Glanon groupoids

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/89090/

Version: Accepted Version

Article:

Lean, M.J., Stiénon, M. and Xu, P. (2015) Glanon groupoids. Mathematische Annalen.

ISSN 0025-5831

https://doi.org/10.1007/s00208-015-1222-z

[email protected] https://eprints.whiterose.ac.uk/ Reuse

Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website.

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

GLANON GROUPOIDS

MADELEINE JOTZ LEAN, MATHIEU STIÉNON, AND PING XU

Abstract. We introduce the notions of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures, and of Glanon algebroids, their infinites-imal counterparts. Both symplectic and holomorphic Lie groupoids are particular instances of Glanon groupoids. We prove that there is a bijection between Glanon Lie algebroids on one hand and source connected and source-simply connected Glanon groupoids on the other. As a consequence, we recover various known integrability results and obtain the integration of holomorphic Lie bialgebroids to holomorphic Poisson groupoids.

1. Introduction

In their study of quantization, Karasev [19], Weinstein [34], and Zakrzewski [40, 41] indepen-dently introduced the notion of symplectic groupoids. By a symplectic groupoid, we mean a Lie groupoid equipped with a multiplicative symplectic 2-form on the space of morphisms. It is a classical theorem that the unit space of a symplectic groupoid is naturally a Poisson manifold [7]. The Lie algebroid of a symplectic groupoid Γ⇉M is naturally isomorphic to

(T∗M)π, the canonical Lie algebroid associated to the Poisson manifold (M, π). Conversely,

Mackenzie-Xu [29] proved that, for a given Poisson manifold (M, π), if the Lie algebroid

(T∗M)π integrates to an s-connected and s-simply connected Lie groupoid Γ⇉M, then Γ is

naturally a symplectic groupoid. As a consequence, they recovered the following theorem of Karasev-Weinstein: every Poisson manifold of dimension n admits a symplectic realization of dimension 2n. The symplectic groupoid structure on Γ was also obtained by Cattaneo-Felder [6] using the Poisson sigma model. The full integrability criterion for Poisson manifolds was obtained later by Crainic-Fernandes [10]. In fact, symplectic groupoids constitute a partic-ular class of a more general type of structures called Poisson groupoids, which were discovered by Weinstein [35] and also comprise Drinfeld’s Poisson groups [11]. In a Poisson groupoid, the Poisson bivector field and the groupoid multiplication are required to be compatible: the Poisson bivector field must be “multiplicative.” It was proved in [28] that a Poisson bivector field on a Lie groupoidG⇉M with Lie algebroidA is multiplicative if and only if it induces a morphism of Lie groupoids from the cotangent groupoidT∗G⇉A∗to the tangent groupoid T G⇉ T M. For instance, a Poisson bivector field on a Lie group G is multiplicative in the sense of Drinfeld [11] if and only if it induces a morphism of Lie groupoids fromT∗G⇉g∗ to

T G⇉_{∗}. A Poisson groupoid whose Poisson bivector field is nondegenerate is a symplectic groupoid as the inverse of the Poisson bivector is a multiplicative symplectic form in the sense of Coste-Dazord-Weinstein [7].

Two of the authors have recently been interested in holomorphic Lie algebroids and holomor-phic Lie groupoids [24]. Finding out which holomorholomor-phic Lie algebroids can be integrated is a

Research partially supported by a Dorothea-Schlözer fellowship of the University of Göttingen, Swiss NSF grants 200021-121512 and PBELP2_137534, NSF grants DMS-0801129 and DMS-1101827, NSA grant H98230-12-1-0234.

very natural problem. In [23], together with Laurent-Gengoux, they studied holomorphic Lie algebroids and their relation with real Lie algebroids. A holomorphic Lie algebroid is a real Lie algebroid structure on a holomorphic vector bundle A_→ X such that (1) the sheaf _A of holomorphic sections ofAis stable under the Lie bracket of (all smooth) sections of Aand (2) the restriction of the Lie bracket to_AisC-linear. Laurent-Gengoux_{et. al.}proved in particular

that a holomorphic Lie algebroid A can be integrated to a holomorphic Lie groupoid if and only if its underlying real Lie algebroidA_Ris integrable as a real Lie algebroid [24].

In this paper, we introduce the notion of Glanon groupoids: Lie groupoids equipped with a multiplicative generalized complex structure. Recall that a generalized complex structure in the sense of Hitchin [15] on a manifold M is a smooth bundle map J:T M _⊕T∗M _→ T M_⊕T∗M such that _J2=₋idT M⊕T∗_M,J preserves the natural nondegenerate symmetric bilinear form on the fibers ofT M_⊕T∗M, and the+i-eigenbundle ofJ is involutive with respect to the Courant bracket (or, equivalently, the Nijenhuis tensor of _J vanishes). A generalized complex structureJ on a Lie groupoidΓ⇉M is said to be multiplicative ifJ is a Lie groupoid automorphism of the Courant groupoidTΓ_⊕T∗Γ⇉T M_⊕A∗ in the sense of Mehta [30]. When

J is the generalized complex structure determined by a symplectic structure on Γ, it is clear that J is multiplicative if and only if the symplectic 2-form is multiplicative. In this case, the Glanon groupoid is simply a symplectic groupoid. On the other hand, when _J is the generalized complex structure determined by a complex structure onΓ,J is multiplicative if and only if the complex structure on Γ is multiplicative. In this case, the Glanon groupoid is simply a holomorphic Lie groupoid. On the infinitesimal level, to each Glanon groupoid corresponds a Glanon Lie algebroid: a Lie algebroid A equipped with a generalized complex structure JA: T A⊕T∗A → T A⊕T∗A which is also an automorphism of the Lie algebroid

T A_⊕T∗A_→T M_⊕A∗. More precisely, we prove the following main result:

Theorem A. If Γ is an s_{-connected and} s_{-simply connected Lie groupoid with Lie algebroid} A, then there is a bijection between Glanon groupoid structures onΓ and Glanon Lie algebroid structures onA.

As a consequence, we recover the following standard results [29, 24]:

Theorem B. Let (M, π) be a Poisson manifold. IfΓ is a s_{-connected and}s_{-simply connected}

Lie groupoid integrating the Lie algebroid (T∗M)π, thenΓ automatically admits a symplectic

groupoid structure.

Theorem C. IfΓis as_{-connected and}s_{-simply connected Lie groupoid integrating the real Lie}

algebroidAR underlying a holomorphic Lie algebroidA, thenΓis a holomorphic Lie groupoid.

When the matrix representation of the generalized complex structureJ on a Glanon groupoid

Γrelative to the direct sum decompositionTΓ⊕T∗Γof the Pontryagin bundle1has the special form

J =

N π♯

0 ₋N∗

,

1_{Yoshimura and Marsden refer to the Whitney sumT M⊕T}∗

Γ is simply a holomorphic Poisson groupoid and it is no surprise that its Lie algebroid is necessarily part of a holomorphic Lie bialgebroid. We prove the following result.

Theorem D. Given a holomorphic Lie bialgebroid (A, A∗), if the real Lie algebroid AR

un-derlying A integrates to a s_{-connected and} s_{-simply connected Lie groupoid Γ, then Γ is a}

holomorphic Poisson groupoid.

This theorem was proved in [24] using a different method in the special case of the holomorphic Lie bialgebroid ((T∗X)π, T X) determined by a holomorphic Poisson manifold (X, π). More

precisely, it was proved that when the underlying real Lie algebroid of (T∗X)π integrates to

as-connected ands-simply connected Lie groupoid _Γ, then_Γ is automatically a holomorphic symplectic groupoid. To the best of our knowledge, the integration problem for arbitrary holomorphic Lie bialgebroids had remained open to this day. Solving it constituted one of the motivations behind our study of Glanon groupoids.

It is known that a generalized complex structure on a manifold determines on it a Poisson bivector field [2, 14]. Therefore a Glanon groupoid is automatically a (real) Poisson groupoid and we can consider the ‘Glanon to Poisson’ forgetful functor. On the other hand, every Glanon Lie algebroidA admits a linear Poisson structure so that its dual A∗ is also a Lie algebroid. We prove that, for any Glanon algebroid A, the pair (A, A∗) automatically constitutes a Lie bialgebroid. Finally, we prove that the ‘groupoid to algebroid’ Lie functor, which takes Glanon groupoids and Poisson groupoids, respectively, to Glanon algebroids and Lie bialgebroids, commutes with the forgetful functor, which takes Glanon groupoids and Glanon algebroids, respectively, to Poisson groupoids and Lie bialgebroids.

