DOI 10.1007/s10440-009-9444-2
Compatible Structures on Lie Algebroids and Monge-Ampère Operators
Yvette Kosmann-Schwarzbach·Vladimir Rubtsov
Received: 15 November 2008 / Accepted: 23 December 2008 / Published online: 3 February 2009
© Springer Science+Business Media B.V. 2009
Abstract We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tan- gent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid.
These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the re- lationships betweenP N-,P Ω- andΩN-structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional mani- folds, such Monge-Ampère operators give rise to generalized complex structures or gener- alized product structures on the cotangent bundle of the manifold.
Keywords Graded Poisson brackets·Lie algebroids·Poisson structures·Symplectic structures·Nijenhuis tensors·Complementary 2-forms·Bi-Hamiltonian structures· P N-structures·P Ω-structures·ΩN-structures·Hitchin pairs·Dorfman bracket· Courant algebroids·Generalized complex structures·Monge-Ampère operators
Mathematics Subject Classification (2000) Primary 53D17·17B70·58J60·Secondary 37K20·37K25·70G45
Introduction
On the tangent bundle of a manifold or, more generally, on a Lie algebroid, we consider pairs of structures, such as the Poisson-Nijenhuis structures which give rise to hierarchies of Poisson structures (also called Hamiltonian structures) that play a very important role in
Y. Kosmann-Schwarzbach
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau, France e-mail:yks@math.polytechnique.fr
V. Rubtsov (
)Département de Mathématiques, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers, France e-mail:Volodya.Roubtsov@univ-angers.fr
the theory of completely integrable systems. These structures are defined by closed 2-forms, Poisson bivectors or(1,1)-tensors with vanishing Nijenhuis torsion. When suitable compat- ibility assumptions are introduced, one obtains composite structures called complementary 2-forms,P N-,P Ω- andΩN-structures. Krasil’shchik contributed to the study of the al- gebraic nature of Hamiltonian and bi-Hamiltonian structures and was the first to underline the cohomological nature of their compatibility condition (see [24] and references therein).
While the first part of this article is a comprehensive survey of the relationships between such composite structures and the related notion of Hitchin pairs, the second part provides numerous examples arising from the theory of Monge-Ampère equations.
Our formulations and proofs make essential use of the big bracket, the even graded bracket on the spaceF of functions on the cotangent bundle of a Lie algebroid consid- ered as a supermanifold. What we call the big bracket was first introduced by Kostant and Sternberg [23]; its use in the theory of Lie bialgebras is due to Lecomte and Roger [26]
and was developed by one of us [15]. Roytenberg extended it to Lie algebroids [33] and Courant algebroids [34]. Recently, it has been used by Antunes [1] in the study of compos- ite structures arising in the theory of sigma-models. In practice, all proofs are reduced to a straightforward use of the graded Jacobi identity, sometimes repeatedly. While many of our results can be found in the literature (see [8], which contains the references to earlier work by Magri, Gelfand and Dorfman, Fokas and Fuchssteiner, see [21,31,32], and the more recent articles [1,10,37,38]), we claim that our method unifies results, generalizing the known properties from the case of manifolds to that of Lie algebroids and Courant al- gebroids. Our main argument is that the big bracket formalism can be applied to problems in the geometric theory of partial differential equations developed in [25,29,30] and [2,3].
We also stress that this theory can be considered in the general framework of Lie algebroids, and we wish to introduce a general abstract theory of Monge-Ampère structures on arbitrary Lie algebroids. In particular, the symplectic Monge-Ampère equations defined byn-forms on the cotangent bundle of a smooth,n-dimensional manifoldM and, more generally, the Jacobi first-order systems, defined by a set of 2-forms on anm+2-dimensional manifoldM, can be viewed as “deformations” of the standard Lie algebroid structure on the tangent bun- dleT (T∗M)ofT∗M. We shall indicate some links between our approach and the approach to the geometric structures developed by Hitchin [14] and Gualtieri [12] in their studies of generalized complex and Kähler structures, a new and fast developing field of differential geometry and of mathematical physics (see, e.g., [28]).
In Sect.1, we introduce the big bracket, we recall the definition of Lie algebroids and give the explicit expression for the Dorfman bracket on the double of a Lie bialgebroid which is a derived bracket [16,18] of the big bracket. The Courant algebroid structure of the double of a Lie bialgebroid is defined by the skew-symmetrized version of the Dorfman bracket, called the Courant bracket. Section2deals with general facts and formulas involv- ing bivectors, forms and (1,1)-tensors that will be used in subsequent sections, and with Grabowski’s formula (2.9) that expresses the Nijenhuis torsion of a(1,1)-tensor in terms of the big bracket [10]. In Sect.3, we show that the adjoint actions of a non-degenerate 2-form and of its inverse bivector induce a representation ofsl2onF, we define the primitive ele- ments and describe a Hodge-Lepage type decomposition of the elements inF.
Section4is a study of the complementary 2-forms introduced by Vaisman [37,38]. We prove that, given a Lie algebroidA, “ωis a complementary 2-form for the Poisson bivector π” is a sufficient condition for the bracket obtained by first dualizing the Lie algebroid struc- ture ofAbyπand dualizing again byωto be a Lie algebroid bracket onA, whose expression we easily derive. A remark concerning the corresponding modular class (Sect.4.4) will be used in Sect.13.3. Sections5to10contain the detailed analysis of the structures introduced
by Magri and Morosi [31,32] defined by a Poisson bivector and a Nijenhuis tensor, called P N-structures (Sect.5), by compatible Poisson tensors (Sect.6), by a closed 2-form and a Poisson bivector, calledP Ω-structures (Sect.7) and by a closed 2-form and a Nijenhuis tensor, calledΩN-structures (Sect.8) and Hitchin pairs introduced by Crainic [6] (Sect.9).
A table and a diagram summarize the relationships between these various structures.
