poisson quasi-nijenhuis structures with background

DOI 10.1007/s11005-008-0272-5Lett Math Phys (2008) 86:33–45

Poisson Quasi-Nijenhuis Structureswith Background

PAULO ANTUNESCMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal.e-mail: [email protected]

Received: 14 August 2008 / Revised: 1 October 2008 / Accepted: 6 October 2008Published online: 16 October 2008 – © Springer 2008

Abstract. We define Poisson quasi-Nijenhuis structures with background on Lie algebroidsand we prove that any generalized complex structure on a Courant algebroid which is thedouble of a Lie algebroid has an associated Poisson quasi-Nijenhuis structure with back-ground. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure withbackground constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair(π,ω) of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure withbackground and we observe that particular cases correspond to already known compatibi-lities between π and ω.

Mathematics Subject Classification (2000). 53D17, (58H05, 53C15, 17B70).

Keywords. Lie algebroid, Courant algebroid, Poisson Nijenhuis structure, generalizedcomplex structure.

0. Introduction

The aim of this work is to define, on a Lie algebroid, the notion of Poissonquasi-Nijenhuis structure with a (closed) 3-form background. The Poisson quasi-Nijenhuis structures (without background) were introduced by Stienon and Xuin [16] on the tangent Lie algebroid and then on any Lie algebroid by Caseiroet al. in [2]. In Physics, the Poisson quasi-Nijenhuis geometry was studied byZucchini [21] as the target space geometry implied by the BV master equationsof a Poisson sigma model. In his paper, Zucchini also treated the case with back-ground but we remarked that a condition is missing in the definition proposedthere. This extra condition was already considered in [16] and appears naturally inour work when we require some structures to be integrable (or some brackets tosatisfy the Jacobi identity).

This paper was presented as a poster in the conference “Poisson 2008”, EPFL, Lausanne, in July2008.

34 PAULO ANTUNES

In this paper we will use a supermanifold approach [14,18] to describe Liealgebroid structures. Let us consider a vector bundle A→ M and change the parityof the fibre coordinates (considering them odd), then we obtain a supermanifolddenoted by �A. The algebra of functions on �A, which are polynomial in thefibre coordinates, is denoted by C∞(�A) and coincides with �(A) := �(

∧• A∗),the exterior algebra of A-forms. Let us consider a Lie algebroid structure (ρ, [., .])on A. As it is known (see [18]), the Lie algebroid structure on A can equiva-lently be given by d, a degree 1 derivation of �(A) such that d2 = 0. In thesupermanifold setting, d is a vector field on �A and can be seen as the deri-vation defined by a hamiltonian on �A, i.e., an element µ ∈ C∞(T ∗�A). Thend = {µ, .} where the so-called big bracket [7], {., .}, is the canonical Poisson bra-cket on the symplectic supermanifold T ∗�A. The condition d2 =0 is equivalent to{µ,µ}=0.

To each f ∈ C∞(T ∗�A) is associated a bidegree (ε, δ). In fact, since usingLegendre transform (see [12]) T ∗(�A)∼= T ∗(�A∗), we can define ε (resp. δ) as thepolynomial degree of f in the fibre coordinates of the vector bundle T ∗(�A)→�A (resp. T ∗(�A)→�A∗). We define the shifted bidegree of f as the pair (ε−1,δ− 1) and the total shifted bidegree as the sum (ε− 1)+ (δ− 1)= ε+ δ− 2. Then,a Lie algebroid structure in A is a hamiltonian µ∈C∞(T ∗�A) of shifted bidegree(0,1) such that {µ,µ}=0.

Instead of the expression “with background” used here, some authors use “twis-ted”, or in Physics, “H-flux”. In this work, our choice was motivated by the resultof the Proposition 4.2. In fact, we prove there that a particular class of Poissonquasi-Nijenhuis structure with background is obtained by twisting, in a way explai-ned in [9,13,17], a Lie algebroid structure by a Poisson bivector and then by a2-form. Therefore, to avoid confusion, we will use the word “twist” only when weare dealing with twisting by a 2-form or a bivector as in [9,13,17].