Note that in this paper we confine ourselves to the standard Courant groupoid TΓ⊕T∗Γ. Instead, one could have considered the twisted Courant groupoid(TΓ_⊕T∗Γ)H, whereH is a

multiplicative closed3-form. This will be discussed elsewhere. Note also that a multiplicative generalized complex structure on a groupoid induces an endomorphismj:T M_⊕A∗ _→T M_⊕ A∗ of the unit space and an endomorphism jA: A⊕T∗M → A⊕T∗M of the core2 of the

VB-groupoidTΓ⊕T∗Γ. Thinking of multiplicative generalized complex structures as pairs of complex conjugate multiplicative Dirac structures, we can conclude from results in [18] that a multiplicative generalized complex structure on a groupoid is equivalent to a pair of complex conjugate Dirac bialgebroids on its space of units. The detailed study of properties of the maps jAand j, the Dirac bialgebroids, and the associated Dorfman connections will be investigated

in the spirit of [23] in a future project.

Acknowledgments. We would like to thank Camille Laurent-Gengoux, Rajan Mehta, and Cristian Ortiz for useful discussions and an anonymous referee for suggesting many improve-ments. We would also like to thank several institutions for their hospitality while work on this project was underway: Penn State (Jotz), and Institut des Hautes Études Scientifiques and Beijing International Center for Mathematical Research (Xu). Special thanks go to the Michéa family and our many friends in Glanon, whose warmth provided us constant inspiration for our work in the past. Our memories of our many friends from Glanon who have unfortunately left mathematics serve as a powerful reminder that our primary role as mathematicians is

2_{The Whitney sum TΓ⊕T}∗

Γ, which is simultaneously a vector bundle with Γas base manifold and a Lie groupoid withT M⊕A∗_{as unit space, is a}

VB-groupoidwith the vector bundleA⊕T∗

to elucidate mathematics while enjoying its beauty. We dedicate this paper to our dear old friends.

Notation. In the following, Γ⇉M will always be a Lie groupoid with set of arrowsΓ, set of objectsM, source and targets_,t_{: Γ→M, object inclusion mapǫ:M →Γand inversion map} i_{: Γ →Γ}. The product of _{g, h∈Γ} withs_{(g) =}t_(h) will be written m_{(g, h) =g ⋆ h} or simply gh.

The Lie functor that sends a Lie groupoid to its Lie algebroid and Lie groupoid morphisms to Lie algebroid morphisms is A. For simplicity, we will write A_{(Γ) =} A. The Lie algebroid q_A:A_→M is identified withTs

MΓ, the bracket[·,·]Ais defined with the right invariant vector

fields and the anchorρA=ρis the restriction ofTttoA. Hence, as a manifold,Ais embedded

inTΓ. The inclusion isι:A_→TΓ. Givena_∈Γ(A), the right-invariant section corresponding to awill simply be written ar, i.e.ar(g) =T Rga(t(g))for all g ∈Γ. We will write al for the

left-invariant vector field defined bya, i.e.al(g) =−T(L_g◦i₎₍a(s₍g)))for all g_∈Γ.

The projection map of a vector bundle A _→ M will always be written qA: A → M, unless

specified otherwise. For a smooth manifold M, we fix once and for all the notation p_M :=

qT M:T M →M and cM := qT∗_M:T∗M →M. We will write P_M for the Pontryagin bundle T M_⊕T∗M over M, andpr_M for the canonical projection qP_M:P_{M →M.}

A bundle morphismP_{M →}P_M, for a manifold_M, will always be meant to be over the identity onM.

2. Preliminaries

2.1. Dirac structures. Let A _→ M be a vector bundle with dual bundle A∗ _→ M. The natural pairingA_⊕A∗_→R,₍a_m, ξ_m)7→ξ_m(a_m)will be written*·,_·+Aor*·,·+qA if the vector

bundle structure needs to be specified. The direct sum A_⊕A∗ is endowed with a canonical fiberwise pairing(·,_·)Agiven by

((am, ξm),(bm, ηm))A=*bm, ξm+A+*am, ηm+A,

for allm_∈M,am, bm∈Am andξm, ηm∈A∗m.

In particular, the Pontryagin bundleP_{M =T M ⊕T}∗_M of a smooth manifold _M is endowed with the pairing(_·,_·)T M, which will be written as usual h·,·iM.

The orthogonal of a subbundle E _⊆ A_⊕A∗ relative to the pairing (·,_·)A will be written

E⊥. An _{almost Dirac structure} [8] on M is a Lagrangian vector subbundle D _⊂ P_M. That is, D _{coincides with its orthogonal relative to h·,·iM,} D ₌ D⊥_{, so its fibers are necessarily} dimM-dimensional.

The set of sectionsΓ(P_M)of the Pontryagin bundle of_Mis endowed with the_{Courant-Dorfman}

bracket, given by

JX+α , Y +βK= [X , Y] + (£_Xβ−i_Yd_α),

for allX+α, Y +β_∈Γ(P_M).

2.2. Generalized complex structures. LetV be a vector space. Consider a linear endo-morphism J of V _⊕V∗ such that J2 _{= −id}

V⊕V∗ and J is orthogonal with respect to the inner product

(X+ξ , Y +η)V =ξ(Y) +η(X), ∀X, Y ∈V, ξ, η∈V∗.

Such a linear map is called alinear generalized complex structure by Hitchin [15]. The com-plexified vector space (V _⊕V∗)_⊗Cdecomposes as the direct sum

(V _⊕V∗)⊗C₌E_+⊕E−

of the eigenspaces of_J corresponding to the eigenvalues _±irespectively, i.e.

E±={(X+ξ)∓iJ(X+ξ)|X+ξ ∈V ⊕V∗}.

Both eigenspaces are maximal isotropic with respect to(_·,_·)V and they are complex conjugate

to each other.

The following lemma is obvious.

Lemma 2.1. The linear generalized complex structures are in bijection with the splittings

(V _⊕V∗)⊗C_{=E+⊕E− with E± maximal isotropic and E−=E+.}

Definition 2.2. Let M be a manifold and _J a bundle endomorphism of P_{M =T M⊕T}∗_M

such that J2_=−id

P_M, and J is orthogonal with respect to h·,·i_M. ThenJ is a generalized almost complex structure. In the associated eigenbundle decomposition

TCM ⊕TC∗M =E+⊕E−,

if Γ(E₊) is closed under the (complexified) Courant bracket, then E₊ is a (complex) Dirac structure onM and one says that_J is a generalized complex structure [15, 14].

If E+ is a Dirac structure, then E− must also be a Dirac structure since E− =E+. Indeed (E+, E−)is a complex Lie bialgebroid in the sense of Mackenzie-Xu [28], in whichE+ andE−

are complex conjugate to each other.

Definition 2.3. Let _J: P_{M →} P_{M be a vector bundle morphism. Then the generalized} Nijenhuis torsionassociated to _J is the map

NJ:P_{M ×M}P_{M →}P_M

defined by

NJ(ξ, η) =J_Jξ ,_JηK+_J2Jξ , ηK_{− J} (J_Jξ , ηK+Jξ ,_JηK)

for all ξ, η_∈Γ(P_{M), where the bracket is the Courant-Dorfman bracket.}

Note that ifJ in the last definition is orthogonal with respect toh·,_·iM and satisfiesJ2 _=−id

(i.e. if_J is an almost complex structure), then its Nijenhuis torsion is a tensor.

The following proposition gives two equivalent definitions of a generalized complex structure.

Proposition 2.4. A generalized complex structure is equivalent to any of the following:

(a) A bundle endomorphism _J of P_{M such that J is orthogonal with respect to h·,·iM,}

J2_=−id

P_M and NJ = 0.