Section11 deals with Nijenhuis tensors on Courant algebroid. We state Grabowski’s theorem [10] that characterizes generalized complex structures by a simple equation in terms of the big bracket.
In Sects.12–14, we describe the geometry of the symplectic Monge-Ampère equations and relate it to the structures discussed in the previous sections, using the formalism of the big bracket. Some of these results are reformulations of results in [25] and [2,3]. Section12 introduces Monge-Ampère structures on manifolds and the associated Monge-Ampère op- erators and equations. We recall the definition of the effective forms and the one-to-one cor- respondence between Monge-Ampère operators and effective forms. Section13is devoted to the case of Monge-Ampère structures on 2-dimensional manifolds, with an emphasis on the non-degenerate case, when the Pfaffian of the defining 2-form is nowhere vanishing. We show that in the integrable case, i.e., when the Monge-Ampère operator is equivalent to an operator with constant coefficients, the Monge-Ampère structure gives rise toP N- andΩN- structures and to a deformed Lie algebroid structure onT (T∗M)which is unimodular. More generally, a non-degenerate Monge-Ampère structure of divergence type defines a general- ized almost complex structure onT∗M. If the defining 2-form is closed, this structure is integrable and corresponds to a Hitchin pair. The von Karman equation is an example where the integrability condition is not satisfied and the associated composite structures do not sat- isfy the compatibility condition. We then consider the first-order Jacobi differential systems which generalize the Monge-Ampère equations, and we describe the associated geometric structures on 2-dimensional manifolds. In Sect.14, we proceed to study Monge-Ampère op- erators on 3-dimensional manifolds, recall the classification of the non-degenerate Monge- Ampère operators, and we prove that when the operator is non-degenerate, i.e., when the Hitchin Pfaffian is nowhere-vanishing, and has constant coefficients, there is either an asso- ciated generalized complex structure or generalized product structure onT∗M. We conclude with a short discussion of two definitions of the generalized Calabi-Yau manifolds.
1 The Big Bracket
WhenA→Mis a vector bundle, letT∗[2]A[1]denote the cotangent bundle of the graded manifoldA[1]obtained fromAby assigning degree 0 to the coordinates on the base and de- gree 1 to the coordinates on the fibers. The space F of smooth functions onT∗[2]A[1]
is a bigraded Poisson algebra [20, 33]. (See [15] for the case where M is a point and therefore A is a vector space.) If (xi, ξa), i=1, . . . ,dimM and a=1, . . . ,rankA, are coordinates on A[1], then coordinates on T∗[2]A[1] are (xi, ξa, pi, θa), with bidegrees (0,0), (0,1), (1,1), (1,0), respectively. If an elementuofF is of bidegree(p+1, q+1), we call|u| =p+q+2 its (total) degree and we call(p, q)its shifted bidegree,p≥ −1, q≥ −1. The spaceFp,q of elements ofF of shifted bidegree(p, q)contains the space of sections of∧p+1A⊗ ∧q+1A∗.
As the cotangent bundle of a graded manifold,T∗[2]A[1]is canonically equipped with an even Poisson structure. We denote the even Poisson bracket onF by{,}, and we call it the big bracket. It satisfies{xi, pj} =δij and{ξa, θb} =δab, so that{f, pj} =∂jf, where f ∈C∞(M)and∂jf= ∂x∂fj. This bracket is of bidegree(−1,−1)and of shifted bidegree
(0,0). It is skew-symmetric,{u, v} = −(−1)|u||v|{v, u}, for alluandv∈F, and it satisfies the Jacobi identity,
{u,{v, w}} = {{u, v}, w} +(−1)|u||v|{v,{u, w}}, for allu,vandw∈F. We often use the Jacobi identity in the form,
{{u, v}, w} = {u,{v, w}} +(−1)|v||w|{{u, w}, v}.
The big bracket satisfies the Leibniz rule,
{u, v∧w} = {u, v} ∧w+(−1)|u||v|v∧ {u, w}, or
{u∧v, w} =u∧ {v, w} +(−1)|v||w|{u, w} ∧v.
The space of sections of a vector bundleEis denoted by E. We call a section of∧•E (resp., ∧•E∗) a multivector (resp., a form) onE. Accordingly, we use the terms vector, bivector,k-form,(p, q)-tensor, etc. All manifolds and maps are assumed to be smooth.
1.1 Lie Algebroids
A Lie algebroid structure onA→Mis an elementμofFof shifted bidegree(0,1)such that
{μ, μ} =0.
The Schouten bracket of multivectors, i.e., sections of∧•A,XandY, is [X, Y]μ= {{X, μ}, Y}.
In particular, this formula defines the Lie bracket ofXandY ∈ Aas well as the anchor ofA,ρ:A→T M, by
ρ(X)f = {{X, μ}, f}, forX∈ Aandf∈C∞(M).
The Lie algebroid differential acting on sections of∧•A∗is denoted bydμ, thus dμ= {μ,·}.
The Lie derivative of forms by X ∈ A is defined to be the graded commutator, LμX= [iX, dμ].
A Lie bialgebroid is defined byμ∈F0,1 andγ ∈F1,0such that{μ+γ , μ+γ} =0.
More generally, a proto-bialgebroid is defined byS=φ+μ+γ+ψ, whereψ∈ (∧3A∗) andφ∈ (∧3A), such that{S, S} =0. Lie quasi-bialgebroids correspond toψ=0, while quasi-Lie bialgebroids correspond toφ=0.
1.2 The Dorfman Bracket
If(A, μ, γ )is a Lie bialgebroid, its double is the vector bundle,A⊕A∗, equipped with the Dorfman bracket defined by
[u, v]D= {{u, μ+γ}, v}, (1.1) foruand v∈ (A⊕A∗). The skew-symmetrized Dorfman bracket is called the Courant bracket andA⊕A∗with the Dorfman bracket is a Courant algebroid. Since the Dorfman bracket (see [8,10,18]) is a derived bracket, it is a Loday-Leibniz bracket and therefore satisfies the (graded) Jacobi identity in the sense that, for eachu∈ (A⊕A∗),[u,·]Dis a derivation of the bracket[,]D(see [16,18]). More generally, formula (1.1) defines a Loday- Gerstenhaber bracket on (∧•A⊗ ∧•A∗).