The paper is organized as follows. In the first section we recall some basicdefinitions such as Nijenhuis tensors, Poisson bivectors and Poisson Nijenhuisstructures on a Lie algebroid and give the corresponding expression in the super-manifold approach. Then, in the second section, we introduce the notion of Pois-son quasi-Nijenhuis structure with a 3-form background H , on a Lie algebroid(A,µ). We prove that any complex structure (or more generally any c.p.s. struc-ture, see Definition 2.2) on (A ⊕ A∗,µ + H) induces such a structure. In thethird section we generalize a result from [2,16] and prove that any Poisson quasi-Nijenhuis structure with background on A induces a Lie quasi-bialgebroid on(A∗, A). Finally, in the last section we study Poisson quasi-Nijenhuis structureswith background defined by a pair (π,ω) of a Poisson bivector and a 2-form.We observe that already known compatible pairs such that complementary2-forms for Poisson bivectors [19], Hitchin pairs [3] and P�-structures or �N -structures [11] are all particular examples of Poisson quasi-Nijenhuis structureswith background.

POISSON QUASI-NIJENHUIS STRUCTURES WITH BACKGROUND 35

1. Basic Definitions

In this section we will recall some well known structures such as Poisson Nijenhuisstructures on a Lie algebroid (A, ρ, [., .]) and give their expression in terms of thebig bracket and polynomial functions on T ∗�A.

Let A be a vector bundle over a smooth manifold M . A Lie algebroid structureon A is a pair (ρ, [., .]) where ρ : A → T M is a vector bundle morphism and [., .]is a Lie bracket on the space of smooth sections �(A), such that the Leibniz ruleis satisfied

[X, f Y ]= f [X,Y ]+ (ρ(X) · f )Y, ∀X,Y ∈�(A), ∀ f ∈C∞(M).

The Lie algebroid structure, (ρ, [., .]), can be seen [12] as a function µ ∈C∞(T ∗�A), of shifted bidegree (0,1), such that {µ,µ}=0. The pair (ρ, [., .]) canbe recovered from µ by using the formulae

• ρ(X) · f ={{X,µ}, f };• [X,Y ]= {{X,µ},Y },for all X,Y ∈�(A) and f ∈C∞(M).

Consider a (1,1)-tensor N ∈�(A⊗ A∗). The Nijenhuis torsion of N is defined by

TN (X,Y )=[N X, NY ]− N ([N X,Y ]+ [X, NY ]− N [X,Y ]) .In terms of the big bracket and elements of C∞(T ∗�A), the Nijenhuis torsion

is given by

TN = 12

({N , {N ,µ}}−

{N 2,µ

}). (1)

If TN = 0, N is said to be a Nijenhuis tensor and in this case we define a newLie algebroid structure on A as follows

{ [X,Y ]N =[N X,Y ]+ [X, NY ]− N [X,Y ], X,Y ∈�(A),ρN =ρ ◦ N .

(2)

In the supermanifold setting, the structure ([., .]N , ρN ) is given by {N ,µ} ∈C∞(T ∗�A). We denote by dN the degree 1 derivation of �(A) induced by this Liealgebroid structure. Then

dN ={{N ,µ} , .} .A bivector π ∈�(∧2 A) is said to be Poisson if [π,π ]SN =0, where [., .]SN is the

Schouten-Nijenhuis bracket naturally defined on �(∧• A). If π is a Poisson bivec-

tor we define a Lie algebroid structure on A∗ by setting{ [α,β]π =Lπ�(α)β−Lπ�(β)α−d(π(α,β)), α,β ∈�(A∗),

ρπ =ρ ◦π�.(3)

In the supermanifold setting, the structure ([., .]π , ρπ ) is given by {π,µ} ∈C∞(T ∗�A).

