Quasi-Lie bialgebroids, Dirac structures, and deformations of Poisson quasi-Nijenhuis manifolds

We show how to deform a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Then we interpret this procedure in the context of quasi-Lie bialgebroids, as a particular case of the so called twisting of a quasi-Lie bialgebroid. Finally, we frame our result in the setting of Courant algebroids and Dirac structures.


Introduction
A tensor N of type (1,1) on a manifold M is called a Nijenhuis operator if its Nijenhuis torsion vanishes -see (11).Such a geometrical object was introduced in [21] and is still the subject of interesting investigations (see [2] and references therein).Moreover, it is very useful in the theory of integrable systems thanks to the notion of Poisson-Nijenhuis (PN) manifold [15,19], i.e., a manifold endowed with a Poisson tensor π and a Nijenhuis operator N fulfilling suitable compatibility conditions -see (6).This approach to integrability was put in a more general context in [8,9], using suitable pairs of Dirac structures.
Poisson quasi-Nijenhuis (PqN) manifolds [24] are a generalization of PN manifolds, where π and N are still compatible but the Nijenhuis torsion of N is just required to be controlled by a suitable 3-form (so it is not necessarily zero).Some preliminary results about the relevance of PqN manifolds in the study of integrable systems have recently been obtained in [10,11].Much remains to be done as far as the construction of a family of functions in involution is concerned.
As shown in [14], one can associate a Lie bialgebroid [18] to a PN manifold.In a similar way [24], a quasi-Lie bialgebroid [22] can be associated to a PqN manifold.A general setting to describe Lie bialgebroids and quasi-Lie bialgebroids is supplied by the notions of Courant algebroid and Dirac structure [7,16] -for the definition of Courant algebroid, see Appendix A and references therein.Indeed, Lie bialgebroids are in one-to-one correspondence with pairs of transversal Dirac structures in a Courant algebroid.If one of the transversal structure is not Dirac but Lagrangian, one obtains a quasi-Lie bialgebroid (see, e.g., [24] and references therein).Moreover, in [13] (following [22]) it is shown how to deform a quasi-Lie bialgebroid (A, A * ) thanks to the so called twisting by a section of 2 A.
The aim of this paper is to present a result about the deformation of a PqN manifold into another PqN manifold, by means of a closed 2-form.This is a first step to generalize the ideas in [9], with the aim of constructing (under suitable assumptions) an involutive family of functions on a PqN manifold.We will give a direct proof of our result in Subsection 2.1, a proof in the setting of twists of quasi-Lie bialgebroids in Subsection 2.2, and a proof in the context of Courant algebroids and Dirac structures in Section 3.

Conventions and notations.
Hereafter all manifolds will be smooth of class C ∞ and defined over the real numbers R. In the same vein, all vector bundles considered will be real and smooth of class C ∞ and their sections will be always considered smooth of the same class.As far as the notation is concerned, a vector bundle whose total space, base and canonical projection are E, M and, respectively, p will be denoted by the triple (E, p, M) or, more simply, by E, if this will not be cause of confusion.The space of (global) sections of E will be denoted by Γ(E).For the general notions about Lie algebroids used hereafter we refer the reader to the monograph [17].
Acknowledgments.We thank Gregorio Falqui for useful discussions.We are grateful to the anonymous referee, whose suggestions helped us to substantially improve the content and the presentation of our manuscript.MDNL thanks the Department of Mathematics and its Applica-

Poisson quasi-Nijenhuis manifolds and quasi-Lie bialgebroids
In the first part of this section we recall the definition of Poisson quasi-Nijenhuis manifold and we present a result, generalizing that in [11], concerning the deformations of PqN manifolds.The second part is devoted to a recollection of definitions and results on quasi-Lie bialgebroids, and to an alternative proof of our result.