Explicitly, forX∈ Aandα∈ (A∗),
[X, α]D= {{X, μ}, α} + {{X, γ}, α} = {X,{μ, α}} + {μ,{X, α}} − {{γ , X}, α}
=iX(dμα)+dμ(iXα)−iα(dγX)=LμXα−iα(dγX), while
[α, X]D= {{α, μ}, X} + {{α, γ}, X} = −{{μ, α}, X} + {α,{γ , X}} + {γ ,{α, X}}
= −iX(dμα)+iα(dγX)+dγ(iαX)=LγαX−iX(dμα).
Therefore, forXandY∈ A,αandβ∈ (A∗),
[X+α, Y+β]D= [X, Y]μ+LγαY−iβ(dγX)+ [α, β]γ+LμXβ−iY(dμα). (1.2) In the case of the standard Courant algebroid,T M⊕T∗M, by assumption,γ=0 anddμ is the de Rham differential,d. Thus, forX∈ (T M)andα∈ (T∗M),
[X, α]D= {{X, μ}, α} = {X,{μ, α}} + {μ,{X, α}} =iX(dα)+d(iXα)=LXα, and
[α, X]D= {{α, μ}, X} = −{{μ, α}, X} = −iX(dα).
In addition, it is clear that, for vector fieldsXandY,[X, Y]Dis the Lie bracket, and for 1-forms,αandβ,[α, β]D=0, this bracket vanishes on pairs of 1-forms. Therefore
[X+α, Y+β]D= [X, Y] +LXβ−iY(dα). (1.3) We compute these brackets on T∗[2]T M[1] in local coordinates, (xi, ξi, pi, θi). Here μ=piξi. LetX=Xiθiandα=αiξi. Then
[X, α]D= {{Xiθi, pjξj}, αkξk} = {Xipi−∂jXiθiξj, αkξk} =Xi∂iαkξk+∂jXiαiξj, which is the expression ofLXαin local coordinates. Similarly,
[α, X]D= {{αiξi, pjξj}, Xkθk} = {−∂jαiξiξj, Xkθk} = −Xk∂kαiξi+Xk∂jαkξj, which is the expression of−iX(dα)in local coordinates.
Remark 1 When μis replaced byμ+H, whereH is adμ-closed 3-form, the equation {μ+H, μ+H} =0 is satisfied, and one obtains the Dorfman bracket with background, [,]D,H, on (A⊕A∗),
[X+α, Y+β]D,H= [X+α, Y+β]D+iX∧YH, (1.4) makingA⊕A∗a twisted Courant algebroid [12,33,35].
Any 2-formBonAdefines a gauge transformation,B:X+α→X+α+iXB, satisfying [B(X +α),B(Y +β)]D,H=B([X +α, Y+β]D,H−dμB).
IfBisdμ-closed, thenBis an automorphism of(A⊕A∗,[,]D,H).
2 Tensors and the Big Bracket
We shall need various preliminary results concerning tensors on a Lie algebroid.
2.1 Bivectors, Forms and(1,1)-tensors
Letπ:A∗→Abe the map defined by a bivectorπ, where πα=iαπ, forα∈ (A∗).
Then
πα= {α, π}. (2.1)
Letω:A→A∗be the map defined by a 2-formω, whereωX= −iXω, forX∈ A.
Then
ωX= {ω, X}. (2.2)
LetNˆ: A→ Abe the linear map induced by a vector bundle endomorphism ofA.
Then Nˆcan be identified with a(1,1)-tensor onA, more precisely with a section N of A∗⊗A, by setting
Nˆ(X)= {X, N}, (2.3)
for allX∈ A. In local coordinates, ifNˆhas componentsNba, thenN=Nbaξbθa. We shall not distinguish betweenNˆandN, and we shall abbreviateNˆtoN.
Lemma 1 The mapN=π◦ω:A→Aconsidered as a section ofA∗⊗Ais N= {π, ω}.
Proof By the Jacobi identity, since{X, π} =0, for allX∈ A,
{X, N} = {X,{π, ω}} = {{ω, π}, X} = {{ω, X}, π} = {ωX, π} =(π◦ω)(X).
This proves the result, in view of (2.3).
In local coordinates, let π= 12πabθaθb and ω= 12ωabξaξb. For α=αaξa∈ (A∗), πα = πabαaθb. For X = Xaθa ∈ A, ωX = ωabXbξa, whence (π ◦ ω)(X)
=πabωacXcθb. On the other hand, {π, ω} =1
4πabωcd{θaθb, ξcξd} =πabωbcθaξc, whence{X,{π, ω}} =πabωacXcθb.
In particular, if π is non-degenerate, and if π and ω are inverses of one another, by definition,π◦ω=IdAandω(X)= −iXω, for allX∈ A. Then the(1,1)-tensor{π, ω}
is the identity ofA, IdA. In Sect.3below, we denote the adjoint action of IdA,{IdA,·}, by I.
In local coordinates,πabωac=δcb and{π, ω} =ξaθa, satisfying{X, ξaθa} =X, for all X∈ A.
This relation is a particular case of a general result, proved in [20]: whenπandωare inverses of one another, foru∈Fp,q, in particular foru∈ (∧p+1A⊗ ∧q+1A∗),
{{π, ω}, u} = {IdA, u} =(q−p)u, (2.4) or, in local coordinates,
{ξaθa, u} =(q−p)u. (2.5) Therefore, for a bracketμ(resp., cobracketγ) onA,
{IdA, μ} =μ (resp.,{IdA, γ} = −γ )
and for a 3-formψ(resp., 3-tensorφ) onA,{IdA, ψ} =3ψ(resp.,{IdA, φ} = −3φ).