36 PAULO ANTUNES

DEFINITION 1.1. A Poisson bivector π and a Nijenhuis tensor N are said to becompatible if

{N ◦π�=π� ◦ t N ,Cπ,N =0,

with

Cπ,N = ([., .]N )π − ([., .]π )t N ,

a C∞(M)-bilinear bracket on �(A∗). When π and N are compatible, the triple(A, π, N ) is called a Poisson Nijenhuis Lie algebroid.

In the supermanifold setting, we have

Cπ,N ={π, {N ,µ}}+{N , {π,µ}}.

THEOREM 1.2 [5]. If (A, π, N ) is a Poisson Nijenhuis Lie algebroid, then (AN , A∗π )

is a Lie bialgebroid, where AN and A∗π are the Lie algebroids defined, respectively, by

(2) and (3).

Remark 1.3. When A = T M and µ is the standard Lie algebroid structure, theimplication of the previous theorem becomes an equivalence (see [8]).

The Lie bialgebroid (AN , A∗π ) induces a Courant algebroid structure on A ⊕

A∗ [10,12] which is given in the supermanifold setting by

S ={π,µ}+{N ,µ}={π + N ,µ} .In the next sections we will weaken the Poisson Nijenhuis Lie algebroid

(A, π, N ) and study the structures that we get on A ⊕ A∗.

2. Poisson Quasi-Nijenhuis with Background and Generalized Geometry

Let S be a Courant algebroid structure on A ⊕ A∗, i.e., S ∈C∞(T ∗�A) is of totalshifted degree 1 and {S, S}=0. Consider also a (1,1)-tensor J on A ⊕ A∗, seen asa map J : A ⊕ A∗ → A ⊕ A∗. We call J orthogonal if

〈J (X ),Y〉+〈X , J (Y)〉=0,

for all X ,Y ∈�(A ⊕ A∗), with 〈., .〉 defined by 〈X +α,Y +β〉=β(X)+α(Y ) for allX,Y ∈�(A), α,β ∈�(A∗).

As in the Lie algebroid case, we can define a new bracket [., .]J deforming by Jthe Courant structure on A ⊕ A∗ by setting

[X ,Y]J =[JX ,Y]+ [X , JY]− J [X ,Y],


for all X ,Y ∈�(A⊕ A∗), where [., .] is the Dorfman bracket on A⊕ A∗. When J isan orthogonal (1,1)-tensor on A⊕ A∗, this deformed bracket is given by the hamil-tonian

SJ := {J, S}∈C∞(T ∗�A).

We also define the Nijenhuis torsion of J ,

TJ (X ,Y)=[JX , JY]− J ([X ,Y]J ) ,

for all X ,Y ∈�(A ⊕ A∗).

PROPOSITION 2.1 [1]. 1. The hamiltonian SJ defines a Courant structure on A⊕A∗ if and only if {S,TJ }=0.

2. J is a Courant morphism from (A⊕ A∗, SJ ) to (A⊕ A∗, S) if and only if TJ =0.

DEFINITION 2.2. An orthogonal (1,1)-tensor J , on A ⊕ A∗, is an almost c.p.s.structure if J 2 =λidA⊕A∗ , with λ∈{−1,0,1}. The almost c.p.s. structure J is inte-grable when TJ =0.

The abbreviation “c.p.s.” is due to Vaisman [20] and corresponds to the threedifferent structures we are considering:1 if λ=−1, J is an almost complex struc-ture [6]; if λ= 1, J is an almost product structure; and if λ= 0, J is an almostsubtangent structure.

As it was noticed in [3,20], J is an almost c.p.s. structure if and only if J canbe represented in a matrix form by

J

(Xα

)

=(

N π�

σ � −t N

)(Xα

)

(4)

for all X ∈�(A) and α ∈�(A∗), where π ∈�(∧2 A), σ ∈�(∧2 A∗) and N ∈�(A ⊗A∗) satisfy

⎧⎨

⎩

N ◦π�=π� ◦ t N ,σ � ◦ N = t N ◦σ �,N 2 +π� ◦σ �=λidA.