(b) A complex Lie bialgebroid (E₊, E−) whose double is the standard Courant algebroid

For a two-formω onM we denote byω♭:T M _→T∗M the bundle mapX _7→iXω, while for a

bivectorπ onM we denote byπ♯:T∗M _→T M the contraction with π. Also ifπ is a Poisson bivector, we denote by[_·,_·]π the Lie algebroid bracket defined on the space of 1-forms on M

by

[ξ, η]π =Lπ♯_ξη−L_π♯_ηξ−dπ(ξ, η)

for allξ, η_∈Ω1(M).

A generalized complex structure_J:P_{M →}P_M can be written

J =

N π♯ ω♭ ₋N∗

with π is a bivector field on M and π♯:T∗M _→ T M, π♯(α) = π(α,_·) is the vector bundle morphism defined byπ,ω is a two-form on M and ω♭:T M _→T∗M, X_7→ i_Xω is the vector

bundle morphism defined byω, andN:T M _→T Mis a bundle map. The geometric structures N,π andωhave to satisfy together a list of identities [9]. In particularπ is a Poisson bivector field.

LetIM:PM →PM be the endomorphism IM =

idT M 0

0 −idT∗_M

.

Then we have

hIM(·),IM(·)iM =−h·,·iM,

JIM(·),IM(·)K=IMJ·,·K

and the following proposition follows.

Proposition 2.5. If _J is a generalized almost complex structure on M, then

¯

J :=_IM◦ J ◦ IM (1)

is a generalized almost complex structure. Furthermore,

N_J¯ =IM ◦ NJ ◦(IM,IM).

Hence,J¯is a generalized complex structure if and only if J is a generalized complex structure.

The following are two standard examples [15].

Examples 2.6. (a) LetJ be an almost complex structure on M. Then

J =

J 0

0 ₋J∗

is _h·,_·iM-orthogonal and satisfiesJ2 =−id. J is a generalized complex structure

if and only if J is integrable.

(b) Let ω be a nondegenerate 2-form on M. Then

J = 0 − ω♭ −1

ω♭ 0

!

2.3. Pontryagin bundle over a Lie groupoid.

The tangent prolongation of a Lie groupoid. LetΓ⇉M be a Lie groupoid. Applying the tangent functor to each of the maps definingΓyields a Lie groupoid structure onTΓwith base T M, sourceTs_{, targetT}t_{, multiplicationT}m_{:T(Γ×MΓ)→TΓ and inversionT}i_:TΓ→TΓ. The identity atv_{p ∈}T_pM is1vp =Tpǫvp. This defines the tangent prolongation TΓ⇉T M of Γ⇉M or the tangent groupoid associated to Γ⇉M.

The cotangent Lie groupoid defined by a Lie groupoid. IfΓ⇉M is a Lie groupoid with Lie algebroidA_→M, then there is also an induced Lie groupoid structure onT∗Γ⇉A∗. The source mapˆs_:T∗_Γ→A∗ _{is given by}

ˆs_(αg)∈A∗_s

(g) for αg ∈Tg∗Γ, ˆs(αg)(a(s(g))) =αg(al(g))

for alla_∈Γ(A), and the target mapˆt_:T∗Γ_→A∗ is given by

ˆt_(αg)∈A∗_t

(g), ˆt(αg)(a(t(g))) =αg(ar(g))

for alla_∈Γ(A). Ifˆs_{(αg) = ˆ}t_(αh), then the product_{αg⋆ αh} is defined by (αg⋆ αh)(vg⋆ vh) =αg(vg) +αh(vh)

for all composable pairs(v_g, v_h)∈T_(g,h)(Γ×M Γ).

This Lie groupoid structure was introduced in [7], see also [32, 27]. Note that the original definition was the following: letΛΓ be the graph of the partial multiplication minΓ, i.e.

ΛΓ={(g, h, g ⋆ h)|g, h∈Γ,s(g) =t(h)}.

The isomorphism ψ: (T∗Γ)3 → (T∗Γ)3, ψ(α, β, γ) = (α, β,₋γ) sends the conormal space

(TΛΓ)◦ ⊆(T∗Γ)3|ΛΓ to a submanifold Λ∗ of (T

∗_Γ)3_{. It is shown in [7] thatΛ}

∗ is the graph of

a groupoid multiplication onT∗Γ, which is exactly the multiplication defined above.

The “Pontryagin groupoid” of a Lie groupoid. If Γ⇉M is a Lie groupoid with Lie algebroidA_→M, according to [30], there is hence an induced VB-Lie groupoid structure on P_Γ=TΓ⊕T∗Γover T M_⊕A∗, namely, the product groupoid, whereTΓ⊕T∗ΓandT M_⊕A∗ are identified with the fiber products TΓ_×ΓT∗Γand T M_×M A∗, respectively. It is called a

Courant groupoid by Mehta [30].

Proposition 2.7. Let Γ⇉M be a Lie groupoid with Lie algebroid A _→ M. Then the Pon-tryagin bundleP_{Γ =TΓ⊕T}∗_{Γ is a Lie groupoid over T M⊕A}∗_{, and the canonical projection} P_{Γ→Γ is a Lie groupoid morphism.}

We will write Ttfor the target map P_{Γ →}T M_⊕A∗ defined byTt₍v_g, α_g) = Tt₍v_g),ˆt₍α_g) ,

Ts for the source map P_{Γ → T M ⊕A}∗, and Tǫ, Ti,Tm for the embedding of the units, the inversion map, and the multiplication of this Lie groupoid.

2.4. Pontryagin bundle over a Lie algebroid. Given any vector bundle q_A:A _−→ M, the map T qA: T A−→ T M has a vector bundle structure obtained by applying the tangent

functor to the operations inA_−→M. The operations inT A_−→T M are consequently vector bundle morphisms with respect to the tangent bundle structures inT A_−→AandT M _−→M andT Awith these two structures is therefore a double vector bundle which we call thetangent double vector bundle ofA_−→M (see [26] and references given there).

If (qA: A → M,[·,·]A, ρA) is a Lie algebroid, then there is a Lie algebroid structure on

For a general vector bundleq:A_−→M, there is also a double vector bundle

T∗A

cA

rA _/ /_A∗

q∗

A _q

A

/

/_M

.

Here the map rA is the composition of the Legendre transformation T∗A → T∗A∗ with the

projection T∗A∗ _→ A∗. Elements of T∗A can be represented locally as (ω, a, φ) where ω _∈ T_m∗M, a_∈Am, φ∈A∗m for somem∈M. The Legendre transformation

T∗A_∋(ω, φ, a)_7→(₋ω, a, φ)_∈T∗A∗

is an isomorphism of double vector bundles preserving the side bundles; that is to say, it is a vector bundle morphism over both A and A∗. An intrinsic definition of the Legendre transformation can be found in [28].

SinceAis a Lie algebroid, its dualA∗ has a linear Poisson structure, and the cotangent space T∗A∗ has a Lie algebroid structure overA∗. Hence, there is a unique Lie algebroid structure on rA: T∗A → A∗ with respect to which the projection cA : T∗A → A is a Lie algebroid

morphism overA∗_→M.

Now one can form the fibered product Lie algebroidT A_×AT∗A→T M ×M A∗.

Proposition 2.8. LetA_→M be a Lie algebroid. Then the Pontryagin bundle PAis naturally

a Lie algebroid. Moreover, the canonical projectionP_A→A is a Lie algebroid morphism.

The double vector bundle(PA, T M⊕A∗, A, M)is a VB-Lie algebroid in the sense of Gracia-Saz

and Mehta [13].