2.2 Deformed Brackets and Torsion
Let(A, μ)be a Lie algebroid. LetN∈ (A∗⊗A)be a(1,1)-tensor onA, an element of shifted bidegree(0,0). Then the deformed structure,
μN= {N, μ},
defines an anchorρ◦Nand a skew-symmetric bracket onAwhich we shall denote by[,]μN, explicitly,
[X, Y]μN= {{X,{N, μ}}, Y}, (2.6) forXandY∈ A.
Lemma 2 The bracket[,]μN is such that, forX,Y∈ A,
[X, Y]μN= [N X, Y]μ+ [X, N Y]μ−N[X, Y]μ. (2.7) Proof By definition,
[X, Y]μN= {{X,{N, μ}}, Y} = {{{X, N}, μ}, Y} + {{N,{X, μ}}, Y}
= [N X, Y]μ+ {N,{{X, μ}, Y}} + {{N, Y},{X, μ}}
= [N X, Y]μ+ [X, N Y]μ−N[X, Y]μ,
where we have used the Jacobi identity and the definition of[,]μ.
The bracket[,]μN is called the deformed [21] (or contracted [4,5,10]) bracket of[,]μ. ByTμNwe denote the Nijenhuis torsion ofNdefined by
(TμN )(X, Y )= [N X, N Y]μ−N ([N X, Y]μ+ [X, N Y]μ)+N2[X, Y]μ, (2.8) for allXandY∈ A. It is clear that(TμN )(X, Y )= [N X, N Y]μ−N ([X, Y]μN).
Proposition 1 In terms of the big bracket, TμN=1
2({N,{N, μ}} − {N2, μ}), (2.9) and
1
2{{N, μ},{N, μ}} = {μ,TμN}. (2.10) Proof See Grabowski [10] for formula (2.9), which is proved by a simple calculation. for- mula (2.10) follows from (2.9) by an application of the Jacobi identity.
Remark 2 Formula (2.9) can also be viewed as a particular case of formulas (5.22) and (5.16) of [16], taking into account the fact that, for vector-valued forms, the big bracket and the Richardson-Nijenhuis bracket coincide up to sign (see [15]), or as a particular case of formula (3.14) of [18]. Formula (2.9) also appears in a slightly different form in [8], Sect. 3.3. It plays an essential role in [1].
2.3 Nijenhuis Structures
LetN∈ (A∗⊗A)be a(1,1)-tensor onA, thusNis an element of shifted bidegree(0,0).
Then the deformed structure bracket is defined by (2.6), and its explicit expression is for- mula (2.7) above. We have denoted the Nijenhuis torsion ofNbyTμN. The following result (see, e.g., [21]) is an immediate corollary of Proposition1.
Theorem 1 A necessary and sufficient condition (resp., a sufficient condition) for the de- formed structureμN= {N, μ}to be a Lie algebroid structure onAis
{μ,TμN} =0 (resp.,TμN=0).
When{μ,TμN} =0, we callTμNadμ-cocycle.
Remark 3 The deformed Lie algebroid structureμN is compatible withμin the sense that μ+μN is a Lie algebroid structure, i.e.,{μ+μN, μ+μN} =0.
The operator on (∧•A∗)associated toμNisdμN = {μN,·} = {{N, μ},·}.
By definition, a(1,1)-tensorN is an almost complex structure ifN2= −IdA, and an almost complex structureNis a complex structure ifTμN=0.
Proposition 2 An almost complex structureNis a complex structure if and only if
{{N, μ}, N} =μ. (2.11)
Proof Equation (2.11) follows from (2.9) and the relation{IdA, μ} =μ, a particular case
of (2.4).
2.4 Bivectors and 3-forms
Lemma 3 Ifπis a bivector andψis a 3-form onA, then, forX,Y∈ A,
{{X,{ψ, π}}, Y} =π(iX∧Yψ ), (2.12) with the conventioniX∧Y =iX◦iY.
Proof Since{X, π} = {Y, π} =0, by the Jacobi identity,
{{X,{ψ, π}}, Y} = {{{X, ψ}, π}, Y} = {{{X, ψ}, Y}, π}.
Now {X, ψ} =iXψ, {{X, ψ}, Y} =iXiYψ. Applying formula (2.1) to the 1-form α =
{{X, ψ}, Y}, we obtain (2.12).
In local coordinates, let π = 12πabθaθb and ψ = 16ψabcξaξbξc. Then {ψ, π}
= 12πcdψabcξaξbθd. For X=Xaθa, {X,{ψ, π}} = πcdψabcXaξbθd, and for Y =Yaθa, {{X,{ψ, π}}, Y} = πdcψabcXaYbθd. On the other hand, iXiYψ = ψabcXbYaξc, and π(iX∧Yψ )=πdcψabcXaYbθd.
3 A Representation ofsl2
Foru∈Fp,q, we shall callw(u)=q−pthe weight ofu. Letπbe a non-degenerate bivector andωa 2-form onAwhich are inverses of one another. Then{π, ω} =IdA. Set adω= {ω,·}
and adπ= {π,·}. Then I= {{π, ω,},·}acts onFby I(u)=w(u)u,
foru∈Fp,q (see formula (2.4)). Let adπ= {., π} = −adπ be the right adjoint action ofπ.
Then
[I,adω] =2adω, [I,adπ] = −2adπ [adω,adπ] =I,
where[,]denotes the commutator of operators. Therefore the operators(adω,adπ,I)define a representation ofsl2on the linear spaceFwhich restricts to the linear space of all tensors, analogous to the representation on forms in [25]. Then
adω(Fp,q)⊂Fp−1,q+1, adπ(Fp,q)⊂Fp+1,q−1, so thatw(adωu)=w(u)+2 andw(adπu)=w(u)−2.
