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Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids.


Introduction
It is well known that the cotangent bundle T * M of any Poisson manifold M admits a Lie algebroid structure and the pair (T M, T * M ) is a Lie bialgebroid over M . As a kind of reciprocal result, any Lie bialgebroid (A, A * ) induces a Poisson structure on its base manifold. A special kind of Lie bialgebroids are the triangular Lie bialgebroids. These are Lie algebroids (A, P ) equipped with an A-bivector field P such that [P, P ] = 0. The A-bivector field P induces a Lie algebroid structure on the dual vector bundle A * so that the pair (A, A * ) is a Lie bialgebroid. Triangular bialgebroids are also called Lie algebroids with a Poisson structure.
If one moves from the Poisson to the Jacobi framework, these statements are not true. In fact, if M is a Jacobi manifold, its cotangent bundle T * M is not, in general, a Lie algebroid. In order to associate a Lie algebroid to a Jacobi manifold, one has to consider the 1-jet bundle T * M × R → M . However, if we take the dual vector bundle T M × R → M endowed with its natural Lie algebroid structure, the pair (T M × R, T * M × R) is not a Lie bialgebroid. Motivated by this, Iglesias and Marrero [7] and Grabowski and Marmo [3] introduced the concepts of Jacobi algebroid, i.e. a Lie algebroid A with a 1-cocycle φ 0 , and of Jacobi bialgebroid, i.e. a pair ((A, φ 0 ), (A * , X 0 )) of Jacobi algebroids in duality satisfying a compatibility condition. Jacobi bialgebroids admit Lie bialgebroids as particular cases and are well adapted to the Jacobi context since every Jacobi manifold (M, Λ, E) has an associated Jacobi bialgebroid, ((T M × R, (0, 1)), (T * M × R, (−E, 0))). Imitating the Poisson case, Iglesias and Marrero introduced in [7] the notion of triangular Jacobi bialgebroid, as follows. If (A, φ 0 ) is a Jacobi algebroid and P is a Jacobi bivector field, i.e. an A-bivector field such that [P, P ] φ 0 = 0, then there exists a Lie algebroid structure on A * with a 1-cocycle such that the pair of Jacobi algebroids in duality is a Jacobi bialgebroid. As it happens in the Poisson case, the base manifold of any Jacobi bialgebroid inherits a Jacobi structure.
On the other hand, Poisson-Nijenhuis structures on Lie algebroids, i.e. Poisson-Nijenhuis algebroids, were introduced by Grabowski and Urbanski in [6], as Lie algebroids equipped with a Poisson structure and a Nijenhuis operator fulfilling some compatibility conditions. In the first part of this paper, we extend this concept to the Jacobi framework and we study Jacobi-Nijenhuis algebroids.
The other main goal of this paper is to study modular classes, including modular classes of Jacobi-Nijenhuis algebroids. The modular class of a Poisson manifold was defined by Weinstein in [19], as an analogue in Poisson geometry of the modular automorphism group of a von Neumann algebra. In [2], Evens, Lu and Weinstein introduced the notion of modular class of a Lie algebroid A over M , using a representation of A on the line bundle X top (A) ⊗ Ω top (M ). For the case of the cotangent Lie algebroid T * M of a Poisson manifold M , they showed that its modular class is twice the modular class of M , in the sense of [19]. Modular classes of triangular Lie bialgebroids were studied in [11], from the point of view of generating operators for Batalin-Vilkovisky algebras.
Regarding the Jacobi context, the first work on modular classes is due to Vaisman, who introduced in [18] the concept of modular class of a Jacobi manifold. Then, in [8], modular classes of triangular Jacobi bialgebroids were studied.
In the second part of this paper, we consider a Jacobi algebroid (A, φ 0 ) and we define, using the 1-cocycle φ 0 , a new representation of A on the line bundle X top (A) ⊗ Ω top (M ), which leads to the definition of modular class of a Jacobi algebroid.