2.5. Canonical identifications. The canonical pairing *·,_·+A:A ⊕A∗ → R induces a

nondegenerate pairing_hh·,_·iiA= pr2◦T*·,·+AonT A×T MT A∗, wherepr2:TR=R×R→R

is the map which ‘forgets’ the base point of a tangent vector (see [27]):

A_⊕A∗

*·,·+A

T A_{×T M} T A∗

hh·,·iiA

&

&

T*·,·+A

R _TR

pr₂ //R

That is, ifξ _∈T A andχ_∈T A∗ are such that T qA(ξ) =T qA∗(χ), thenξ = d

dt

 

t=0a(t)∈T A

and χ = _dtd 

t=0ϕ(t) ∈T A

∗ _{for some curves a: (−ε, ε) → A and ϕ: (−ε, ε) → A}∗ _{such that}

q_A∗◦ϕ=q_A◦aandhhξ, χii_A= d

dt

 

t=0*a(t), ϕ(t)+A. For instance, ifX ∈Γ(A)andα∈Γ(A ∗_),

thenT X _∈ΓT M(T A)andT α∈ΓT M(T A∗)are such thatT qA(T X) = idT M =T qA∗(T α)and we have for allvp = ˙c(0)∈TpM:

hhT X(v_p), T α(v_p)iiA=

d dt

   

t=0

If(T qA)∨: (T A)∨ →T M is the vector bundle that is dual to the vector bundle T qA:T A→

T M, there is an induced isomorphismI

T A∗

T qA∗

I //_{(T A)}∨ (T qA)∨

T M //_{T M}

that is defined by

*ξ, I(χ)+T qA =hhξ, χiiA

for all χ _∈ T A∗ and ξ _∈ T A such that T qA∗(χ) = T q_A(ξ). That is, the following diagram commutes:

T A_{×T M} T A∗ T*·,·+A // (id,I)

hh·,·iiA

) ) TR pr2

T A_{×T M} (T A)∨

*·,·+_{T qA}

/

/_R

Applying the construction ofI above to the caseqA=pT M, we get an isomorphism

T(T∗M)

T cM

I //_{(T T M)}∨ (T pM)∨

T M //_{T M}

We also have the canonical involution (see for instance [27])

T T M

T pM

σ //_{T T M}

pT M

T M //_{T M}

Recall that forV _∈X(M)the mapT V :T M _→T T M is a section ofT pM:T T M →T M and

σ(T V) is a section ofp_{T M}:T T M _→T M, i.e. a vector field on T M. We get an isomorphismς :=σ∗_◦I:T(T∗M)→T∗(T M)

T(T∗M)

T cM

ς _//

T∗(T M)

cT M

T M //_{T M}

Proposition 2.9. The map Σ := (σ, ς) :TP_{M →}P_{T M} TP_M

TprM

Σ //_P

T M prT M

T M //_{T M}

,

2.6. Lie functor from P_{Γ to} P_A.

Proposition 2.10. Let Γ⇉M be a Lie groupoid with Lie algebroidA. Then the Lie algebroid of P_{Γ is canonically isomorphic to}P_A.

Moreover, the pairing _h·,_·iΓ is a groupoid morphism: P_Γ×ΓP_{Γ →} R_{. Its corresponding Lie}

algebroid morphism coincides under the canonical isomorphism A₍P_Γ) ∼₌P_{A with the pairing}

h·,_·iA:P_A×AP_A→R_.

This is a standard result, but we recall its proof because it will be useful later on.

For any Lie groupoid Γ⇉M with Lie algebroid A _→ M, the tangent bundle projection p_Γ: TΓ → Γ is a groupoid morphism over p_M:T M _→ M and applying the Lie functor gives a canonical morphism A_{(pΓ) :} A_(TΓ)→ A. This acquires a vector bundle structure by applyingA_(·) to the operations inTΓ→Γ. This yields a system of vector bundles

A_{(TΓ) T M}

A M

qA(TΓ)

A_(pΓ) pM

qA

in which A_{(TΓ) has two vector bundle structures, the maps defining each being morphisms} with respect to the other. In other words,A₍TΓ)is a double vector bundle.

Associated with the vector bundleq_A:A_−→M is the tangent double vector bundle T A

pA

T qA _//

T M

pM

A _q

A

/

/_M

.

It is shown in [28] that the canonical involution σ: T(TΓ)_→ T(TΓ)restricts to a canonical mapσΓ:A(TΓ)→T A which is an isomorphism of double vector bundles preserving the side

bundles. The tangent prolongation T A_→ T M of the Lie algebroidA and the Lie algebroid A(TΓ)_→T M of TΓ⇉T M are isomorphic viaσΓ [28].

Similarly, the cotangent groupoid structure T∗Γ⇉A∗ is defined by maps which are vector bundle morphisms and, reciprocally, the operations in the vector bundle cΓ:T∗Γ −→ Γ are

groupoid morphisms. Taking the Lie algebroid ofT∗Γ⇉A∗ we get a double vector bundle

A_(T∗_{Γ) A}∗

A M

qA(T∗_Γ)

A_(cΓ) pM

qA

where the vector bundle operations in A₍T∗Γ)_→A are obtained by applying the Lie functor to those inT∗Γ_→Γ.

It follows from the definitions of the operations in T∗Γ⇉A∗ that the canonical pairing

*_·,_·+TΓ:TΓ×ΓT∗Γ→R

is a groupoid morphism into the additive group(oid)R_{. Hence*·,·+TΓinduces a Lie algebroid}

morphism

Note thatA_(*·,·+TΓ) is the restriction toA_(TΓ)×AA_(T∗_Γ)of

hh·,_·iiTΓ:T(TΓ)×TΓT(T∗Γ)→R.

As noted in [28],A_(*·,_·+TΓ)is nondegenerate, and therefore induces an isomorphismIΓ:A(T∗Γ)→

A_(TΓ)∨ of double vector bundles, where A_(TΓ)∨ is the dual of A_(TΓ)→ A. Now dualizing σ−1_Γ :T A_→A_(TΓ)over A, we define

ςΓ= (σ_Γ−1)∗◦IΓ:A(T∗Γ)→T∗A.

This is an isomorphism of double vector bundles preserving the side bundles. The Lie alge-broidsT∗A_→A∗ and A₍T∗Γ)→A∗ are isomorphic viaςΓ [28].

The Lie algebroid of the direct sumP_{Γ =TΓ⊕T}∗_Γ is equal to

A₍P_{Γ) =TU}TsP_Γ,

where we writeU for the unit space of P_Γ; i.e.U :=T M_⊕A∗. By the considerations above, we have a Lie algebroid morphismΣΓ = (σΓ, ςΓ):

A A

A₍P_Γ) P_{A=T A⊕T}∗_A

M M

T M_⊕A∗ T M_⊕A∗

id

ΣΓ

id

id

preserving the side bundlesA and T M_⊕A∗.

Recall that we also have a map

Σ = (σ, ς) :TP_Γ→P_TΓ.

Lemma 2.11. Let Γ⇉M be a Lie groupoid and u an element of A₍P_{Γ) ⊆T}P_{Γ projecting to} am ∈ A and (vm, αm) ∈ TmM ×A∗m. Then, if ΣΓ(u) = (vam, αam) ∈ PA(am) and Σ(u) = (˜v_am,α˜_am)_∈P_TΓ(a_m), we have T ιv_am = ˜v_am and α_am = ˜α_am|TamA.

See Remark 3.32 below for an interpretation of this lemma.

Proof. The first equality follows immediately from the definition ofσΓ.

Choose TamA ∋ wam =σΓ(y) for some y ∈ A(TΓ)⊆T(TΓ) and w˜am := T ιwam. Write also

u = (x, ξ) with x _∈ A_(TΓ) and _{ξ ∈} A_(T∗_{Γ) ⊆ T(T}∗_Γ), i.e. _αa

m = ςΓ(ξ). Then we have A_{(pΓ)(y) =}A_{(cΓ)(ξ) =am and we can compute}

*wam, αam+T A=*y, IΓ(ξ)+A(TΓ)=A(*·,·+TΓ)(y, ξ)

=_hhy, ξ_iiTΓ=*y, I(ξ)+T pΓ

=*σ(y), ς(ξ)+pTΓ =*w˜am,α˜am+pTΓ =*wam, ι ∗_α˜

The pairing _h·,_·iΓ is a groupoid morphism P_Γ×ΓP_{Γ →} R. Hence, we can consider the Lie

algebroid morphism

A_{(h·,·iΓ) :}A₍P_Γ)×AA₍P_Γ)→A₍R_{) =}R_.

We have

A_{(h·,·iΓ) = (pr2◦Th·,·iΓ)|A(P}

Γ)×AA(PΓ).

We can see from the proof of the last lemma thatA_(*·,·+TΓ)coincides with_{*·,·+T A}under the isomorphismΣΓ. Hence, A(h·,·iΓ) coincides with the pairingh·,·iA:PA×APA→ R, under

the canonical isomorphismA₍P_Γ)∼₌P_A.