Poisson quasi-Nijenhuis manifolds
First of all, recall (see, e.g., [15]) that any bivector π on a manifold M induces a bracket on the space Γ(T * M) of 1-forms, given by where π ♯ : T * M → T M is defined as β, π ♯ α = π(α, β), and that (1) is a Lie bracket if and only if π is Poisson.Suppose that it is so, and extend (1) to a degree −1, R-bilinear bracket on Γ( • T * M), still denoted by [•, •] π , such that for all η ∈ Γ( q T * M) and η ′ ∈ Γ( q ′ T * M) and for all 1-forms α; It follows that the graded Jacobi identity, holds (see, e.g., [20], Proposition 5.4.9),where q i is the degree of η i , and that, for any differential form η and for any f Remark 1.The bracket on Γ( • T * M) introduced above is an instance of the so called generalized Schouten bracket, which can be defined on the exterior algebra of the vector space Γ(A) of any Lie algebroid (A, [•, •] A , ρ A ), see pages 418-419 in [18].The generalized Schouten bracket associated to the Lie algebroid (T M, [•, •], id) is the usual Schouten bracket defined on Γ( • T M), while the one stemming from (1) is associated to the Lie algebroid defined by the Poisson bivector field π.Note that the sign conventions in the above formulas are the same as in [18] and in agreement with the ones adopted in [9], see Proposition 2.13.Indeed, if read at the level of the generalized Schouten bracket [•, •] π , it becomes, for 1-forms α i and β j , that can be obtained applying iteratively (K1) and (K3) to its left-hand side.
The following example points to a useful identity.
which follows by applying to our setting Formula (2.31) and Proposition 2.15 in [9].
Recall now that a Poisson tensor π and a (1, 1) tensor field N : T M → T M on M are said to be compatible [15,19] if where N * : T * M → T * M is the transpose of N and π N is the bivector field defined by π ♯ N = N π ♯ (notice that it is a bivector thanks to the first condition).
Finally, given a p-form α with p ≥ 1, the p-form i N α is defined as while i N f = 0 for all functions f .After introducing it can be proved [14] that the compatibility between a Poisson tensor π and a (1, 1) tensor field N is equivalent to requiring that d N is a derivation of [•, •] π , a fact that we will often use in the rest of the paper.Moreover, for any k-form η one has that where In [24] a Poisson quasi-Nijenhuis (PqN) manifold was defined as a quadruple (M, π, N, φ) such that: • the Poisson bivector π and the (1, 1) tensor field N are compatible; • the 3-forms φ and i N φ are closed; • T N (X, Y ) = π ♯ (i X∧Y φ) for all vector fields X and Y , where i X∧Y φ is the 1-form defined as i X∧Y φ, Z = φ(X, Y, Z), and is the Nijenhuis torsion of N .
A slightly more general definition of PqN manifold was recently proposed in [4] -see also [5], where the PqN structures are recast in the more general framework of the Dirac-Nijenhuis ones.
Another interesting generalization, given by the so called PqN manifolds with background, was considered in [1,6].
If φ = 0, then the torsion of N vanishes and M becomes a Poisson-Nijenhuis (PN) manifold (see [15] and references therein).In this case, π N is a Poisson tensor compatible with π, so that (M, π, π N ) is a bi-Hamiltonian manifold.
The following theorem generalizes a result in [11], where the starting point was a PN manifold.
then (M, π, N , φ) is a PqN manifold.In particular, if Ω is a solution of the non homogeneous Maurer-Cartan equation then (M, π, N ) is a PN manifold.
Proof.It is similar to the one given in [11], corresponding to the case φ = 0. We present here only the main points, focussing on the differences.
To prove the compatibility between π and N , we notice that dΩ = 0 implies In fact, both d π ♯ Ω ♭ and [Ω, •] π are graded derivation of (Ω • (M), ∧), anticommuting with d and coinciding on C ∞ (M) -see [11] for details.The previous identity entails that •] π , yielding the compatibility between N and π.
The closedness of φ easily follows from that of φ and Ω, recalling that d •] π , the commutation rule (K1), and the graded Jacobi identity (3).Furthermore, observe that d N φ = 0 as a consequence of (8).Then As shown in Lemma 3.7 of [24], this implies that T N (X, Y ) = π ♯ i X∧Y φ for all vector fields is the (additive) group of closed 2-forms on M, the theorem above implies that the set of the PqN-structures on M carries the following G Ω 2 c -action: where the last equality was obtained observing that The aim of the rest of the paper is two give two alternative proofs of Theorem 3, in the frameworks of quasi-Lie bialgebroids (see next subsection) and Dirac structures (see Section 3).

Quasi-Lie bialgebroids
Suppose that p : A → M is a Lie algebroid (see, e.g., [17]) with anchor ρ A : A → T M and Lie bracket [•, •] A , defined on the space of sections Γ(A) and then extended to Γ( • A), see Remark 1.