Modular classes of Poisson-Nijenhuis algebroids were defined in [1]. Inspired in [1], we define modular class of a Jacobi-Nijenhuis algebroid. We obtain a hierarchy of vector fields on the Jacobi algebroid that covers a hierarchy of Jacobi structures on the base.
The paper is divided into five sections. Section 2 is devoted to Jacobi algebroids. We recall how to obtain a Lie algebroid structure on A × R over M × R from a Jacobi algebroid (A, φ 0 ) over M [7,3]. The notion of compatibility of two Jacobi bivectors on a Jacobi algebroid is introduced, and we prove that these Jacobi bivectors cover two compatible Jacobi structures on the base manifold. In Section 3 we define Jacobi-Nijenhuis algebroid and we show that a Jacobi-Nijenhuis algebroid defines a hierarchy of compatible Jacobi bivectors on the Jacobi algebroid and a hierarchy of compatible Jacobi structures on the base manifold. Moreover, the dual vector bundle also inherits a hierarchy of Jacobi algebroid structures that provides the existence of a family of triangular Jacobi bialgebroids. As a particular case of this construction, we recover the notion of strong (or strict) Jacobi-Nijenhuis manifold [9,17]. In Section 4, we introduce the notion of modular class of a Jacobi algebroid and we discuss the relation between modular class of a Jacobi algebroid (A, φ 0 ) over M and modular class of the Lie algebroid A×R over M × R. Relations between modular forms of A * and A * × R, in the triangular case, as well as duality between modular classes of A and A * are also discussed. At this point we relate our results with those obtained in [8]. In Section 5, we give the definition of modular class of a Jacobi-Nijenhuis algebroid and we prove a result which generalizes the corresponding one of [1]: there exists a hierarchy of A-vector fields that defines two hierarchies of vector fields, one on M × R and another on M . These hierarchies determine a family of Jacobi structures on the manifold M .
Notation and conventions: Let (A, ρ, [ , ]) be a Lie algebroid over M . We denote by X k (A) (resp. Ω k (A)) the C ∞ (M )-module of A-k-vector fields (resp. A-k-forms), by X(A) = ⊕ k X k (A) (resp. Ω(A) = ⊕ k Ω k (A)) the corresponding Gerstenhaber algebra of A-multivector fields (resp. A-forms) and by X top (A) the top-degree sections of A. The De Rham differential is denoted by d while d stands for the Lie algebroid differential.
Regarding the conventions of sign for the Schouten bracket and for the interior product by a multivector field, we use the same conventions of [3,11], which are different from those of [7,8].

Jacobi algebroids
We begin by recalling some well known facts about Jacobi algebroids.
2.1. Jacobi algebroids. A Jacobi algebroid [3] or generalized Lie algebroid [7] is a pair (A, φ 0 ) where A = (A, [ , ] , ρ) is a Lie algebroid over a manifold M and φ 0 ∈ Ω 1 (A) is a 1-cocycle in the Lie algebroid cohomology with trivial coefficients, dφ 0 = 0. A Jacobi algebroid has an associated Schouten-Jacobi bracket on the graded algebra X(A) of multivector fields on A given by (1) [ This bracket [ , ] φ 0 satisfies the following properties (in fact it is totaly defined by them), with X, Y ∈ X 1 (A), P ∈ X p (A), Q ∈ X q (A) and f ∈ C ∞ (M ): (2) [ In property (2), ρ φ 0 is the representation of the Lie algebra X 1 (A) on C ∞ (M ) given by The cohomology operator d φ 0 associated with this representation is called φ 0 -differential of A and is given by With the φ 0 -differential we can define a φ 0 -Lie derivative: , ω ∈ Ω(A). In [7,4] we can find a construction which allow us to obtain a Lie algebroid over M × R from a Jacobi algebroid over M . This construction is very useful when we speak about Jacobi algebroids, in fact it contains the essence of philosophy adopted in the proofs in this paper, so we will explain it now.
Consider the natural vector bundleÂ = A × R over M × R. The sections ofÂ may be seen as time-dependent sections of A and this space is generated as a C ∞ (M × R)-module by the space of sections of A, which are simply the time-independent sections ofÂ.
