,Φ), and integrable quasi-Poisson g-bialgebroids(A, ρ,D). Under this correspondence, the Lie algebroid ofΓisA∗and Φintegrates the Lie algebroid morphismµρ:A∗→g.
We also have:
Proposition 4. Suppose(Γ, ρΓ, πΓ,Φ)is a Hamiltonian quasi-Poissong-groupoid corresponding to the quasi-Poisson g-bialgebroids (A, ρ,D). Then the source map s: Γ→M is a quasi-Poisson morphism onto the induced quasi-Poissong-structure (M,a◦ρ, πD)described in Proposition 2. Meanwhile the target map t: Γ→M is anti-quasi-Poisson.
We will provide a proof of both theorems in the next section using graded Poisson-Lie groups.
2. Quasi-Poisson structures and graded Poisson geometry In this section we make use of graded geometry (super geometry) to prove the results from §1. Some good references for super geometry are [17, 26, 53–55]. For the additional structure of graded manifolds one may look at [34, 43, 48, 56].
2.1. g-differential algebras. Letgbe a Lie algebra, and ˆg=g[1]⊕g⊕R[−1] be the graded Lie algebra with bracket given by
[Iξ, Iη] = 0
[Lξ, Iη] =I[ξ,η]g [Lξ, Lη] =L[ξ,η]g
[D, Iη] =Lη [D, Lη] = 0 [D, D] = 0
Here D is the generator of R[−1], and Lξ ∈ g⊂ˆg and Iξ ∈g[1]⊂ ˆgdenote the elements corresponding toξ∈g.
Ifρ:g→Γ(T M) is a morphism of Lie algebras, then the graded Lie algebra ˆg acts on the graded algebra Ω(M) by derivations. Dacts by the de Rham differential d,Lξ acts by the Lie derivativeLρ(ξ), andIξ acts by the interior productιρ(ξ).
Generally, a graded algebra with a graded action of ˆgby derivations is called a g-differential algebra [22, 35]. Note that, by a graded action, we mean that a degree kelement of ˆgacts by a degreekderivation.
Example 3. As a generalization of Ω(M), supposeA → M is any Lie algebroid, and ρ : g→ Γ(A) is any Lie algebra morphism. Then Γ(∧A∗) is a g-differential algebra. The action of ˆgis given as follows
• D·α=dAα, where α∈Γ(∧A∗) anddA is the Lie algebroid differential.
• Iξ·α=ιρ(ξ)αfor anyξ∈g.
• Lξ·α=ιρ(ξ)(dAα) +dA(ιρ(ξ)α) forξ∈gandα∈Γ(∧A∗).
Remark 18. IfA→M is any vector bundle, the following are equivalent:
• A→M is a Lie algebroid, and there is a Lie algebra morphismρ:g→Γ(A)
• Γ(∧A∗) is ag-differential algebra.
We can think ofΓ(∧A∗) as the algebra of functions on the graded manifoldA[1], hence another equivalent formulation is
• The graded Lie algebra ˆgacts on the graded manifoldA[1].
2.2. The quadratic graded Lie algebraQ(g). Suppose a Lie algebragpossesses an invariant non-degenerate symmetric bilinear form h·,·ig. We can associate tog the quadratic graded Lie algebraQ(g), anR[2] central extension of ˆg,
0→R[2]→ Q(g)→ˆg→0,
which plays a central role in the theory of quasi-Poisson structures.
As a graded vector space,Q(g) =R[2]⊕ˆg. LetT denote the generator ofR[2].
The central extension is given by the cocycle
c(Iu, Iv) =hu, viT, c(D,·) =c(Lu,·) = 0, i.e.
[a, b]Q(g)= [a, b]ˆg+c(a, b) fora, b∈ˆg.
The quadratic formh·,·iQ(g) of degree 1 is given by
(13a) ha, biQ(g)= 0 for anya, b∈ Q(g) such that deg(a) + deg(b) + 16= 0, and (13b) hT, DiQ(g)= 1, hIξ, LηiQ(g)=hξ, ηig.