3. Multiplicative generalized complex geometry

3.1. Glanon groupoids.

Definition 3.1. Let Γ⇉M be a Lie groupoid with Lie algebroid A _→ M. A generalized almost complex structure_J onΓ is multiplicativeif it is an automorphism of the Lie groupoid

TΓ⊕T∗Γ⇉T M_⊕A∗. We call the pair (Γ⇉M,_J) a generalized almost complex groupoid. We use the symbolj to denote the induced automorphism of the unit spaceT M_⊕A∗:

TΓ⊕T∗Γ J //

Ts

Tt

TΓ⊕T∗Γ

Ts

Tt

T M_⊕A∗

j

/

/_{T M ⊕A}∗

.

If _J is a generalized complex structure, then we call the pair (Γ⇉M,_J) a Glanon groupoid.

This is equivalent toD_J and D_J, the eigenspaces of_{J: (}TΓ⊕T∗Γ)⊗C_→(TΓ⊕T∗_Γ)⊗C

corresponding to the eigenvaluesiand₋ibeing multiplicative Dirac structures onΓ⇉M [31]. Note also that_J is a Lie groupoid morphism if and only if the mapsN:TΓ_→TΓ,π♯:T∗Γ_→

TΓ,N∗:T∗Γ→T∗Γ andω♭:TΓ→T∗Γ such that

J =

N π♯ ω♭ ₋N∗

are all Lie groupoid morphisms. In particular, we have the following proposition.

Proposition 3.2. If Γ is a Glanon groupoid, then Γ is naturally a Poisson groupoid.

Example 3.3 (Glanon groups). LetG⇉_{∗}be a Glanon Lie group. Since any multiplicative two-form must vanish (see for instance [17]), the underlying generalized complex structure on Gis equivalent to a multiplicative holomorphic Poisson structure. Therefore, Glanon Lie groups are in bijection with complex Poisson Lie groups.

Example 3.4 (Symplectic groupoids). Consider a Lie groupoidΓ⇉M equipped with a non-degenerate two-formω. Then the map

Jω= 0 − ω ♭−1

ω♭ 0

!

,

Example 3.5(Holomorphic Lie groupoids). LetΓbe a holomorphic Lie groupoid andJΓ:TΓ→

TΓ its complex structure. Then the map

J =

J_Γ 0 0 ₋J_Γ∗

,

defines a Glanon groupoid structure onΓ.

3.2. Glanon Lie algebroids. LetA be a Lie algebroid overM. Recall thatP_{A=T A⊕T}∗_A has the structure of a Lie algebroid overT M_⊕A∗.

Definition 3.6. A Glanon Lie algebroid is a Lie algebroid A endowed with a generalized complex structureJA:PA→PA that is a Lie algebroid morphism.

The induced Poisson structureπ on Ain that case is linear, and the mapπ♯:T∗A_→T Ais a Lie algebroid morphism over some mapA∗_→T M. The linear Poisson structure onA is then equivalent to a Lie algebroid structure onA∗, such that (A, A∗) is a Lie bialgebroid [28].

Proposition 3.7. If A is a Glanon Lie algebroid, then (A, A∗) is a Lie bialgebroid.

Example 3.8(Glanon Lie algebras). Letgbe a Lie algebra. ThenP_g=g×(g⊕g∗_{)is a Lie}

algebroid overg∗. Hence, a map

J:P_{g →}P_g,

J(x, y, ξ) = (x,_Jx,g(y, ξ),Jx,g∗(y, ξ))

can only be a Lie algebroid morphism over j: g∗ _→ g∗ if _Jx,g∗ does not depend on the g

-component. That is, the mapJx:{x} ×g×g∗→ {x} ×g×g∗ has the matrix

nx πx♯

0 −n∗_x

.

It thus follows that it must be equivalent to a complex Lie bialgebra.

Example 3.9 (Symplectic Lie algebroids). Let (M, π) be a Poisson manifold. LetA be the cotangent Lie algebroidA= (T∗M)π. Then the map

0 ω_A♭−1 ω_A♭ 0

!

whereω_Ais the canonical cotangent symplectic structure onA, defines a Glanon Lie algebroid structure onA.

Example 3.10(Holomorphic Lie algebroids). LetA be a holomorphic Lie algebroid. LetAR

be its underlying real Lie algebroid andj:T A_R→T A_Rthe corresponding complex structure. Then the map

j 0

0 ₋j∗

3.3. Main theorem. Now we are ready to state the main theorem of this paper.

Theorem 3.11. If Γ is a Glanon groupoid with Lie algebroid A, then A is a Glanon Lie algebroid.

Conversely, given a Glanon Lie algebroid A, if Γ is a s_{-connected and} s_{-simply connected Lie}

groupoid integrating A, then Γ is a Glanon groupoid.

Remark 3.12. Ortiz shows in his thesis [31] that multiplicative Dirac structures on a Lie groupoidΓ⇉M are in one-one correspondence with morphic Dirac structures on its Lie alge-broid, i.e. Dirac structures D_A⊆P_{A such that}D_A is a subalgebroid of P_A→T M _⊕A∗ over a setU _⊆T M_⊕A∗.

By extending this result to complex Dirac structures and using the fact that a multiplicative generalized complex structure onΓ⇉M is the same as a pair of tranversal, complex conjugate, multiplicative Dirac structures in the complexified P_{Γ, one finds an alternative method for}

proving our main theorem. This alternative proof relies crucially on the integration of VB-algebroids to VB-groupoids described in[5]and hence is far more technical than our approach.

Applying Theorem 3.11 to Example 3.4 and Example 3.9, we obtain immediately the following

Theorem 3.13. Let (P, π) be a Poisson manifold. If Γ is a s_{-connected and} s_-simply

con-nected Lie groupoid integrating the Lie algebroid(T∗P)π, then Γadmits a symplectic groupoid

structure.

Similarly, applying Theorem 3.11 to Examples 3.5 and 3.10, we obtain immediately the fol-lowing theorem.

Theorem 3.14. If Γ is an s_{-connected and} s_{-simply connected Lie groupoid integrating the}

underlying real Lie algebroid AR of a holomorphic Lie algebroid A, then Γ is a holomorphic

Lie groupoid.

A Glanon groupoid is automatically a Poisson groupoid, while a Glanon Lie algebroid must be a Lie bialgebroid. The following result reveals their connection

Theorem 3.15. Let Γ be a Glanon groupoid with its Glanon Lie algebroid A, (Γ, π) and

(A, A∗)their induced Poisson groupoid and Lie bialgebroid respectively. Then the corresponding Lie bialgebroid of (Γ, π) is isomorphic to (A, A∗).

3.4. Tangent Courant algebroid. In [3], Boumaiza-Zaalani proved that the tangent bun-dle of a Courant algebroid is naturally a Courant algebroid. In this section, we study the Courant algebroid structure onP_{T M} in terms of the isomorphism _{Σ :T}P_{M →}P_{T M} defined in Proposition 2.9.

First we need to introduce some notations. For everyV _∈X(M), setT_{V =σ(T V)∈X(T M)}.

For everyα_∈Ω1_{(M), setTα=ς(T α)∈Ω}1_{(T M). For everyf ∈C}∞_{(M), setTf = pr}

2◦T f ∈

C∞(T M,R₎.

Note that ifvm = ˙c(0)∈TmM, then

T_{f(vm) =}T_{f( ˙c(0)) =} d

dt    

t=0

(f_◦c(t)), (3)

Now introduce the mapT_{: Γ(}P_M)→Γ(P_{T M)} given by

T_{(V, α) = Σ(T V, T α) = (}T_V,T_α),

for all(V, α)∈Γ(P_M).

The main result of this section is the following:

Proposition 3.16. For any e1, e2∈Γ(PM), we have

JTe₁,Te₂K=T_Je₁, e₂K, (4)

hT_e1,T_{e2iT M =}T_{(he1, e2iM).} (5)

The following results show that Tf _∈ C∞(T M), TV _∈ X(T M) and Tα _∈ Ω1(T M) are the

complete lifts of f _∈C∞(M),V _∈X(M) and α_∈Ω1_{(M) in the sense of [39].}

Lemma 3.17. For allf _∈C∞(M) andV _∈X(M), we have T_V(T_{f) =}T_(V(f)).