Definition 1 An elementu∈Fp,q is called primitive ifu∈ker(adπ), i.e.,{u, π} =0.
The next statement follows from the definitions.
Lemma 4 Forπandωinverses of one another, ker(adπ)∩ker(adω)⊂
p≥−1Fp,p.
The inverse inclusion is not valid since counter-examples are furnished by(1,1)-tensors Nof shifted bidegree(0,0)such that the 2-form{adω, N}or the bivector{adπ, N}does not vanish, e.g., whenNis a multiple of the identity.
The following theorem is an analogue of the Hodge-Lepage decompositions in Kähler [41] and symplectic [25,27] geometry. We first prove a lemma.
Lemma 5 Letu∈Fbe of weightw(u). Then, for anyk≥0, I(adkωu)=(w(u)+2k)adkωu.
If u is primitive, then
adπ(adkωu)= −k(w(u)+k−1)adk−1ω u.
Proof The first formula follows fromw(adkωu)=w(u)+2k. The second is proved by re-
cursion onk.
From the complete reducibility of finite-dimensional representations of semi-simple Lie algebras, and from Lemma5we obtain the following result [25,29].
Theorem 2 Any elementu∈Fp,q admits the decomposition,
u=u0+adωu1+ad2ωu2+ · · · +adkωuk+ · · ·, (3.1) where each uk, k ≥0, is a uniquely defined primitive element of Fp+k,q−k of weight w(u)−2k.
4 Complementary 2-forms for Poisson Structures
The complementary 2-forms with respect to a Poisson structure on a Lie algebroid were defined and studied by Vaisman [37,38]. We shall describe the complementary 2-forms on a Lie algebroidA and their properties by means of the big bracket onF. The method of proof using the big bracket gives a clear view of their nature and properties.
4.1 Poisson Bivectors
We recall several well known facts concerning the Poisson structures on Lie algebroids [19,33].
Lemma 6 Let(A, μ)be a Lie algebroid. Ifπ∈ (∧2A), then γπ= {π, μ}
is of shifted bidegree(1,0), andγπ is a Lie algebroid structure onA∗if and only if
{γπ, γπ} =0. (4.1)
The next lemma gives conditions for the construction, from a bivector on a Lie algebroid, of a Lie algebroid structure on the dual vector bundle. Since{μ, μ} =0,
{γπ, γπ} = {{π, μ},{π, μ}} = {{{π, μ}, π}, μ} = {[π, π]μ, μ}.
Therefore
Lemma 7 A necessary and sufficient condition forγπto be a Lie algebroid structure onA∗ is
{μ,[π, π]μ} =0, (4.2)
while a sufficient condition is
[π, π]μ=0, (4.3)
i.e.,πis a Poisson bivector.
The bracket defined byγπ= {π, μ}on (∧•A∗)is usually denoted simply by[,]π. Thus, by definition,
{{α,{π, μ}}, β} = [α, β]π,
for allαandβ∈ (∧•A∗). The following lemma is proved in [19,33].
Lemma 8 The bracket defined byγπ= {π, μ}on (∧•A∗)is the Koszul bracket of forms.
In particular, for allf∈C∞(M),α, β∈ (A∗), {{α,{π, μ}}, f} =((ρ◦π)α)·f,
{{α,{π, μ}}, β} =Lμπαβ−Lμπβα−dμ(π(α, β)).
Remark 4 A bivectorπ is Poisson if and only ifγπ = {π, μ}is primitive in the sense of Definition1. Assume thatπis a non-degenerate bivector, with inverseω. We consider the decomposition (3.1) of the structureμ∈F0,1,
μ=μ0+adωμ1,
whereμ0∈F0,1andμ1∈F1,0are primitive, adπμi=0, i=0,1, and of weight 1 and−1, respectively. Then, using Lemma5, we obtain
γπ= {π, μ} = {π, μ0+adωμ1} = −adπμ0−adπadωμ1=μ1. Thusμ=μ0+adωγπ, whereμ0andγπ are primitive.
4.2 Dualization and Composition
We now dualize the construction of Sect. 4.1. Let (A∗, γ ) be a Lie algebroid. If ω∈ (∧2A∗), thenμ= {γ , ω}is of shifted bidegree(0,1)andμis a Lie algebroid structure onAif and only if
{[ω, ω]γ, γ} =0, (4.4)
while a sufficient condition is
[ω, ω]γ=0. (4.5)
We shall now combine the two preceding constructions and consider the following scheme,
(A, μ)(π )(A∗, γπ= {π, μ})(ω)(A,μ= {γπ, ω} = {{π, μ}, ω}).
The following definition is due to Vaisman [37].
Definition 2 A 2-form satisfying (4.5) whenγ =γπ= {π, μ}is called a complementary 2-form forπ.
Since, in this case,γ = {π, μ}, by Lemma8,[ω, ω]γ= {{ω, γ}, ω}is equal to[ω, ω]π, where[,]πis the Koszul bracket.
Letπ be an arbitrary bivector andω an arbitrary 2-form. Let us determine sufficient conditions forμ= {γπ, ω}to be a Lie algebroid structure onA, i.e., to satisfy
{μ,μ} =0. (4.6)
Proposition 3
(i) Letπbe a bivector on(A, μ)such thatγπ= {π, μ}satisfies{γπ, γπ} =0. A necessary and sufficient condition forμ= {γπ, ω} = {{π, μ}, ω}to be a Lie algebroid structure on Ais{[ω, ω]π, γπ} =0.
(ii) Letπ be a bivector on(A, μ). A sufficient condition forμ= {γπ, ω} = {{π, μ}, ω}to be a Lie algebroid structure onAis
[π, π]μ=0 (πis Poisson),
[ω, ω]π=0 (ωis a complementary 2-form forπ ).