In the supermanifold setting,

J =π + N +σ

1In [4], the notion of irreducible Courant algebroid is introduced, as a Courant algebroid whereeach orthogonal Nijenhuis tensor is proportional to a c.p.s. structure. It is proved there that, forexample, the classical Courant algebroid structure on T M ⊕ T ∗M is irreducible.

38 PAULO ANTUNES

in the sense that J (.)={., π + N +σ }. Moreover, the integrability condition of analmost c.p.s. structure, TJ =0, is expressed by (see [4])

{{J, S} , J }+λS =0. (5)

Let us now consider the case S =µ+ H , where µ ∈ C∞(T ∗�A) defines a Liealgebroid structure on A, and H ∈�(∧3 A∗) is a closed 3-form. Then {S, S}=0 andS defines a Courant algebroid structure on A ⊕ A∗.

The goal of this section is to relate c.p.s. structures on (A ⊕ A∗,µ+ H) with thePoisson quasi-Nijenhuis structures with background which we now define.

DEFINITION 2.3. A Poisson quasi-Nijenhuis structure with background on a Liealgebroid A is a quadruple (π, N ,ψ, H) where π ∈�(∧2 A), N ∈�(A ⊗ A∗), ψ ∈�(

∧3 A∗) and H ∈�(∧3 A∗) are such that N ◦π� =π� ◦ t N , dψ = 0, d H = 0 andthe following conditions hold:

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

π is a Poisson bivector,

Cπ,N (α,β)=2 iπ�α∧π�βH,

TN (X,Y )=π�(iN X∧Y H − iNY∧X H + iX∧Yψ),

dNψ=dH,

(6)

for all X,Y ∈�(A), α, β ∈�(A∗) and where H is the 3-form defined by

H(X,Y, Z)=�X,Y,Z H(N X, NY, Z), (7)

for all X,Y, Z ∈�(A).

Remark 2.4. 1. In terms of the big bracket and elements of C∞(T ∗�A), theconditions (6) correspond to

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

{{π,µ} , π}=0,

{{π,µ} , N }+{{N ,µ} , π}+{{π, H} , π}=0,

{{N ,µ} , N }+{N 2,µ

}−2 {π,ψ}+{{π, H} , N }+{{N , H} , π}=0,

2 {{N ,µ} ,ψ}={µ, {N , {N , H}}−{

N 2, H}}.

(8)

2. If H =0 we recover the Poisson quasi-Nijenhuis structures defined in [2,16].3. The last condition of (6) is missing in the definition proposed by Zucchini

[21]. In our study this condition appears naturally and is necessary in orderto include the case without background, described in [2,16].

THEOREM 2.5. If an endomorphism J , defined by (4), is a c.p.s. structure on(A ⊕ A∗,µ + H) then (π, N ,−dσ, H) is a Poisson quasi-Nijenhuis structure withbackground on A.


Proof. The result follows directly by writing the integrability condition (5) withJ =π+ N +σ and S =µ+ H . Using the bilinearity of {., .} and taking into accountthe bidegree of each term we obtain the following system of equations

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

{{π,µ} , π}=0,{{π,µ} , N }+{{N ,µ} , π}+{{π, H} , π}=0,{{N ,µ} , N }+2 {π, {µ,σ }}+{{π,σ } ,µ}+{{π, H} , N }+{{N , H} , π}+λµ=0,{{N ,µ} , σ }+{{σ,µ} , N }+{{N , H} , N }+{{π,σ } , H}+λH =0.

In the last two equations of the system we now use the algebraic conditions forJ to be a c.p.s. structure and more precisely the condition N 2 + π� ◦ σ � = λidA

which is written in terms of the big bracket and elements of C∞(T ∗�A) as

{π,σ }= N 2 −λidA.

We obtain⎧⎪⎪⎪⎨

⎪⎪⎪⎩

{{π,µ} , π}=0,{{π,µ} , N }+{{N ,µ} , π}+{{π, H} , π}=0,{{N ,µ} , N }+{

N 2,µ}+2 {π, {µ,σ }}+{{π, H} , N }+{{N , H} , π}=0,

{{N ,µ} , σ }+{{σ,µ} , N }+{{N , H} , N }+{N 2, H

}−2λH =0.