Note that (13a) is equivalent to saying the quadratic form is of degree 1.
Remark 19. The Lie algebra Q(g) was first introduced in [5], where the so called non-commutative Weil algebra was defined as a quotient of the enveloping algebra ofQ(g).
2.3. Quasi-Poisson g-manifolds revisited. It is easy to check that R[2]⊕g[1]
andg⊕R[−1] are transverse Lagrangian subalgebras ofQ(g). Therefore (Q(g),R[2]⊕
g[1],g⊕R[−1]) forms a Manin triple [18, 30, 31]. The corresponding Lie bialgebra R[2]⊕g[1] integrates to the Poisson Lie group
Gsmall=R[2]×g[1], where multiplication is given by
(t, ξ)·(t0, ξ0) = (t+t0+1
2hξ, ξ0ig, ξ+ξ0).
Since the quadratic form onQ(g) is of degree 1, the Poisson bracket onGsmall
is of degree −1. To describe the Poisson bracket, note that linear functions on g[1] may be identified with elements of g (using the quadratic form). If we let t denote the standard coordinate on R[2] then we see that there is a canonical algebra isomorphism C∞(R[2]×g[1]) ∼= (∧∗g)[t]. Under this isomorphism the Poisson bracket is simply
{t, t}=φ {t, ξ}= 0 {ξ, η}= [ξ, η]g
Proposition 5. A quasi-Poisson g-structure on M is equivalent to a graded Pois- son mapT∗[1]M →Gsmall.
Proof. A quasi-Poisson g-manifold (M, ρ, π) is equivalent to a morphism of Ger- stenhaber algebrasρ0 : (∧∗g)[t]→Γ(∧∗T M). Hereρ0 is defined on the generators byρ0(t) =πandρ0(ξ) =ρ(ξ) for anyξ∈g.
The standard symplectic form onT∗[1]M induces a degree−1 Poisson bracket on C∞(T∗[1]M). As Gerstenhaber algebras C∞(T∗[1]M) ∼=Γ(∧∗T M), canonically.
Paired with the isomorphism C∞(Gsmall) ∼= (∧∗g)[t], we see that ρ0 defines a morphism of Poisson algebras
C∞(Gsmall)→C∞(T∗[1]M).
This is equivalent to a Poisson morphism
T∗[1]M →Gsmall.
Proof of Theorem 1. If (M, ρ, π) is a quasi-Poissong-manifold, then we have a Pois- son mapf :T∗[1]M →Gsmall(f∗t=π,f∗ξ=ρ(ξ)). Therefore, the dual Lie alge- brag⊕R[−1] acts onT∗[1]M. To describe the action explicitly, recall [30] that the left invariant one forms on Gsmall form a subalgebra ofΓ(T∗Gsmall) isomorphic tog⊕R[−1]∼=Te∗[1]Gsmall (evaluation at the identity provides the isomorphism).
The left-invariant 1-form onGsmall corresponding toD is dt+1
2 X
i
ξidξi,
whereξi andξi refer to coordinates ong[1] induced by the basis vectorsei andei, respectively. The corresponding vector field onT∗[1]M is thus
{f∗t,·}+1 2
X
i
f∗ξi{f∗ξi,·},
i.e. the differential dT∗M (3). Since [D, D] = 0, this shows that d2T∗M = 0. The action of g on T∗[1]M preserves dT∗M (since g⊕R[−1] is a direct sum) and it is just the natural lift of the action ρ on M (the left-invariant 1-form on Gsmall
corresponding toξ∈gisdξ).
The dressing action ofD∈ˆgonGsmall is given by {t,·}+1
2 X
i
ξi{ξi,·}=φ∂t+dg,
where dg is the Lie algebra differential ofg. The projectionGsmall →g[1] is thus R[−1]-equivariant, with respect to the R[−1] actions generated by D ∈ˆgand dg, respectively. Since the map f is also R[−1]-equivariant, so is their composition T∗[1]M →g[1], i.e. we have a Lie algebroid morphismT∗M →g.