Proof Using the Jacobi identity we compute
{μ,μ} = {{γπ, ω},{γπ, ω}} = {γπ,{ω,{γπ, ω}}} + {{γπ,{γπ, ω}}, ω}
= {γπ,{ω,{γπ, ω}}} +1
2{{{γπ, γπ}, ω}, ω}.
Let us assume thatγπ= {π, μ}, and thatπsatisfies{γπ, γπ} =0, which is equivalent to (4.2). Condition (4.6) becomes
{γπ,{{ω, γπ}, ω}} =0, i.e.,
{γπ,[ω, ω]π} =0. (4.7)
This proves part (i), and part (ii) follows immediately.
4.3 Lie Algebroid Structure Defined by a Complementary 2-form
Let us determine an explicit expression for the anchor and bracket of(A,μ). By the Jacobi identity,
μ= {{π, μ}, ω} =μ1+μ2, (4.8) where we have set
μ1= {{π, ω}, μ} and μ2= {π,{μ, ω}}. We set
N= {π, ω},
then μ1 = {N, μ} and we write μ2 = {π,{μ, ω}} = {ψ, π}, where ψ is the 3-form
−{μ, ω} = −dμω.
By definition, the anchor of(A,μ)isρsuch thatρ(X)f = {{X,μ}, f}, for allX∈ A and f∈C∞(M). Thenρ=ρ◦N, whereρis the anchor ofA. In fact, forX∈ Aand f∈C∞(M),
{{X, μ1}, f} = {{X,{N, μ}}, f} =ρ(N X)·f and
{{X, μ2}, f} = {{X,{ψ, π}}, f} =0.
Let us consider the bracket defined, forX,Y∈ A, by [X, Y]μ= {{X,μ}, Y}.
By Lemma2, the bracket[,]μ1 is the bracket[,]μN recalled in (2.7). The theorem below follows from Lemmas1,2and3.
Theorem 3 Letπbe a bivector andωa 2-form on(A, μ). Thenπandωsatisfy (4.6) if and only if the bracket of sections ofAdefined by
[X, Y]μ= [X, Y]μN−π(iX∧Ydμω), (4.9) for allX,Y∈ A, whereN=π◦ω, is a Lie algebroid bracket with anchorρ◦N.
In order to compare (4.9) with formula (3.3) in [37], we remark thatB= −N, so that [,]E is the opposite of[,]μ. As a corollary of Proposition3(ii) and Theorem3, we obtain the following results which were proved in [37].
Corollary 1 Ifπis a Poisson bivector andωis a complementary 2-form forπ, then (i) formula (4.9) defines a Lie bracket on the space of sections ofA, and
(ii) if, in addition,dμω=0, then bracket[,]μN , whereN=π◦ω, is a Lie bracket.
In part (ii) of this corollary the assumption thatωbedμ-closed can be replaced by the weaker assumption that, for allXandY∈ A,iX∧Ydμω∈ker(π).
Remark 5 Ifπ is a non-degenerate Poisson bivector, its inverseωis a complementary 2- form forπ. In fact [21],[ω,·]π=dμand therefore[ω, ω]π=dμω=0. In this caseN=IdA andμ2=0, thereforeμ=μ.
4.4 The Modular Class of(A,μ)
Consider a Poisson bivectorπonAand a complementary 2-formωwith respect toπ. As- sume thatAis orientable and letλbe a nowhere-vanishing section of∧top(A∗)that defines an isomorphism,∗λ, from multivectors to forms. Letdωπ= −[ω, .]π be the Lie algebroid cohomology operator of A with structure μ= {{π, μ}, ω}. Each of the operators on the sections of∧•A,
∂ωπ= [dπ, iω], and
∂ω,λπ = −(∗λ)−1dωπ∗λ
generates[,]μand also has square 0 sinceωis a complementary 2-form with respect toπ.
The 1-formξπ,ω,λonAdefined by
∂ω,λπ −∂ωπ=iξπ,ω,λ
is adμ-cocycle. Its class is the modular class of the Lie algebroid(A,μ)[40].
In the following sections,Adenotes a vector bundle over a manifoldM, and we let(A, μ) be a Lie algebroid, so that, by assumption,{μ, μ} =0. We will sometimes abbreviate(A, μ) byA.
5 What Is aP N-structure on a Lie Algebroid?
We have reviewed the Nijenhuis structures in Sect.2.3. We now consider Nijenhuis struc- tures on Lie algebroids equipped with a Poisson structure.
5.1 Compatibility
Given a bivectorπand a(1,1)-tensorNon(A, μ), we consider both μN= {N, μ},
which defines an anchorρ◦Nand a bracket[,]μN onA, and γπ= {π, μ},
which defines an anchor ρ◦π and a bracket on A∗ that we have denoted by [,]π. We assume that
N◦π=π◦N∗, (5.1)
whereN∗denotes the transpose ofN satisfyingN (X), α = X, N∗(α), for allX∈ A andα∈ (A∗), so thatN◦πdefines a bivectorπN byπN =N◦π. Then,
πN=1 2{π, N}.
We introduce a compatibility condition for π and N by requiring that the bracket [,]μN twisted byπ, which is{π,{N, μ}}, be equal to the bracket[,]π deformed byN∗, which is {{π, μ}, N}. Thus we set
Cμ(π, N )= {π,{N, μ}} + {N,{π, μ}}, (5.2) which is a section of∧2A⊗A∗.
Definition 3 A bivectorπ and a(1,1)-tensorN on(A, μ)are called compatible if they satisfy (5.1) and
Cμ(π, N )=0.
AP N-structure on(A, μ)is defined by a Poisson bivector and a Nijenhuis tensor on(A, μ) which are compatible.
5.2 P N-structures and Lie Bialgebroid Structures
A necessary and sufficient condition for(μN, γπ)to define a Lie bialgebroid structure on (A, A∗) is {μN+γπ, μN +γπ} =0. When N is a Nijenhuis (1,1)-tensor, {μN, μN} = 0, and when π is a Poisson bivector,{γπ, γπ} =0. Therefore in this case the condition {μN+γπ, μN+γπ} =0 is equivalent to{μN, γπ} =0.