Proof. Letφbe the flow of V. For any vm = ˙c(0)∈TmM, we have

(T_V(T_{f)) (vm) = (}T_V)(vm)(T_f) =σ d dt     t=0 d ds     s=0

φs(c(t))

T_f = d ds     s=0 d dt     t=0

φs(c(t))

(T_f)

= d ds     s=0 Tf d dt     t=0

φ_s(c(t)) (3) = d ds     s=0 d dt     t=0

(f _◦φs) (c(t))

= d dt     t=0 d ds     s=0

(f_◦φs) (c(t))

= d dt     t=0

V(f)(c(t))(3)= T_(V(f))(vm).

The following lemma characterizes the sectionsT_α of _Ω1_{(T M)}.

Lemma 3.18. (a) For allα_∈Ω1(M) andV _∈X(M), we have

*T_V,T_{α+T T M =}T_{(*V, α+T M).} (6)

(b) Given ξ_∈Ω1(T M), we have ξ = 0 if and only if

*TV, ξ+T T M = 0, ∀V ∈X(M).

Proof. (a) Usingσ2 = idT T M, we get:

*T_V,T_α+p

T M =*σ(T V),(σ

∗

◦I)(T α)+pT M =*T V, I(T α)+T pM

=hhT V, T α_iiT M (2)

(b) Let ξ _∈Ω1_{(T M) be such that ξ(TV) = 0for all V ∈X(M). For anyu ∈T M with}

u ₆= 0andv _∈Tu(T M), there exists a vector field V ∈X(M) such thatTV(u) =v.

This yields ξ(v) = 0. Therefore,ξ vanishes at all points of T M except for the zero

section of T M. By continuity, we getξ = 0.

Using this, we can show the following formulas.

Lemma 3.19. (a) For allV, W _∈X(M), we have

[T_V,T_{W] =}T_{[V, W].}

(b) For any α, β_∈Ω1(M) andV, W _∈X(M), we have

£T_VTβ−iT_WdTα=T(£_Vβ−i_Wdα).

Proof. (a) This is an easy computation, using the fact that ifφis the flow of the vector field V, thenT φis the flow of T_{V (alternatively, see [27]).}

(b) For any U _∈X(M), we can compute

*T_U,£_TVT_β−i_TWdT_α+p

T M =T_{V *}T_β,T_U+p

T M −*Tβ,T[V, U]+pT M −TW *Tα,TU+pT M

+T_U*T_α,T_{W +p}

T M +*Tα,T[W, U]+pT M

=TV (T_*β, U+pM)−T*β,[V, U]+pM −TW(T*α, U+pM) +T_U(T_{*α, W+p}

M) +T*α,[W, U]+pM

=T_{(V *β, U+p}

M)−T*β,[V, U]+pM −T(W *α, U+pM)

+T_{(U*α, W+p}

M) +T*α,[W, U]+pM

=T_*U,£_Vβ−i_Wd_α+p M = *TU,T₍£_Vβ−i_Wd_α)+p

T M .

We get

*T_U,T₍£_Vβ−i_Wd_α)−(£_TVT_β−i_TWdT_α)+p

T M = 0

for all U _∈X(M)and we can conclude using Lemma 3.18.

Proof of Proposition 3.16. Equation (5) follows immediately from (6).

Formula (4) for the Dorfman bracket on sections of P_{T M =}T(T M)⊕T∗(T M) follows from

Lemma 3.19.

3.5. Nijenhuis torsion. Now let_J:P_{M →}P_M be a vector bundle morphism over the iden-tity. Consider the map T_J:TP_{M →} TP_M and the map T_{J defined by the commutative}

diagram

TP_M TJ // Σ

TP_M

Σ

P_{T M} TJ

/

/P_{T M} ,

i.e.T_{J = Σ◦}T_{J ◦}Σ−1. Then, by definition, we get, for alle_∈Γ(P_M),

T_{J (}T_{e) = (Σ◦TJ) (T e) = Σ(T(J(e))) =}T_(J(e)). (7)

Lemma 3.20. (a) T_(idP_M) = idP_{T M}.

(b) We have T_(J2_{) = (}T_(J))2 _{for every base-preserving endomorphismJ of the vector}

bundle P_M.

If_J:P_{M →}P_M is now a generalized complex structure, the Nijenhuis torsion is a tensor and hence can be seen as a vector bundle map

NJ:P_{M ×M} P_{M →}P_M. We consider as above

T_NJ:TP_{M ×T M} TP_{M →}TP_M.

DefineT_NJ:P_{T M ×T M} P_{T M →}P_{T M} by the following commutative diagram: TP_{M ×T M} TP_M TNJ //

Σ×Σ

TP_M

Σ

P_{T M ×T M} P_{T M}

TNJ

/

/P_{T M} .

An easy computation using (7) and (4) yields

T_NJ(T_e1,T_{e2) =NTJ(}T_e1,T_e2)

for alle1, e2 ∈Γ(PM). As in the proof of Lemma 3.18, this implies thatTNJ andNT_J coincide at all points of T M outside the zero section of T M. By continuity, we obtain the following theorem.

Theorem 3.21. Let J:P_{M →}P_{M be a vector bundle morphism. Then} T_{NJ =NTJ.}

LetΓ⇉M andΓ′⇉M′be Lie groupoids. Recall that a mapΦ : Γ→Γ′is a groupoid morphism if and only if the map Φ_×Φ_×Φ restricts to a map ΛΓ → ΛΓ′, where Λ_Γ and Λ_Γ′ are the graphs of the multiplications inΓ⇉M and respectivelyΓ′⇉M′.

Consider the graphs of the multiplications onTΓand T∗Γ:

ΛTΓ={(vg, vh, vg⋆ vh)|vg, vh ∈TΓ, Tt(vh) =Ts(vg)}=TΛΓ

and

ΛT∗_Γ={(α_g, α_h, α_g⋆ α_h)|α_g, α_h ∈T∗Γ,ˆt(α_h) = ˆs(α_g)}. Recall that if

(ΛT∗_Γ)op ={(α_g, α_h,−(α_g⋆ α_h))|α_g, α_h ∈T∗Γ,ˆt(α_h) = ˆs(α_g)}, i.e.

ΛTΓ⊕ΛΓ(ΛT∗Γ)

op _{= (id}

P_Γ×idP_Γ×IΓ) (Λ_TΓ⊕Λ_ΓΛ_T∗_Γ), then

(ΛT∗_Γ)op = (TΛ_Γ)◦.

A map J:P_{Γ →}P_Γ is a groupoid morphism if and only if _ΛTΓ⊕Λ

ΓΛT∗Γ is stable under the

map_{J × J × J}. This yields:

Lemma 3.22. The mapJ is a groupoid morphism if and only if ΛTΓ⊕ΛΓ(ΛT∗Γ) op

Since ΛTΓ = TΛΓ and (ΛT∗_Γ)op ∼= (TΛ_Γ)◦, we get that J is multiplicative if and only if TΛΓ⊕(TΛΓ)◦ is stable underJ × J ×J¯.

Further, ifJ is orthogonal relative toh·,_·iΓand satisfiesJ2 _=−id

P_Γ (i.e. ifJ is a generalized almost complex structure), then we say for short that the Nijenhuis torsion_NJ:PΓ×ΓPΓ→PΓ

ismultiplicativeif it is a Lie groupoid morphism. Similarly as above, we find that the Nijenhuis torsionNJ is multiplicative if and only ifNJ × NJ × NJ restricts to a map

(ΛTΓ⊕ΛΓΛT∗Γ)×ΛΓ(ΛTΓ⊕ΛΓΛT∗Γ)→ΛTΓ⊕ΛΓΛT∗Γ.

Lemma 3.23. The mapNJ is multiplicative if and only if NJ× NJ× N_J¯ restricts to a map (TΛΓ⊕ΛΓ(TΛΓ)

◦_)×Λ

Γ(TΛΓ⊕ΛΓ(TΛΓ)

◦_)→TΛ

Γ⊕ΛΓ(TΛΓ) ◦_.