Definition 5 ([24]
).A quasi-Lie bialgebroid is a triple (A, d A * , φ), where , both with respect to the wedge product and Under these assumptions, one can define a morphism ρ A * : and show that d A * is explicitly given by a formula which is analog to (18).If φ = 0, then ) is also a Lie algebroid, and (A, A * ) turns out to be a Lie bialgebroid (see, e.g., [24]).
The first alternative proof of Theorem 3 hinges on the following two results.
This statement can be found, for example, at the beginning of Section 4.4 of [13].Its proof consists in showing that the new defined triple satisfies what is needed to form a quasi-Lie bialgebroid, i.e., d A * φ = 0 and d 2 A * = [ φ, •] A , which can be both checked by a direct computation completely analogous to (15) and, respectively, (16).
Remark 8.Note that if φ = 0 Proposition 6 reduces to the correspondence between Lie bialgebroids and PN manifolds described in [14].
Going back to the proof Theorem 3, let ((T * M) π , d N , φ) be the quasi-Lie bialgebroid associated to the PqN manifold (M, π, N, φ), and let Ω be any 2-form on M. Then we obtain the twist If dΩ = 0, we know from ( 14) that [Ω, The last step is to realize that the quasi-Lie bialgebroid (T * M) π , d N , φ comes from a PqN manifold, i.e., that φ is closed.This follows, as already observed just above Formula (15), from the fact that both φ and Ω are closed.

Ω.((T
The latter restricts to an action of G

Courant algebroids and Dirac structures
In this section we give a second alternative proof of Theorem 3 in the framework of Dirac structures.
To this end, we start by recalling how quasi-Lie bialgebroids are related to Courant algebroids, see [22,24] -more precisely, how quasi-Lie bialgebroids correspond to the pairs formed by a Dirac structure and a complementary Lagrangian subbundle in a given Courant algebroid, see the proof of part (ii) of Theorem 2.6 in [24].For the reader convenience, we recall the definition of a Courant algebroid in Appendix A.
Given a quasi-Lie bialgebroid (A, d A * , φ), and using the notations in and after Definition 5, we consider E = A * ⊕ A together with the non degenerate symmetric pairing the bundle map ρ : E → T M given by and the bracket in Γ(E) defined by for all X, Y ∈ Γ(A * ) and α, β ∈ Γ(A), where the bracket [•, •] A * was defined in (20) while d A is the differential defined on Γ( • A * ) by the algebroid A.
algebroid, A is a Dirac structure (i.e., it is maximal isotropic and its space of sections is closed under •, • ), and A * is a Lagrangian (i.e., maximal and isotropic) subbundle of E. Note that, if φ = 0, i.e., in the case of a Lie bialgebroid, A * is a Dirac structure too.
Example 10.Consider the quasi-Lie bialgebroid (T * M, d, φ), where A = T * M with its trivial Lie algebroid structure, i.e., with zero bracket and zero anchor, d is the Cartan differential and φ is any closed 3-form on M. Applying (22) to the this setting, one finds •] is the usual Lie bracket defined on Γ(T M).Computing the bracket so obtained on a pair of general sections of E, one has Ω is any closed 2-form in M , its graph is a Lagrangian subbundle of TM φ , transversal to T * M. A simple computation, using ( 24) and ( 25), shows that the quasi-Lie bialgebroid corresponding to (Gr(Ω), Remark 11.It is worth recalling that every closed 2-form Ω defines an automorphism of TM φ via the formula see for example [12], just above Proposition 2.2, the proof of Lemma 3.1 in [3] and, in a more general setting, Remark 4.2 in [22].In particular, the image of a Dirac structure under (28) is still a Dirac structure.More in general, if Ω is any 2-form on M, (28) sends a Lagrangian subbundle to a Lagrangian subbundle and a pair of transversal Lagrangian subbundles to a pair of transversal Lagrangian subbundles.Note that, in the previous example, the pair (Gr(Ω), T * M) is obtained applying (28) to the pair of transversal Lagrangian subbundles (T M, T * M).On the other hand, since dΩ = 0 and the Poisson structure π is trivial, i.e., identically zero, (21) yields As opposed to what is stated in part (ii) of Theorem 2.6 in [24], this example seems to suggest that the correspondence between quasi-Lie bialgebroids and pairs of a Lagrangian subbundle and of a transversal Dirac structure of a given Courant algebroid, described in the first part of this section is, in general, not one-to-one.On the other hand, the above example and its generalization, contained in Theorem 12, propound the existence of a one-to-one correspondence between the Ω 2 c (M)-orbits of quasi-Lie bialgebroids of the type ((T * M) π , d N , φ) and the Ω 2 c (M)-orbits of pairs of a Lagrangian subbundle and of a transversal Dirac structure of the type (Gr(Ω), T * M) in a Courant algebroid of the type E = (T M) N ⊕ (T * M) π , where Ω 2 c (M) is the (additive) group of the closed 2-forms on M acting, on these sets, via (21) and, respectively, (28).
We are now ready to present the third proof of Theorem 3. We start from a PqN manifold  (12).
The first step to prove this theorem is the computation of the bracket (22) between two sections of L. To this aim, we need two preliminary results.Lemma 13.For any closed 2-form Ω, the following relation holds: A direct proof of this lemma is given in Appendix B. Another proof can be spelled out along the lines of the following Remark 14.For any 2-form Ω, not necessarily closed, a tedious but straightforward computation which uses (5) shows that where Using Lemma 13 for the first term and Lemma 15 for the last three, the sum (35) turns out to be We can now prove Theorem 12, i.e., that the quasi-Lie bialgebroid (( First of all, we notice that the identification (23) in this case is simply X = X + Ω ♭ X, where X ∈ Γ(T M).By definition (25) of ϕ and using Lemma 16, we have that for all X, Y, Z ∈ Γ(T M).So we are left with showing that d L acts as d N .But this immediately follows from