The proof is achieved after interpreting the previous system of equations as⎧⎪⎪⎪⎨

⎪⎪⎪⎩

π is a Poisson bivector,

Cπ,N (α,β)=2 iπ�α∧π�βH,

TN (X,Y )=π�(iN X∧Y H − iNY∧X H − iX∧Y dσ),

2iN dσ −d(iNσ)=2(H+λH),

(9)

for all X,Y ∈�(A) and α,β ∈�(A∗) and H defined by Equation (7).

Remark 2.6. In [20], Vaisman studied the integrability of almost c.p.s. structures onT M ⊕ T ∗M considering both the usual Courant bracket, and also the case with a3-form background. The conditions obtained in Remark 1.5 of [20] coincide withthe system of conditions (9).

Note that, in the previous proof, the last equation of (9) is only a sufficientcondition for the last equation of (6). We can get an equivalence if we imposeadditional conditions on the quadruple (π, N , σ, H).

THEOREM 2.7. An endomorphism J , defined by (4), is a c.p.s. structure on (A ⊕A∗,µ+ H) if and only if (π, N ,−dσ, H) is a Poisson quasi-Nijenhuis structure onA with background H such that

⎧⎪⎪⎨

⎪⎪⎩

N 2 +π� ◦σ �=λidA,

σ � ◦ N = t N ◦σ �,2(iN dσ −H)=d(iNσ)+2λH.

40 PAULO ANTUNES

3. Poisson Quasi-Nijenhuis with Background and Lie Quasi-Bialgebroids

In this section we will generalize a result proved for structures without backgroundin [2,16]. Let (A,µ) be a Lie algebroid over a smooth manifold M .

DEFINITION 3.1. A Lie quasi-bialgebroid is a triple (A, δ, ϕ) where A is a Liealgebroid, δ is a degree one derivation of the Gerstenhaber algebra (�(

∧• A),∧, [., .]) and ϕ∈�(∧3 A) is such that δ2 =[ϕ, .] and δϕ=0.

The main result of the section is the following

THEOREM 3.2. If (π, N ,ψ, H) is a Poisson quasi-Nijenhuis structure with back-ground on A then (A∗

π , d HN , ψ + iN H) is a Lie quasi-bialgebroid, where d H

N (α)=dN (α)− iπ�(α)H, for all α∈�(A∗).

Proof. The hamiltonian on C∞(T ∗�A) which induces the structure (A∗π , d H

N ,

ψ + iN H) is S = {π + N ,µ+ H} +ψ . Considering the bidegree of each term, theequation

{S, S

}=0 is equivalent to

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{{π,µ} , {π,µ}}=0,

{{π,µ} , {π, H}}+{{π,µ} , {N ,µ}}=0,

{{π, H} , {π, H}}+{{N ,µ} , {N ,µ}}+2 {{π,µ} , {N , H}}++2 {{π, H} , {N ,µ}}+2 {{π,µ} ,ψ}=0,

{{π, H} , {N , H}}+{{N ,µ} , {N , H}}+{{π, H} ,ψ}+{{N ,µ} ,ψ}=0.

(10)

It is now straightforward to observe that the system of equations (8) implies thesystem (10).

COROLLARY 3.3. If (π, N ,−dσ, H) is a Poisson quasi-Nijenhuis structure withbackground on A, then {π + N +σ,µ+ H} is a structure of Lie quasi-bialgebroid on(A∗, A) or equivalently a Courant algebroid structure on A ⊕ A∗.

Proof. If we consider ψ=−dσ in the previous proof, we obtain S ={π + N +σ ,µ+ H}. Then, as we have already seen,

{S, S

}=0.