The relation (16) can be generalized to multisections of A * if we consider the gauging in Ω(A): One should also notice that the structure of Lie algebroid onÂ * does not coincide with Lie algebroid structure induced from A * by the 1-cocycle X 0 (at least not in the same way it was done with A and φ 0 ). In fact, the bracket of two time independent sections onÂ * , α, β ∈ Ω 1 (A), is given by and the anchor ofÂ * is defined bŷ Any Jacobi bialgebroid ((A, φ 0 ), (A * , X 0 )) gives to M a structure of Jacobi manifold, i.e., it equips M with a bivector field P M and a vector field E M satisfying or, equivalently, it defines a Jacobi bracket on C ∞ (M ) given by: 2.2. The triangular Jacobi bialgebroid of a Jacobi manifold. Let (M, Λ, E) be a Jacobi manifold, i.e, a manifold equipped with a bivector Λ and a vector field E such that The vector bundle T * M × R is endowed with a Lie algebroid structure over M [10]. The Lie bracket and the anchor are defined by In this Lie algebroid the differential is given by . Now consider the canonical vector bundle T M × R over M with its structure of Lie algebroid given by the Lie bracket and the anchor ρ(X, f ) = X.
The differential d of this Lie algebroid is in T * M × R coincides with the Lie algebroid structure defined by the Jacobi bivector (Λ, E). In fact one can check that and So we may conclude that the pair ( (T M × R, (0, 1)), (T * M × R, (−E, 0)) ) is a triangular Jacobi bialgebroid [7]. Moreover, the Jacobi structure induced on the base manifold coincides with the initial one.

Compatible Jacobi bivectors.
With the construction presented in the section 2.1 the notion of compatible Jacobi bivectors appears naturally. Definition 2. Let (A, φ 0 ) be a Jacobi algebroid. Two Jacobi bivectors P 1 and P 2 on A are said to be compatible if Due to relation (12), compatible Jacobi bivectors P 1 and P 2 on A are obviously associated with compatible Poisson bivectors onÂ,P 1 = e −t P 1 andP 2 = e −t P 2 : Moreover, they cover compatible Jacobi structures on the base manifold M . Recall that two compatible Jacobi structures on a manifold M (see [16]) is a pair of Jacobi structures (Λ 1 , is also a Jacobi structure, or, equivalently, they satisfy the following two conditions: Theorem 3. Let P 1 and P 2 be compatible Jacobi bivectors on a Jacobi algebroid (A, φ 0 ). These bivectors cover two compatible Jacobi structures on the base manifold M .
Since ρ is a Lie algebroid morphism, we have So the given Jacobi structures on M are compatible.

Jacobi-Nijenhuis algebroids
We begin this section exposing some well known results about Nijenhuis operators and compatible Poisson structures on Lie algebroids.

Poisson-Nijenhuis Lie algebroids. Let (A, [ , ]
, ρ) be a Lie algebroid over a manifold M . Recall that a Nijenhuis operator is a bundle map N : A → A (over the identity) such that the induced map on the sections (denoted by the same symbol N ) has vanishing torsion: Let us set ρ N := ρ • N . For a Nijenhuis operator N , one easily checks that the triple A N = (A, [ , ] N , ρ N ) is a new Lie algebroid, and then N : Since N is a Lie algebroid morphism, its transpose gives a chain map of complexes of differential forms N * : Hence we also have a map at the level of algebroid cohomology N * : When the Lie algebroid A is equipped with a Poisson structure P and a Nijenhuis operator N which are compatible, it is called a Poisson-Nijenhuis Lie algebroid.
The compatibility condition between N and P means that N P is a bivector field and [ , ] N P = [ , ] N P , where [ , ] N P is the bracket defined by the bivector field N P ∈ X 2 (A), and [ , ] N P is the bracket obtained from the Lie bracket [ , ] N by the Poisson bivector P .