The fusion also appears in a natural way from this perspective. A Poisson morphism
f :T∗[1]M →Gsmall×Gsmall
defines a quasi-Poissong⊕g-structure onM. SinceGsmall is a Poisson Lie group, the multiplication map
mult :Gsmall×Gsmall→Gsmall
is a Poisson morphism. The map
mult◦f :T∗[1]M →Gsmall
defines a quasi-Poissong-structure onM. Since mult∗t=t1+t2+12P
i(ξi)1(ξi)2
(where the sub-indices (·)1and (·)2indicate which factor ofGsmall×Gsmallthe co- ordinates parametrize), the bivector onMis modified by the termf∗(12P
i(ξi)1(ξi)2) (note the similarity to (10)). It is easy to check that this is the same quasi-Poisson g-structure onM given by the fusion (11).
2.4. Hamiltonian quasi-Poissong-manifolds revisited. LetGbe a Lie group with Lie algebra g. Suppose h·,·ig is a quadratic form on g. Let ¯g denote the quadratic Lie algebra whose quadratic form ish·,·i¯g =−h·,·ig. We let d=g⊕¯g, and diag(g)⊂ddenote the diagonal subalgebra. Then
(14a) R[2]⊕g[1]⊕¯g
and
(14b) ˆg∼= diag(g)[1]⊕diag(g)⊕R[−1]
are two Lagrangian subalgebras of the quadratic Lie algebra Q(d) =R[2]×d[1]×d×R[−1]
defined in§2.2. The corresponding Lie bialgebra R[2]⊕g[1]⊕¯gintegrates to the Poisson Lie group
Gbig =R[2]×g[1]×G, where multiplication is given by
(t, ξ, g)·(t0, ξ0, g0) = (t+t0+1
2hξ, ξ0i, ξ+ξ0, g·g0).
The groupGbig is thus the direct product ofGwith the Heisenberg groupGsmall
described above. As in § 2.3, there is a canonical identification C∞(Gbig) ∼= (∧∗g)[t]⊗C∞(G). Using this identification, we may describe the Poisson bracket (of degree−1) onGbig by
{t, t}=φ (15)
{t, ξ}= 0 {ξ, η}= [ξ, η]g (16)
{t, f}=b∗df {ξ, f}= (ξL−ξR)·f {f, g}= 0 (17)
where f, g ∈C∞(G),ξ, η ∈g, ξL and ξR denote the corresponding left and right invariant vector fields onGandb is given by (7).
We have
Proposition 6. A Hamiltonian quasi-Poisson g-structure onM is equivalent to a graded Poisson map T∗[1]M →Gbig.
Proof. The proof of Proposition 5 shows thatM is a quasi-Poissong-manifold. The mapT∗[1]M →Gbig restricts to define a map
(18) Φ :M →G;
and the formulas for the brackets in (17) show that Φ defines a moment map
(Definition 3).
As in §2.3, fusion can be described in terms of composition with the multipli- cation Poisson morphism
mult :Gbig×Gbig →Gbig, this is precisely equivalent to the explanation given in [1].
Proof of Theorem 2. A Poisson morphismF:T∗[1]M →Gbig induces an action of diag(g)[1]⊕diag(g)⊕R[−1]∼= ˆgonT∗[1]M,
%: ˆg→Γ T(T∗[1]M) .
Forξ∈g, and the corresponding elementIξ ∈ˆg, let us describe this action explic- itly.
The standard symplectic form onT∗[1]M induces a degree−1 Poisson bracket on C∞(T∗[1]M). As Gerstenhaber algebras C∞(T∗[1]M) ∼=Γ(∧∗T M), canonically.
A functionf ∈C∞(M) acts onΓ(∧∗T M) by contraction with df. It follows that the Hamiltonian vector field generated by a one formα∈Ω1(M) acts onΓ(∧∗T M) by contraction withα.