The following lemma is easy to prove.

Lemma 3.24. If M is a smooth manifold and N a submanifold of M, then the Courant-Dorfman bracket onP_{M restricts to sections of} T N_⊕(T N)◦.

Note thatT N _⊕(T N)◦ is a generalized Dirac structure in the sense of [1]. We get the following theorem.

Theorem 3.25. Let(Γ⇉M,_J)be a generalized almost complex groupoid. Then the Nijenhuis tensorNJ is a Lie groupoid morphism NJ:P_Γ×ΓP_Γ→P_Γ.

Proof. Choose sections ξ1, ξ2, ξ3, η1, η2, η3 ∈ Γ(PΓ) such that (ξ1, ξ2, ξ3)|ΛΓ and (η1, η2, η3)|ΛΓ

belong to Γ(TΛΓ⊕ΛΓ (TΛΓ) ◦

). Then (_Jξ1,Jξ2,J¯ξ3)|ΛΓ and (Jη1,Jη2,J¯η3)|ΛΓ belong to

Γ(TΛΓ⊕ΛΓ(TΛΓ)

◦_{). From Lemma 3.24, it follows that}

(_{NJ × NJ × N}J¯) ((ξ1, ξ2, ξ3),(η1, η2, η3))

takes values in

TΛΓ⊕ΛΓ(TΛΓ) ◦

onΛΓ. By Lemma 3.23, the proof is complete.

3.6. Infinitesimal multiplicative Nijenhuis tensor.

Definition 3.26. Let J:P_Γ→P_{Γ be a Lie groupoid morphism. The map}

A(J) :P_A→P_A

is defined by the commutative diagram

A(P_Γ) P_A

A₍P_Γ) P_A

ΣΓ

A_{(J) A(J)}

ΣΓ

.

The following lemma can be found in [5].

Lemma 3.27. Let J :P_Γ→P_{Γ be a multiplicative map. ThenA(J) is an endomorphism of}

Now assume that _J is orthogonal relative to _h·,_·iΓ and that _J2 = ₋idP_Γ. Since the map

NJ: P_{Γ ×Γ} P_{Γ →} P_Γ is then also a Lie groupoid morphism by Theorem 3.25, we can also consider

A(NJ) :P_A×AP_A→P_A defined by

A₍P_Γ)×AA₍P_Γ) P_A×AP_A

A₍P_Γ) P_A

A_(NJ)

Σ2 Γ

A(NJ)

ΣΓ

.

The main result of this section is the following.

Theorem 3.28. Suppose that a map_J:P_Γ→P_{Γ is simultaneously a vector bundle morphism}

and a Lie groupoid morphism.

(a) Then the map A(_J) is_h·,_·iA-orthogonal if and only if J ish·,·iΓ-orthogonal.

(b) Moreover,if J is orthogonal w.r.t.h·,_·iΓ and J2 _=−id

P_Γ, thenA(NJ) =NA(J).

For the proof, we need a couple of lemmas.

Definition 3.29. Let M be a smooth manifold and ι:N ֒_→M a submanifold of M.

(a) A section e_N =: X_N +α_N is ι-related to e_M = X_M +α_M if ι∗XN = XM|N and

αN =ι∗αM. We write then eN ∼ι eM.

(b) Two vector bundle morphisms JN:PN → PN and JM:PM → PM are said to be

ι-related if for each section eN of PN, there exists a section eM ∈ PM such that

eN ∼ι eM andJN(eN)∼ι JM(eM).

(c) Two vector bundle morphisms NN: PN ×N PN → PN and NM: PM ×M PM →

P_{M are} ι-related _{if for each pair of sections eN, fN ∈ Γ(}P_{N), there exist sections} e_M, f_{M ∈}Γ(P_{M) such that} e_{N ∼}ι eM, fN ∼ι fM and NN(eN, fN)∼ι NM(eM, fM).

Lemma 3.30. Let M be a manifold and N _⊆ M a submanifold. If eN, fN ∈ Γ(PN) are

ι-related to e_M, f_{M ∈}Γ(P_{M), then J}e_N, f_NK∼ι JeM, fMK, for the Courant-Dorfman bracket.

Proof. This is an easy computation, see also [33].

Lemma 3.31. Let M be a manifold, N _⊆ M a submanifold and JN: PN → PN and JM:PM → PM two ι-related generalized almost complex structures. Then the generalized

Nijenhuis tensors NJN andNJM are ι-related.

Proof. This follows immediately from Lemma 3.30 and the definition.

Recall that the Lie algebroidA of a Lie groupoid Γ⇉M is an embedded submanifold of TΓ, ι:A ֒_→TΓ.

Remark 3.32. Note that Lemma 2.11 states that ifu_∈ΓA(A(PΓ))andu˜∈ΓTΓ(TPΓ) is an

extension ofu, thenΣΓ◦u∼ι Σ◦u˜.

Lemma 3.33. Let Γ⇉M be a Lie groupoid and J:P_{Γ →} P_{Γ a vector bundle morphism. If}

J is multiplicative, then

(a) A(J) :P_A→P_{A and}T_J:P_TΓ→P_{TΓ are}ι-related, and

Proof. (a) Choose a sectioneAofPA. Then we haveΣ−1_Γ (eA) =:u∈Γ(A(PΓ))and since

A₍P_Γ)⊆TP_Γ, we find a sectionu_˜ofTP_Γsuch thatu_˜restricts tou. SeteTΓ:= Σ◦u.˜

By Lemma 2.11, we have then eA ∼ι eTΓ. Furthermore, by construction of A(J),

we know that A_{(J)◦u= (TJ)◦u˜ onT M⊕A}∗ _{=T M⊕T M}◦ _⊆P_{Γ. We have then}

A(_J)(eA) = ΣΓ◦A(J)◦u∼ιΣ◦TJ ◦u˜=TJ(eTΓ).

(b) By definition of A(_NJ), this can be shown in the same manner.

Proof of Theorem 3.28. (a) The mapA(_J)is _h·,_·iA-orthogonal if and only if h·,_·iA◦(A(J),A(J)) =h·,_·iA.

We have

h·,_·iA=A(h·,_·iΓ) =A_(h·,_·iΓ)◦(Σ−1_Γ ,Σ−1_Γ )

by definition and so

h·,_·iA◦(A(J),A(J))

=A_(h·,_·iΓ)◦(Σ_Γ−1,Σ−1_Γ )◦(ΣΓ,ΣΓ)◦(A(J),A(J))◦(Σ−1Γ ,Σ−1Γ ) =A _{h·,·iΓ◦(J,J)}

◦(Σ−1_Γ ,Σ−1_Γ ).

Since ΣΓ:A(PΓ)→PA is an isomorphism, we get thatA(J) ish·,·iA-orthogonal if

and only if

A_{(h·,·iΓ) =}A _{h·,·iΓ◦(J,J)}

and we can conclude.

(b) Choose sections eA, fA ∈Γ(PA) andu, v ∈ΓA(A(PΓ))such thateA= ΣΓ◦u, fA=

ΣΓ◦v. Choose as in the proof of Lemma 3.33 two extensions u˜ and v˜∈ΓTΓ(TPΓ)

of u andv and seteTΓ:= Σ◦u˜and fTΓ:= Σ◦v˜∈Γ(PTΓ). Then we have

e_A∼ι eTΓ, fA∼ι fTΓ,

A(_J)(e_A)_∼ι TJ(eTΓ), A(J)(fA)∼ι TJ(fTΓ), (8)

and A(_NJ)(eA, fA)∼ι TNJ(eTΓ, fTΓ).

But by Lemma 3.31, (8) yields also

NA(J)(eA, fA)∼ι NTJ(eTΓ, fTΓ).

Since _NTJ =TNJ by Theorem 3.21, we get that

A(NJ)(e_A, f_A)∼ι NTJ(eTΓ, fTΓ) and NA(J)(eA, fA)∼ι NTJ(eTΓ, fTΓ).

This yields

A(NJ)(e_A, f_A) =NA(J)(eA, fA)

and the proof is complete.

The following corollary is immediate.