Lemma 9 LetCμ(π, N )=2{μN, γπ}. ThenCμ(π, N )= {μ, Cμ(π, N )}. Proof By the Jacobi identity,
Cμ(π, N )=2{{N, μ},{π, μ}} = {{N, μ},{π, μ}} + {{π, μ},{N, μ}}
= {{{N, μ}, π}, μ} + {{{π, μ}, N}, μ} = {μ, Cμ(π, N )}. Theorem 4 LetNbe a Nijenhuis(1,1)-tensor andπa Poisson bivector on(A, μ).
(i) The vanishing of{μ, Cμ(π, N )}is a necessary and sufficient condition for(μN, γπ)to define a Lie bialgebroid structure on(A, A∗). In particular, ifπandNare compatible, then(μN, γπ)is a Lie bialgebroid structure.
(ii) If the dμ-exact 1-forms generate (A∗) locally as aC∞(M)-module, then a Poisson bivectorπand a Nijenhuis tensorNdefine aP N-structure on(A, μ)if and only if the pair(μN, γπ)defines a Lie bialgebroid structure on(A, A∗).
Proof Only (ii) needs to be proved. FromCμ(π, N )= {μ, Cμ(π, N )}we obtain {Cμ(π, N ),{μ, f}} = {{Cμ(π, N ), μ}, f} = −{Cμ(π, N ), f},
for all f ∈ C∞(M). Thus Cμ(π, N ) = 0 implies Cμ(π, N )(dμf,·) = 0, for all f∈C∞(M). Under the assumptions of part (ii) of the theorem,Cμ(π, N )vanishes identi-
cally since it isC∞(M)-linear.
The equivalence stated in the theorem was proved in [17] for the case whenA is the tangent bundle of a manifold. This equivalence may fail for Lie algebroids which are not tangent bundles, a fact observed by Grabowski and Urbanski [11].
Remark 6 The compatibility condition Cμ(π, N )=0 implies that the brackets [,]πN, ([,]μN)π and ([,]π)N∗ coincide. In fact, from {π,{N, μ}} = {{π, μ}, N}, we obtain {μ,{N, π}} = {{μ, N}, π} + {N,{μ, π}} = 2{{μ, N}, π} = 2{N,{μ, π}} or {μ, πN}
= {μN, π} = {{π, μ}, N}.
Remark 7 The conditionCμ(π, N )=0 is equivalent to each of the following:
• The operatordN= {{N, μ},·} = [iN, dμ]is a derivation of[,]π,
• ( (∧•A∗),[,]π, dN)is a differential Gerstenhaber algebra,
• The operatordπ= {{π, μ},·} = [π,·]μis a derivation of[,]μN,
• ( (∧•A),[,]μN, dπ)is a differential Gerstenhaber algebra.
6 On Compatible Poisson Structures
For some of the results in this section, see [8] and earlier articles cited there.
Two Poisson bivectors on(A, μ)are said to be compatible, or to form a Hamiltonian pair or to define a bi-Hamiltonian structure, if their sum is a Poisson bivector. Thus Poisson bivectorsπandπ1are compatible if and only if{{π, μ}, π1} =0.
Letπ be a bivector on(A, μ)andNa(1,1)-tensor. Assume thatN◦π is a bivector, i.e.,N◦π=π◦N∗. We have setπN =N◦π. Then
πN=1 2{π, N}.
In particular, if a bivectorπis non-degenerate and has inverseω, thenN= {πN, ω}.
Proposition 4 Assume thatπis a non-degenerate Poisson bivector on(A, μ)with inverse ω, andNis a(1,1)-tensor such thatπandπN satisfy{{π, μ}, πN} =0. Then
{{μ, π}, N} + {{μ, N}, π} =0, (6.1) and
{{N, μ}, ω} =0. (6.2)
If, in addition,πNis a Poisson bivector, then
{{μ, πN}, N} + {{μ, N}, πN} =0. (6.3) Proof Sinceωis the inverse ofπ,{π, ω}is the identity ofAand, by (2.4), foru∈Fp,q,
{u,{π, ω}} =(p−q)u. (6.4)
By assumption,{{π, μ}, π} =0 and{{π, μ}, πN} =0.
(i) From the compatibility ofπandπN and the Jacobi identity, we derive 0= {{{πN, μ}, π}, ω} = {{{πN, μ}, ω}, π} + {{πN, μ},{π, ω}}.
Becauseπis a Poisson bivector,{μ, ω} =0. Therefore
0= {{{πN, ω}, μ}, π} + {{πN, μ},{π, ω}}.
Using the relationsπN=12{π, N},N= {πN, ω}, and (6.4), we obtain 0= {{N, μ}, π} +1
2{{π, N}, μ} = {{N, μ}, π} +1
2{π,{N, μ}} +1
2{{π, μ}, N}
=1
2{{N, μ}, π} +1
2{{π, μ}, N}.
(ii) To prove (6.2), we compute
{{π,{N, μ}}, π} = {{{π, N}, μ}, π} + {{N,{π, μ}}, π} =4{πN,{μ, π}} =0, and we know that the vanishing of{{π,{N, μ}}, π}is equivalent to{{N, μ}, ω} =0.
(iii) To prove (6.3), we use the assumption{{πN, μ}, πN} =0 to obtain 0= {{{πN, μ}, πN}, ω} = {{πN, μ}, N} + {{{πN, μ}, ω}, πN}
= {{πN, μ}, N} + {{N, μ}, πN},
thus proving (6.3).
From (6.1), it follows that {πN, μ} =1
2{{π, N}, μ} = {{π, μ}, N} = {{μ, N}, π}, (6.5) and from (6.3), it follows that
1
2{{πN, N}, μ} = −{{μ, πN}, N} = {{μ, N}, πN}. (6.6) Lemma 10 WhenN= {σ, τ}, whereσis a bivector andτ a 2-form, then
N2= −1
2{{N, σ}, τ}.