LetλIξdenote the left invariant one form onGbig corresponding toIξ. λIξ is just the pullback of hξ, θLi ∈ Ω1(G) toGbig. Consequently, %(Iξ) acts by contraction with Φ∗hξ, θLi, where Φ :M →Ggiven by (18). ForX ∈Γ(T M),
%(Iξ)·X = Φ∗hξ, θLi
(X) =i(ξ)(X), wherei:g→Ω1(M) is given by (8).
As stated in Remark 18, an action of ˆgon T∗[1]M is equivalent to Γ(∧∗T M) being ag-differential algebra. The proof of Theorem 1 shows that the differential is given by (3). From this perspective, Theorem 2 is just a special case of Example 3.
2.5. Hamiltonian quasi-Poissong-groupoids revisited. Let Γ be any groupoid.
Recall that T∗[1]Γ has a natural groupoid structure (see Appendix A, Page 35).
Combining this structure with the canonical symplectic structure on the cotangent bundle,T∗[1]Γ becomes a symplectic groupoid.
Since T∗[1]Γ is a symplectic groupoid, it integrates a Poisson manifold [59].
We can describe this Poisson manifold explicitly. Let A∗ denote Lie algebroid corresponding to the groupoid Γ (we denote the Lie algebroidA∗ (and notA) for later convenience). A[1] has a linear Poisson structure on it (of degree−1) defining the Lie algebroid structure onA∗[52]. T∗[1]Γ is the symplectic groupoid integrating A[1].
Proposition 7. A compatible Hamiltonian quasi-Poissong-structure onΓis equiv- alent to a morphism of Poisson groupoids
(19) F :T∗[1]Γ→Gbig.
Proof. First we introduce some notation. If M is a graded manifold and f ∈ C∞(M), let (f)idenote the pullback off to theithfactor of the direct powerMn. IfM is a graded Poisson manifold with Poisson bracket{·,·}M, let ¯M denote the same graded manifold with the Poisson bracket
(20) {·,·}M¯ =−{·,·}M.
By Proposition 6,F defines a Hamiltonian quasi-Poissong-structure on Γ. The moment map Φ : Γ→Gis given by restricting F to the subgroupoid Γ⊂T∗[1]Γ.
Consequently Φ is a morphism of Lie groupoids.
Under the isomorphismC∞(Gbig)∼= (∧∗g)[t]⊗C∞(G),η∈gdefines a function onGbig. We notice that the functions
(η)1+ (η)2−(η)3, and t1+t2+1
2(ξi)1(ξi)2
−t3
vanish on the graph of the multiplication GrmultGbig ∈Gbig×Gbig×Gbig. Since F is a groupoid morphism, it follows that the functions
(F∗η)1+ (F∗η)2−(F∗η)3, and (F∗t)1+ (F∗t)2+1 2
X
i
(F∗ξi)1(F∗ξi)2
−(F∗t)3
vanish on the graph of the multiplication GrmultT∗[1]Γ. In the first case, this shows that action ofgon Γ×Γ×Γ is tangent to the graph of the multiplication. In other words,gacts on Γ by groupoid automorphisms. In the latter case, this shows that the bivector field on Γ is fusion multiplicative.
2.6. The Lie bialgebra ˆg is quasi-triangular. Recall that (14) makes ˆginto a Lie bialgebra.
Proposition 8. The elementrˆ=P
iIei⊗Lei ∈ˆg⊗ˆgis anr-matrix for the graded Lie bialgebraˆg.
Proof. We may view the degree−1 element ˆr∈ˆg⊗ˆgas a degree 0 map ˆr: ˆg∗[1]→ˆg.
Using (14), we identify ˆg∗[1] and ˆgwith the transverse Lagrangian subalgebras R[2]⊕g[1]⊕¯g⊂ Q(d)
and
diag(g)[1]⊕diag(g)⊕R[−1]⊂ Q(d)
respectively. Then the graph of ˆr: ˆg∗[1]→ˆgis identified with the subspace Gr(ˆr)∼=R[2]⊕g[1]⊕g.
Thus Gr(ˆr) is an ideal of the Drinfeld double of ˆg. Equivalently ˆris anr-matrix.