As a consequence, N P defines a new Poisson structure on A, compatible with P : [P, N P ] = [N P, N P ] = 0, and one has a commutative diagram of morphisms of Lie algebroids: In fact, we have a whole hierarchy N k P (k ∈ N) of pairwise compatible Poisson structures on A.
3.2. Jacobi-Nijenhuis algebroids. Let (A, φ 0 ) be a Jacobi algebroid and N a Nijenhuis operator on A. The definition of the Lie algebroid structure onÂ = A × R given by (9) and (10) allows us to say that N is also a Nijenhuis operator onÂ. So we have an additional Lie algebroid structure onÂ,Â N . Proof. First notice that, since N : Since X(Â), as C ∞ (M × R)-module, is generated by X(A), we conclude that A N and the Lie algebroid structure onÂ induced from A N by φ 1 are the same.
In fact we have a whole sequence of Lie algebroid structures onÂ given by N k or, equivalently, by the 1-cocycle of A N k , φ k = N * k φ 0 : Now suppose P ∈ X 2 (A) is a Jacobi bivector, i.e., a bivector field such that [P, P ] φ 0 = 0. If N P is a bivector on A, we can consider the bracket on A * obtained from (A, φ 0 ) by N P : . On the other hand, we can also consider the bracket on A * obtained from (A N , φ 1 = N * φ 0 ) by the Jacobi bivector P : Definition 5. The Jacobi bivector P and the Nijenhuis operator N are compatible if the following two conditions are satisfied: (1) N P = P N * ; (2) the brackets [ , ] N P and [ , ] N P , given by (33) and (34), coincide. In this case, the Jacobi algebroid (A, φ 0 ) is said to be a Jacobi-Nijenhuis algebroid and is denoted by (A, φ 0 , P, N ).
The compatibility between N and P can be expressed by the vanishing of a suitable concomitant.
On a Jacobi algebroid (A, φ 0 ) consider a Nijenhuis operator N and a Jacobi bivector P such that N P is a bivector. Following [12], we define the concomitant of P and N as where [ , ] N P and [ , ] N P are the brackets on A * given by (33) and (34), respectively. We immediately see that Condition (2) on Definition 5 is equivalent to C(P, N ) = 0.
A direct computation gives the following equalities, with α, β ∈ Ω 1 (A): P is the bracket onÂ * obtained fromÂ N byP and [ , ] NP is the bracket onÂ * obtained fromÂ by the bivector NP .
The Poisson-compatibility between N andP implies NP ,P Â = 0 (see [12]). From [N P, P ] φ 0 = e 3t NP ,P Â , we conclude that the Jacobicompatibility between N and P implies [N P, P ] φ 0 = 0. Proof. Consider the Poisson bivectorP = e −t P onÂ . As we have already seen, the Jacobi-compatibility between N and P is equivalent to Poissoncompatibility between N andP and we have a hierarchy of compatible Poisson bivectors N kP , k ∈ N, onÂ. This hierarchy induces a hierarchy of compatible Jacobi bivectors on A, N k P : Proof. This is an immediate consequence of the above proposition and theorem 3.
Also the compatibility conditions define a sequence of Lie algebroid structures on A * . Theorem 8. Let (A, φ 0 , P, N ) be a Jacobi-Nijenhuis algebroid. Then A * has a hierarchy of Jacobi algebroid structures (A * , X k ), such that ((A, φ i ), (A * , X k )), φ i = N * i φ 0 and X k = N k X 0 , i ≤ k, k ∈ N, are triangular Jacobi bialgebroids.
Proof. Last proposition guarantees that N i P , i ∈ N, is a hierarchy of compatible Jacobi bivectors.
Each one of the Poisson bivectors N kP = N k P = e −t N k P defines a Lie algebroid structure onÂ * , , and a Lie algebroid structure on A * , [α, β] N k P = e −t e t α, e t β N kP , α, β ∈ X 1 (A * ).