Proof This formula is proved by a simple calculation.
The following essential result in the theory of bi-Hamiltonian systems was proved by Magri and Morosi in [31] and also by Gelfand and Dorfman [8] in the algebric framework of Hamiltonian pairs and by Fuchssteiner and Fokas [9] in their study of Hamiltonian structures for evolution equations. See [21] for the case of Lie algebroids.
Theorem 5 Let π and π1 be compatible Poisson structures, with π non-degenerate. Set N=π1◦(π)−1.
(i) The Nijenhuis torsion of the(1,1)-tensorNvanishes.
(ii) The pair(π, N )is aP N-structure.
(iii) The pair(π1, N )is aP N-structure.
Proof (i) Letωbe the inverse ofπ. ThenN= {π1, ω}. Applying Lemma10toσ=π1and τ=ω, we obtain from formula (2.9):
2TμN= {N,{N, μ}} − {N2, μ}
= {N,{N, μ}} +1
2{{{N, π1}, ω}, μ}. Using (6.2) and (6.3) yields
2TμN= {N,{N, μ}} + {{{N, μ}.π1}, ω}
= {N,{N, μ}} + {{N, μ}, N} =0.
(ii) Equation (6.1) expresses the vanishing ofCμ(π, N ).
(iii) Equation (6.3) expresses the vanishing ofCμ(π1, N ).
In [21] and in the references cited above, it is proved more generally that eachπkdefined byπk=Nk◦π,k∈N, is a Poisson bivector, and the bivectorsπkare pairwise compatible.
The preceding theorem admits a converse.
Theorem 6 Ifπandπ1are non-degenerate Poisson bivectors, and ifN=π1◦(π)−1has vanishing Nijenhuis torsion, thenπandπ1are compatible.
Proof Assume thatπis non-degenerate and letωbe its inverse. Sinceπis a Poisson bivec- tor,{μ, ω} =0. From this fact, the fact thatπ1is Poisson and the formulaN= {π1, ω}, we obtain (6.3) which implies
{{N, π1}, μ} =2{{N, μ}, π1}. (6.7) From Lemma10applied toN= {π1, ω}, we obtain
2TμN= {N,{N, μ}} +1
2{{{N, π1}, ω}, μ}.
Using{μ, ω} =0 and (6.7), we obtain
2TμN= {N,{N, μ}} + {{{N, μ}, π1}, ω}, whence
TμN=1
2{{{N, μ}, ω}, π1}. (6.8) We now assume that π1 also is non-degenerate, and we denote its inverse byω1. Then 2{TμN, ω1} = {{N, μ}, ω}and the vanishing ofTμN implies the vanishing of{{N, μ}, ω}.
We then remark that the vanishing of {{N, μ}, ω} is equivalent to the vanishing of {{π,{N, μ}}, π}. Since{{π1, μ}, π} =14{{π,{N, μ}}, π}, the vanishing ofTμNimplies that
πandπ1are compatible.
Remark 8 Letπbe a non-degenerate Poisson bivector with inverseωand letNbe a(1,1)- tensor. Assume thatπNdefined by(πN)=N◦πis a Poisson bivector. In view of Lemma3 and Lemma11below, formula (6.8) means that
(TμN )(X, Y )=1
2(πN)(iX∧Y(dNω)), for allXandY∈ A.
In the next sections we shall review and compare theP Ω- andΩN-structures of Magri and Morosi [31] and the Hitchin pairs of Crainic [6].
7 What Is aP Ω-structure on a Lie Algebroid?
In [31], Magri and Morosi defined theP Ω- andΩN-structures on manifolds and, more recently, in his study of generalized complex structures, Crainic defined Hitchin pairs on manifolds [6]. These notions admit straightforward generalizations to the case of Lie alge- broids which we now define and study. When the Lie algebroid isT Mwith its standard Lie algebroid structure, these definitions recover the classical case. Most of the results in this section are particular cases of the general theorem of Antunes1on Poisson quasi-Nijenhuis structures with background, see [1], Theorem 4.1.
IfNis a(1,1)-tensor on a Lie algebroid(A, μ), letμN= {N, μ}be the deformed bracket satisfying (2.7). Define the operator on formsiN= {N,·}and letdN be the operator consid- ered in Remark7,
dN= [iN, dμ],
where[,]is the graded commutator. In particular, if a formαisdμ-closed, thendμ(αN)=
−dNα, whereαN=iNα. The following simple lemma was proved in [21]. We present an alternative proof.
Lemma 11 LetNbe a(1,1)-tensor on(A, μ). The operators on formsdN= [iN, dμ]and dμN= {μN,·}coincide.
Proof For any formα,dμNα= {μN, α} = {{N, μ}, α} = {N,{μ, α}} − {μ,{N, α}}, while dNα= [iN, dμ](α)=iN{μ, α} − {μ, iNα} = {N,{μ, α}} − {μ,{N, α}}.
LetNbe a(1,1)-tensor andωa 2-form on(A, μ)such that
ω◦N=N∗◦ω. (7.1)
ThenωN defined byωN=ω◦Nis a 2-form and ωN=1
2iNω=1 2{N, ω}.
Letπbe a bivector and letωbe a 2-form on the Lie algebroid(A, μ). SetN=π◦ω. Then (7.1) is satisfied and
N= {π, ω}.
We shall now prove identities relatingπ,ωandNwhenN= {π, ω}.
Lemma 12 Letπbe a bivector andωa 2-form on(A, μ), and letN be the(1,1)-tensor N=π◦ω. The 2-formωsatisfies
1
2[ω, ω]π+dNω= −dμ(ωN). (7.2)
1The convention for the definition ofωin [1] is the opposite of ours.