Remark 3.4. In the corollary above, J = π + N + σ is not necessarily integrable,i.e., the Nijenhuis torsion TJ may not vanish (see necessary conditions in Theo-rem 2.7). But the previous corollary proves that the deformed structure SJ

(= S)

defines a Courant algebroid structure in A ⊕ A∗, i.e., that {S,TJ } = 0 (seeProposition 2.1).


4. Poisson Quasi-Nijenhuis with Background and Compatible Second-OrderTensors

In this section we shall consider π ∈�(∧2 A) a Poisson bivector and a 2-form ω∈�(

∧2 A∗). Let us denote

π�(α)=π(α, .), ∀α∈�(A∗), ω�(X)=ω(X, .), ∀X ∈�(A),N =π� ◦ω�, ωN =ω(N ., .).

Then, the main result of this section is the following:

THEOREM 4.1. The quadruple (π, N ,dωN ,−dω) is a Poisson quasi-Nijenhuisstructure with background on A.

Proof. Let us set ψ = dωN and H =−dω. In terms of elements of C∞(T ∗�A),we have the following correspondences

⎧⎪⎪⎨

⎪⎪⎩

N ={ω,π} ,ψ= 1

2 {µ, {N ,ω}} ,H ={ω,µ} .

We easily check that ψ and H are closed and that N ◦π�=π� ◦ t N . To prove that(π, N ,ψ, H) is a Poisson quasi-Nijenhuis structure with background we need toverify the set of conditions (6) [or equivalently the conditions (8)].

1. π is a Poisson bivector by assumption.2. Considering the fact that π is a Poisson bivector, i.e., that

{{π,µ} , π}=0

and applying {ω, .} to both sides, we get

{ω, {{π,µ} , π}}=0.

Then, we use the Jacobi identity to obtain

{{ω, {π,µ}} , π}+{{π,µ} , {ω,π}}=0.

Using once more the Jacobi identity in the first term of the l.h.s. we have

{{{ω,π} ,µ} , π}+{{π, {ω,µ}} , π}+{{π,µ} , {ω,π}}=0,

which is the second condition of (8)

{{N ,µ} , π}+{{π, H} , π}+{{π,µ} , N }=0.

3. As above, we start from the previous condition

{{N ,µ} , π}+{{π, H} , π}+{{π,µ} , N }=0,

42 PAULO ANTUNES

and apply {ω, .} to both sides. We obtain

{ω, {{N ,µ} , π}}+{ω, {{π, H} , π}}+{ω, {{π,µ} , N }}=0,

and using the Jacobi identity twice we get the required equation

{{N ,µ} , N }+{

N 2,µ}

−2 {π,ψ}+{{π, H} , N }+{{N , H} , π}=0.

4. The way of proving this condition is the same as above. We start from the pre-vious condition and apply {ω, .} to both sides. Then, using the Jacobi identity,we get

{{N , H} , N }+{

N 2, H}

−2 {N ,ψ}−{{

N 2,ω},µ

}=0. (11)

Finally, applying {µ, .} we obtain

{µ, {{N , H} , N }+

{N 2, H

}}−2 {µ, {N ,ψ}}=0.

Using again the Jacobi identity and the fact that ψ is closed we get

2 {{N ,µ} ,ψ}={µ, {N , {N , H}}−

{N 2, H

}}.

The proof of the previous theorem suggests that starting from a Poisson bivectorand composing iteratively, in a certain way, with a 2-form we get all the conditionsof the definition of a Poisson quasi-Nijenhuis structure with background. The pre-cise way to describe this fact is using the twist of a structure by a bivector or a2-form as in [9,13,17].

PROPOSITION 4.2. If we denote by S the Lie quasi-bialgebroid structure inducedby the Poisson quasi-Nijenhuis structure with background (π, N ,dωN ,−dω), thenS = e−ω ◦ (e−πµ−µ), or equivalently

S = e−ω(µπ),

where µπ is the Lie algebroid structure defined by (3).

In the next proposition we will see that the Poisson quasi-Nijenhuis structurewith background (π, N ,dωN ,−dω) is induced (as shown in Theorem 2.5) by asubtangent structure.