Proof of Proposition 1. By Proposition 8, ˆgis a quasi-triangular Lie bialgebra. Re- call from§1.3, the functor
Fˆg: Ham-qPoisˆg→Ham-Poisgˆ,
from the category of Hamiltonian quasi-Poisson ˆg-manifolds to the category of Hamiltonian Poisson ˆg-manifolds, and its inverse Fˆg−1, which describe an equiva- lence of categories.
LetM andM0be two Hamiltonian quasi-Poissong-manifolds. By Proposition 6, T∗[1]M and T∗[1]M0 are Hamiltonian Poisson ˆg-manifolds. Now Fˆg is strongly monoidal, and we have the natural transformation
(21) J :Fˆg Fgˆ−1(T∗[1]M)~Fˆg−1(T∗[1]M0)
→(T∗[1]M)×(T∗[1]M0)
described in Remark 13. Note that the map j : Gbig → Gˆ from Remark 13 is just the projection Gbig → g[1]. Therefore (21) is just the map J described in Proposition 1.
The right hand side of (21) describes the fusion of T∗[1]M and T∗[1]M0 as Hamiltonian Poisson ˆg-manifolds. As shown in § 2.4, (T∗[1]M)×(T∗[1]M0) = T∗[1](M~M0), where the right hand side is the fusion ofM andM0as Hamiltonian quasi-Poissong-manifolds.
The left hand side of (21) describes the fusion ofFˆg−1(T∗[1]M) andFgˆ−1(T∗[1]M0) as Hamiltonian quasi-Poisson ˆg-manifolds. Therefore the action of ˆgon (T∗[1]M)~ (T∗[1]M0) is by definition diagonal, and the functorFˆg preserves this action.
Therefore the Lie algebroid structure on the left-hand side (given by the action of D∈ˆg) is the direct sum of the Lie algebroidsT∗M andT∗M0 (as the action of ˆgis diagonal), while the right-hand side corresponds to the Lie algebroidT∗(M~M0).
2.7. Quasi-Poisson g-bialgebroids revisited. Before proving Theorem 5, it is important to formulate a description of quasi-Poissong-bialgebroids in terms of the Manin triple (Q(g⊕¯g),ˆg,R[2]⊕g[1]⊕¯g). In fact, the description is quite natural, namely:
Proposition 9. SupposeA→M is a vector bundle. The following are equivalent:
• A is a quasi-Poissong-bialgebroid.
• There is a Poisson structure of degree −1 on A[1], and a Lie bialgebra action ofˆgonA[1].
Proof. Suppose we are given a degree−1 Poisson structureπonA[1] and an action ˆ
ρ of the Lie bialgebra ˆg on A[1]. We need to show that A is a quasi-Poissong- bialgebroid. Recalling Definition 5, we must show that
• We have a Lie algebroid structure onAand a Lie algebra morphism ρ:g→Γ(A).
(This follows directly from Remark 18.)
• There is a degree+1derivationDof the Gerstenhaber algebraΓ(V
A), such that
– Dρ(ξ) = 0 for anyξ∈g, and – D2= 12[ρ(φ),·].
By Proposition 8 and the graded version of Remark 2, an action of the Lie bialgebra ˆg on A[1] is equivalent to a quasi-Poisson action of ˆg on A[1]. Let us describe this explicitly, to avoid possible sign problems (as ˆgis a graded Lie bialge- bra with cobracket of degree−1). A Poisson structure of degree −1 on the graded manifold A[1] is, by definition, a functionπ on the bigraded symplectic manifold T∗[1,1]A[1,0] of degree (1,2) such that{π, π}= 0. An action ˆρof the graded Lie algebra ˆgcan be seen as a map ˆρ: ˆg→C∞(T∗[1,1]A[1,0]) (as vector fields can be seen as linear functions on the cotangent bundle) shifting degrees by (1,1).