As in the Poisson case, N * is a Nijenhuis operator of A * and we have a commutative relation between duality by P and deformation along N * . Proposition 9. Let (A, φ 0 , P, N ) be a Jacobi-Nijenhuis algebroid and consider the Lie algebroid structure on A * given by (17) and (18). The operator N * is a Nijenhuis operator on A * .
This way A * can be deformed by N * into A * N * and one can easily check that this is exactly the Lie algebroid A * N P . Proposition 10. The Lie algebroid A * N k P coincides with the Lie algebroid A * N * k , obtained from A * by deformation along N * k .
We finish this section, showing that the definition of strong (or strict)Jacobi-Nijenhuis structure defined for Jacobi manifolds in [9] and [17] can be recovered in this framework.
(ii) The concomitant of (Λ, E) and N , C((Λ, E), N ), identically vanishes. The concomitant C((Λ, E), N ) is given in [9,17] by The modular form of the Lie algebroid A with respect to η ⊗ µ (see [2]) is the 1-form ξ η⊗µ A ∈ Ω 1 (A), defined by . This is a 1-cocycle of the Lie algebroid cohomology of A. If one makes a different choice of sections η ′ and µ ′ , then η ′ ⊗ µ ′ = f η ⊗ µ, for some nonvanishing smooth function f ∈ C ∞ (M ). One checks easily that the modular form ξ η ′ ⊗µ ′ A associated with this new choice is given by:  [2] to define the modular class of the Lie algebroid A. Obviously Moreover, since D is a representation and φ 0 a 1-cocycle of A, Definition 13. The modular form of the Jacobi algebroid (A, φ 0 ) with respect to η ⊗ µ is the A-form ξ φ 0 , η⊗µ A defined by Again, the cohomology class of a modular form is independent of the section of Q A chosen. Definition 14. The modular class of the Jacobi algebroid (A, φ 0 ) is the cohomology class of a modular form. It will be denoted by Obviously mod φ 0 A = mod A if and only if φ 0 is exact.

4.3.
Relation between the modular classes of A andÂ and of A * andÂ * . Let (A, φ 0 ) be a Jacobi algebroid of rank n. In this section we compute modular forms ofÂ andÂ * (in the triangular case) and we establish relations between them and the modular forms of A and A * . Let η ∈ X n (A) and µ ∈ Ω top (M ), then η is also a n-section ofÂ andμ = µ ∧ dt is a volume form of M × R. The Lie bracket onÂ coincides with the Lie bracket on A for timeindependent multivectors, so Now consider the section of ∧ nÂ ,η = e −(n−1)t η. Using relation (12) and definition of the Schouten-Jacobi bracket [ , ] φ 0 , we find that the modular form ofÂ with respect toη ⊗μ is given by Proposition 15. Let (A, φ 0 ) be a Jacobi algebroid, then It is clear that the cohomology considered in the previous proposition is theÂ-cohomology. InÂ the 1-form φ 0 is exact, φ 0 =dt and, generally, this is not the case in A. Now suppose we also have a Jacobi bivector P on (A, φ 0 ). We saw that it induces a Poisson structure onÂ, a Lie algebroid structure onÂ * and another one on A * . Consider ν ∈ X n (A * ) a top-section on A * and µ a volume form on M .
Proposition 16. The modular form of the Lie algebroid A * with respect to ν ⊗ µ is given by withν = e nt ν andμ = µ ∧ dt.

4.4.
Relation with the modular vector field of a triangular Jacobi bialgebroid. The definition of modular class of a triangular Jacobi bialgebroid was given in [8]. In this section we will present this definition using the approach we have chosen, relating it with the modular field of the triangular bialgebroid associated with the Jacobi bialgebroid. Let (Â,P ) be the triangular Lie bialgebroid associated with the triangular Jacobi algebroid (A, φ 0 , P ) of rank n and ν a section of ∧ n A * . The modular field of the triangular Lie bialgebroid (Â,P ) with respect toν = e nt ν (see [11]) is the sectionXν ofÂ given bŷ Comparing with the definition of M ν (A,φ 0 ,P ) , the modular vector field of the triangular Jacobi bialgebroid (A, φ 0 , P ) given in [8], we notice that Moreover (see (43)), Using the definition of modular form of a Lie algebroid (37): ξν ⊗μ A * (α)ν ⊗μ = [α,ν]P ⊗μ +ν ⊗ Lρ (P ♯ (α))μ and relation (41), we obtain On the other hand, notice that relation (45) implies d f,Xν ν = d f,ν P , where X T (M ×R) is the modular vector field of the Poisson manifold M × R (endowed with the Poisson bivector induced from the triangular Lie bialgebroid (Â,P )).