Appendix A: Definition of Courant algebroid
In this appendix we will recall the definition of Courant algebroid following [16] -see also [7], where this definition appeared for the first time.To this end, let M be a manifold.
The last six terms are equal to where we have used again the fact that dΩ = 0.So we conclude that 2A(X, Y, Z) = (X,Y,Z) The comparison between this formula and (B1) ends the proof.
tions of the University of Milano-Bicocca and the Department of Management, Information and Production Engineering of the University of Bergamo for their hospitality, and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -CAPES, Brazil, for supporting him with the grants 88887.506201/2020-00,88887.695676/2022-00and 88887.911537/2023-00.MP thanks the ICMC-UPS, Instituto de Ciências Matemáticas e de Computação of the University of São Paulo, for its hospitality and Fundação de Amparo à Pesquisa do Estado de São Paulo -FAPESP, Brazil, for supporting his visit to the ICMC-USP with the grant 2022/02454-8.We also gratefully acknowledge the auspices of the GNFM section of INdAM, under which part of this work was carried out, and the financial support of the MMNLP project -CSN4 of INFN, Italy.

Remark 9 .
Proposition 7 and a computation very similar to the one in Remark 4 show that the (additive) group G Ω 2 of all 2-forms on M acts on the set of all quasi-Lie bialgebroids of the form ((T * M) π , d N , φ) by twisting them, i.e., Ω 2 c , see Remark 4, on the set of all quasi-Lie bialgebroids of the form ((T * M) π , d N , φ) with dφ = 0. Finally, the bijection between the set of PqN structures (M, π, N, φ) and the quasi-Lie bialgebroids of the form ((T * M) π , d N , φ) invoked in Proposition 6 intertwines (21) with (17).
.e., E is the Courant algebroid TM φ , obtained twisting the standard one by the closed 3-form φ, see Example 19.Note that T M is a Lagrangian subbundle of E transversal to the Dirac structure T * M ⊂ E. In particular (T M, T * M) is the pair corresponding to (T * M, d, φ).Note now that if

(
M, π, N, φ) to construct the quasi-Lie bialgebroid ((T * M) π , d N , φ) as recalled in Subsection 2.2, and therefore the Courant algebroid E = (T M) N ⊕ (T * M) π with its Lagrangian subbundle A * = T M and its Dirac structure A = T * M. Given a closed 2-form Ω, we apply Formula (28) to each member of the pair (T M, T * M), see Remark 11, to get (L = Gr(Ω), T * M), where L is a non-integrable Lagrangian subbundle of E. Note that the action of Ω on the original pair leaves T * M fixed.Applying the construction presented in the first part of this section to the new pair (L, T * M), we obtain the quasi-Lie bialgebroid ((T * M) π , d L , ϕ) which, thanks to Theorem 12, we are able to identify with ((T * M) π , d N , φ).Finally, applying Proposition 6 to this quasi-Lie bialgebroid we conclude that the quadruplet (M, π, N , φ) is a Poisson quasi-Nijenhuis manifold, yielding our (third) proof of Theorem 3. In this way we are left with proving the following Theorem 12.If the 2-form Ω is closed, the quasi-Lie bialgebroid ((T * M) π , d L , ϕ) coincides with ((T * M) π , d N , φ), where N = N + π ♯ Ω ♭ and φ is given by A Courant algebroid (over M) is a quadruplet (E, •, • E , •, • , ρ), where the identity as anchor map, while T * M carries the trivial Lie algebroid structure, i.e., with zero Lie bracket and zero anchor map.The bracket (A2) can be modified by twisting it with the term i X∧Y φ, where φ is any closed 3-form on M. The resulting structure is called twisted Courant algebroid and it is denoted with TM φ .Twisted Courant algebroids were introduced by Severa in[23], who proved that a Courant algebroid E fits into the exact sequence0 −→ T * M ρ * −→ E ρ −→ T M −→ 0,if and only if E is isomorphic to TM φ for some closed 3-form.In the exact sequence ρ denotes the anchor of E.which vanishes if Ω is closed.Now, 2A(X, Y, Z) = A(X, Y, Z) − A(X, Z, Y )