Corollary 3.34. Let _J: P_{Γ →} P_{Γ be a multiplicative generalized almost complex structure.}

3.7. Proof of the integration theorem.

Proof of Theorem 3.11. By Lemma 3.27, the map A(_J) :P_{A →} P_A is a vector bundle mor-phism. Since _J2 _{= −id}

P_Γ, it follows from Lemma 3.20 that (A(J))2 = −idP_A. From Corollary 3.34 and Theorem 3.28, we infer thatNA(J) = 0 andA(_J) is h·,_·iA-orthogonal.

Since

A₍P_Γ)

A_(J)

/

/

A₍P_Γ)

T M_⊕A∗ //_{T M⊕A}∗

is a Lie algebroid morphism and

A₍P_Γ)

Σ|A₍P Γ) //

P_A

T M_⊕A∗ //_{T M⊕A}∗

is a Lie algebroid isomorphism over the identity, the map

A(_J) = Σ_|A₍P_Γ)◦A(J)◦ Σ|A₍P_Γ) −1

= Σ_|A₍P_Γ)◦A(J)◦Σ−1|P_A, is a Lie algebroid morphism

P_A A(J) //

P_A

T M_⊕A∗ //_{T M⊕A}∗

.

For the second part, consider the map

AJ := Σ−1|P_A◦ J_A◦Σ|A₍P_Γ):A(P_Γ)→A(P_Γ).

SinceJA:PA→ PA is a Lie algebroid morphism, AJ is a Lie algebroid morphism and there

is a unique Lie groupoid morphism _J:P_{Γ →} P_Γ such that _{AJ =}A_(J). By Lemma 3.27, _J is a morphism of vector bundles.

We get then immediatelyJA=A(J). SinceJA2 =−idA, we getA(J2) =−idA=A(−idP_Γ)

and we can conclude by Theorem 3.28.

4. Application

4.1. Holomorphic Lie bialgebroids. Given a complex manifoldX, letΘX denote the sheaf

of holomorphic vector fields onX.

Let A _→ X be a holomorphic vector bundle and let ρ: A _→ T_X be a holomorphic vector bundle map, which we call anchor. When the sheaf_A of holomorphic sections ofA_→X is a sheaf of complex Lie algebras, the anchor mapρ induces a homorphism of sheaves of complex Lie algebras from_Ato ΘX, and the Leibniz identity

holds for all V, W _{∈ A}(U), f _{∈ O}X(U), and all open subsets U of X, we say that A is a

holomorphic Lie algebroid. Holomorphic Lie algebroids were studied in various contexts, see for instance [4, 12, 23, 16, 36].

Since the sheaf A locally generates the C∞(X)-module of all smooth sections of A, each holomorphic Lie algebroid structure on a holomorphic vector bundle A _→ X determines a unique smooth real Lie algebroid structure onA.

Proposition 4.1 ([23]). Let A _→ X be a holomorphic vector bundle and let ρ:A _→ T_X be a holomorphic vector bundle map. Given a structure of holomorphic Lie algebroid on A with anchor ρ, there exists a unique structure of smooth real Lie algebroid on A with the same anchor ρ such that the inclusion of the sheaf of holomorphic sections into the sheaf of smooth sections is a morphism of sheaves of Lie algebras overR_.

Conversely, given a real Lie algebroid A _→ X, it is a holomorphic Lie algebroid if A _→ X is a holomorphic vector bundle, with the sheaf of holomorphic sections denotedA, such that the Lie bracket on smooth sections induces aC-linear bracket on _A(U), for all open subsets

U _⊂X. We writeARto denote the real Lie algebroid underlying a holomorphic Lie algebroid

A.

Assume that (A _→ X, ρ,[_·,_·]) is a holomorphic Lie algebroid. Multiplication by the scalar

√

−1 in each fiber of A determines an automorphism j of the vector bundle A. It is simple to see that the Nijenhuis torsion of j vanishes [23]. Hence one can define a new (real) Lie algebroid structure onA (see [21]), with anchorρ_◦j and Lie bracket

[V, W]j := [jV, W] + [V, jW]−j[V, W] =−j[jV, jW], ∀V, W ∈Γ(A), (9)

which we call underlying imaginary Lie algebroid of A and write AI. It follows immediately

thatj:AI →AR is an isomorphism of Lie algebroids [21].

Given a holomorphic vector bundleA_→X, we use the symbols_A,_Ak, and _A• to denote the sheaves of holomorphic sections of the holomorphic vector bundle A _→ X, its k-th exterior power_∧k_{A→X, and the Whithney sum} L

k≥0 ∧kA)respectively.

Definition 4.2. If a holomorphic vector bundleA (with sheaf of holomorphic sectionsA) and its dualA∗ are both holomorphic Lie algebroids and the Chevalley-Eilenberg differentiald∗ of

the Lie algebroidA∗ is a derivation of the (sheaf of ) complex Lie algebras A•_{, i.e.}

d∗[V, W] = [d∗V, W] + [V, d∗W], ∀V, W ∈ A•(U) (10)

for all open subsets U of the base manifold X, we say that the pair (A, A∗) is a holomorphic Lie bialgebroid.

As in the smooth case, a holomorphic vector bundleA_→X is a holomorphic Lie algebroid if and only if(_A•,_∧,[_·,_·])is a sheaf of Gerstenhaber algebras [23].

Proposition 4.3. Let A _→ X be a holomorphic vector bundle. The pair (A, A∗) is a holo-morphic Lie bialgebroid if and only if (_A•,_∧,[_·,_·], d∗) is a sheaf of differential Gerstenhaber

algebras overX.

respect to the exterior multiplication, it follows immediately, as in [20], from the compatibility condition (10) that

d∗[X, f] = [d∗X, f] + [X, d∗f], ∀X∈ A(U), f ∈ OX(U). (11)

Therefore, since the exterior algebraA• _{is generated by its homogeneous elements of degree 0}

and 1, we have

d∗[X, Y] = [d∗X, Y] + [X, d∗Y], ∀X, Y ∈ A•(U).

Thus(A•_{,∧,[·,·], d}

∗) is a sheaf of differential Gerstenhaber algebras over X. The converse is

obvious.

Proposition 4.4. Let(A, A∗) be a holomorphic Lie bialgebroid with anchorsρ:A_→TX and

ρ∗:A∗ →TX. Then

(a) LdfV =−[d∗f, V] for any f ∈ OX(U) and V ∈ A(U);

(b) [d∗f, d∗g] =d∗{f, g},∀f, g∈ OX(U);

(c) ρ◦(ρ∗)∗ =−ρ∗◦ρ∗. Therefore,the holomorphic bundle map

π_X#=ρ_◦(ρ∗)∗:TX∗ →TX.

is skew-symmetric and defines a holomorphic Poisson bivector on X.

Proof. The proof is similar to the proofs of Proposition 3.4, Corollary 3.5 and Proposition 3.6

in [28].

4.2. Associated real Lie bialgebroids. Given a holomorphic Lie algebroid A, we denote its underlying real and imaginary Lie algebroids byARandAI and their respective

Chevalley-Eilenberg differentials by dR and dI. When the dual A∗ of A is also a holomorphic Lie algebroid, its underlying real and imaginary Lie algebroids are written A∗_R and A∗_I and their respective Chevalley-Eilenberg differentialsdR_∗ and dI_∗.

Lemma 4.5. Let A be a holomorphic Lie algebroid over a complex manifold X. Then

(a) dIα=₋(j∗◦dR◦j∗)α, for all α∈Γ(A∗);

(b) dIf = (j∗◦dR)f, for all f ∈C∞(M). Proof. For all V, W _∈Γ(A), we have

(dIf)(V) =ρI(V)(f) =ρ(jV)(f) = (dRf)(jV)

and

(dI(j∗α))(V, W) =ρI(V)α(jW)−ρI(W)α(jV)−α(j[V, W]j)

=ρ(j(V))α(jW)₋ρ(j(W))α(jV)₋α([j(V), j(W)])

= (dRα)(j(V), j(W)).

If the dualA∗ of a holomorphic Lie algebroidA_→X is also endowed with a holomorphic Lie algebroid structure, we can conclude that

dI_∗V =₋(j◦dR_∗◦j)V, ∀V ∈Γ(A), (12)

dI_∗f = (j◦dR_∗)f, ∀f ∈C∞(X),