Since the 1-form φ 0 is closed, we have On another hand,ρ(Xν ) = e −t ρ(M ν (A,φ 0 ,P ) ) + M ν (A,φ 0 ,P ) (φ 0 ) ∂ ∂t and equation (47) can be rewritten as Let (P M , E M ) be the Jacobi structure on M induced by the triangular Jacobi algebroid (A, φ 0 , P ), i.e., The modular field of the Jacobi manifold (M, P M , E M ), V (P M ,E M ) , was introduced in [18] and is defined as

So, equation (48) is equivalent to
4.5. Duality between modular classes of A and A * . Following the philosophy of this paper, we will find a relation between the modular classes of the Jacobi algebroids (A, φ 0 ) and (A * , X 0 ) using relations on the associated Lie bialgebroid. So we begin by presenting some results about duality of modular classes on Lie bialgebroids.
Proof. Since ν is a top-section of A * , using Cartan's formula, we have Substituting in (49) we have Again because ν is a top-section, we have that i P (dα ∧ ν) = 0, so i P (dα)ν = dα ∧ i P ν and (50) On the other hand, using the fact that ∂ P = [d, i P ] is a generator of the Gerstenhaber algebra of A * , we have where µ is a volume form of M , η ∈ X top (A) and ν ∈ Ω top (A) such that ν, η = 1.
Proof. Since ν, η = 1, we have Now let (A, φ 0 ) be a Jacobi algebroid of rank n and P a Jacobi bivector on A. The pair (Â,P ) is a triangular Lie bialgebroid and we can use the previous proposition to relate the modular classes ofÂ andÂ * .
Consider η ∈ X n (A) and ν ∈ Ω n (A) such that ν, η = 1, then we havẽ Relations (40) and (42) imply that The previous equation is obviously equivalent to the duality equation written in [8]. It can also be rewritten as or as
Definition 19. The modular vector field of the Jacobi-Nijenhuis algebroid (A, φ 0 , P, N ) is defined by and is independent of the section of Q A chosen. Its cohomology class is called the modular class of (A, φ 0 , P, N ) and is denoted by mod (N,P ) A = X (N,P ) .
Remark 20. In fact, the modular class defined above is mod(N * ), the relative modular class of the Lie algebroid morphism N * : A * N * → A * [14]. As in the Poisson case, mod(N ) and mod(N * ) are related by P : P ♯ mod(N ) = − mod(N * ).
Following [1], if N is non-degenerated, we have a hierarchy ofÂ-vector fields:X These hierarchies cover two hierarchies, one on M × R and another one on M : The hierarchy on M is given by Remark 22. Some remarks should be made at this point. First, one should notice that even if N is degenerated the hierarchy exists but only for i+j > 1, i.e., X i+j (N,P ) = d N i P h j = d N j−1 P h i+1 , (0 ≤ i < j, 1 < j). In case N is degenerated we can always consider a non-degenerated Nijenhuis operator of the form N + λI, λ constant, and we obtain the same algebra of commuting integrals.
It is also important to observe that although the hierarchy of vector fields on A is defined by a Nijenhuis operator, we may not have a Nijenhuis operator on M nor on M ×R that generates neither one of the covered hierarchies.
We will finish with a relation between the sequence of modular vector fields of the Jacobi-Nijenhuis algebroid and the sequence of modular vector fields of the Jacobi bialgebroid (in the sense of [8]).