The action ˆρis a Lie bialgebra action on (A[1,0], π). Therefore, by Proposition 8 and Remark 2,
(22) π˜=π−1
2 X
i
ˆ
ρ(Iei) ˆρ(Lei) is ˆg-invariant and (A[1,0],π) is a quasi-Poisson ˆ˜ g-space:
{˜π,˜π}= 1 4
X
ijk
cijkρ(Iˆ ei) ˆρ(Iej) ˆρ(Lek),
wherecijk=h[ei, ej], ekiare the structure constants ofg. We can rewrite it as (23) {π,˜ π}˜ ={ρ(D),ˆ ρ(Iˆ φ)}.
Using the canonical symplectomorphism
(24) T∗[1,1]A[1,0]∼=T∗[1,1]A∗[0,1],
˜
π becomes a vector field onA∗[0,1] (since it is a function linear on the fibers of T∗[1,1]A∗[0,1]), i.e. a derivation D of the algebra Γ(∧A) of degree 1. Since ˜π is ˆg-invariant, D preserves the Gerstenhaber bracket on Γ(∧A) and Dρ(ξ) = 0 for everyξ∈g. Finally, Equation (23) becomesD2= 12[ρ(φ),·].
We have shown that, (A,D, ρ) is a quasi-Poissong-bialgebroid. To establish the converse, just reverse the procedure.
Proof of Proposition 3. By Proposition 9, a quasi-Poisson g-bialgebroid structure on A defines a degree −1 Poisson structure on A[1]. This is equivalent to a Lie algebroid structure onA∗ [52].
A careful examination of the proof of Proposition 9 allows us to describe the Lie algebroid differential explicitly. The bivector corresponding to the Poisson structure onA[1] is a functionπon the bigraded symplectic manifoldT∗[1,1]A[1,0] of degree (1,2). Under the canonical symplectomorphism (24) it becomes a degree +1 vector field onA∗[1]. By (22) this vector field is
D+1 2
X
i
ˆ
ρ(Iei) ˆρ(Lei).
The corresponding Lie algebroid differential is given by (12).
The remaining details in Proposition 3 follow from a similar examination of the
proof of Proposition 9.
Remark 20. One can also describe quasi-Poisson g-bialgebroids in the spirit of Proposition 5. IfAis a Lie algebroid, so thatA∗[1] is a graded Poisson manifold, a Poisson mapA∗[1]→Gsmall would give us a quasi-Poissong-bialgebroid structure onA, but with the additional property thatDis Hamiltonian. In general, a quasi- Poisson g-bialgebroid structure on a Lie algebroid A is equivalent to a principal PoissonR[2]-bundleP →A∗[1] with a Poisson R[2]-equivariant mapP →Gsmall. 2.8. Proof of Theorem 5. The proof of [62, Theorem 5.5] (see also [20, 21]) goes through in the graded setting to show that the existence of a morphism of Poisson groupoids
F :T∗[1]Γ→Gbig
is equivalent to the action of the Lie bialgebra ˆg on the Poisson manifold A[1].
By Proposition 7, the former describes a compatible Hamiltonian quasi-Poissong- structure on Γ, while the latter describes a quasi-Poisson g-bialgebroid structure (A, ρ,D) (see Proposition 9). This proves Theorem 5.
Proof of Proposition 4. Let (Γ, ρΓ, πΓ) be the quasi-Poisson structure on Γ, and (M, ρM, πD) be the quasi-Poisson structure onM induced by (A, ρ,D) (see Propo- sition 2). We must show that the source map s0 : Γ → M is a quasi-Poisson morphism. Let DA and DΓ be the homological vector fields on A[1] and T∗[1]Γ defined by the respective actions ofD∈ˆg. Since the source maps:T∗[1]Γ→A[1]
is ˆg-equivariant, we have
s∗(DAf) =DΓs∗f,
for every f ∈ C∞(M), where we view f as an element of C∞(A[1]). Since s : T∗[1]Γ→A[1] is also a Poisson map, it follows that
(25) s∗{DAf, g}A[1]={DΓs∗f, s∗g}T∗[1]Γ,