PROPOSITION 4.3. The (1,1)-tensor J =(

N π�

−ωN� −t N

)

is a subtangent structure

(i.e., a c.p.s. structure with λ=0) on (A ⊕ A∗,µ−dω).


Proof. Using the Theorems 4.1 and 2.7 we only need to prove⎧⎪⎪⎨

⎪⎪⎩

N 2 −π� ◦ωN�=0,

ωN� ◦ N = t N ◦ωN

�,

2(iN dωN +H)=d(iN ωN ).

But the verification of the two first conditions is straightforward and, using thefact that iN ωN = iN 2ω, the last condition is equivalent to (11).

In the remaining part of this section, we will see that if we impose somerestrictions on the 2-form ω, in Theorem 4.1, we get already known structuresstronger than Poisson quasi-Nijenhuis with background. We also notice that thepairs (π,ω), (π, N ) and (ω, N ) thus obtained correspond to (or slightly generalize)already known compatible pairs.

COROLLARY 4.4. (Poisson Nijenhuis) If π ∈ �(∧2 A) is a Poisson bivector andω∈�(∧2 A∗) is a closed 2-form such that dωN =0, then (π, N ) is a Poisson Nijen-huis structure on A.

Remark 4.5. 1. A pair (π,ω) in the conditions of the corollary above is exactlywhat is called a P�-structure in [11].

2. The condition dωN = 0 is the compatibility condition for (ω, N ) to be a Hit-chin pair as it is defined in [3] for A = T M . The pair (ω, N ) above is moregeneral because ω is not necessarily symplectic.

3. Using the fact that ω is a closed form, we can prove that the compatibilitycondition dωN =0 is equivalent to two other known compatibility conditions:• ω is a complementary 2-form for π as in [19];• (ω, N ) is a �N -structure as in [11].

Let us justify briefly the last remark. In [19], Vaisman defines ω as a complemen-tary 2-form for π when

[ω,ω]π =0,

where [., .]π is the natural extension to �(∧• A∗) of the bracket [., .]π defined in

(3). But in terms of the big bracket and elements of C∞(T ∗�A), we have

[ω,ω]π ={{ω, {π,µ}} ,ω} ,and using the Jacobi identity twice we obtain

[ω,ω]π =2 {N , {µ,ω}}−{µ, {N ,ω}} ,which corresponds to

[ω,ω]π =2iN dω−2d(ωN ). (12)

44 PAULO ANTUNES

In [11], Magri and Morosi define a pair (ω, N ) to be a �N -structure if a particular3-form S(ω, N ) vanishes. But we can write

S(ω, N )=−iN dω+d(ωN ). (13)

Therefore, using (12) and (13) the vanishing of dωN is equivalent, when dω=0, tothe vanishing of [ω,ω]π or the vanishing of S(ω, N ).

COROLLARY 4.6. (Poisson quasi-Nijenhuis) If π ∈�(∧2 A) is a Poisson bivectorand ω∈�(∧2 A∗) is a closed 2-form then (π, N ,dωN ) is a Poisson quasi-Nijenhuisstructure on A (without background).

We can also define a Poisson Nijenhuis structure with background (π, N , H) byconsidering ψ = 0 in the Definition 2.3. Up to our knowledge, this structure wasnever studied before. We have the following result.

COROLLARY 4.7. (Poisson Nijenhuis with background) If π ∈�(∧2 A) is a Pois-son bivector and ω∈�(∧2 A∗) is a 2-form such that dωN =0, then (π, N ,−dω) is aPoisson Nijenhuis structure with background on A.

OBSERVATION 4.8. In the above results, the bivector π is a true Poisson bivec-tor. So the last structure we obtain is different from a possible compatibility bet-ween a Poisson structure with background [15] and a Nijenhuis tensor.

Acknowledgements

I would like to thank Yvette Kosmann-Schwarzbach for suggesting these topicsand for always useful discussions about this work.

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poisson quasi-nijenhuis structures with background

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