E extracta mathematicae Vol. 31, Núm. 2, 199 – 225 (2016) On Generalized Lie Bialgebroids and Jacobi Groupoids Apurba Das Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, West Bengal, India apurbadas348@gmail.com Presented by Juan C. Marrero Received November 23, 2016 Abstract: Generalized Lie bialgebroids are generalization of Lie bialgebroids and arises nat- urally from Jacobi manifolds. It is known that the base of a generalized Lie bialgebroid carries a Jacobi structure. In this paper, we introduce a notion of morphism between gen- eralized Lie bialgebroids over a same base and prove that the induce Jacobi structure on the base is unique up to a morphism. Next we give a characterization of generalized Lie bialgebroids and use it to show that generalized Lie bialgebroids are infinitesimal form of Jacobi groupoids. We also introduce coisotropic subgroupoids of a Jacobi groupoid and these subgroupoids corresponds to, so called coisotropic subalgebroids of the corresponding generalized Lie bialgebroid. Key words: Jacobi manifolds, coisotropic submanifolds, (generalized) Lie bialgebroids, Ja- cobi groupoids. AMS Subject Class. (2010): 17B62, 53C15, 53D17. 1. Introduction The notion of Lie bialgebroid was introduced by Mackenzie and Xu [11] as a generalization of Lie bialgebra and infinitesimal version of Poisson groupoid. Roughly, a Lie bialgebroid (A,A∗) over M is a Lie algebroid A over M such that its dual vector bundle A∗ also carries a Lie algebroid structure which is compatible in a certain way with that of A. As an example, if (M,π) is a Poisson manifold, then (TM,T∗M) forms a Lie bialgebroid over M, where TM is the usual tangent Lie algebroid and T∗M is the cotangent Lie algebroid of the Poisson manifold M. A duality theorem for a Lie bialgebroid was shown in [11], that is, if (A,A∗) satisfy the criteria of a Lie bialgebroid, then (A∗,A) also satisfies a similar criteria. It was also proved in [11] that the base space of a Lie bialgebroid carries a natural Poisson structure. In [13], Kosmann- Schwarzbach gave a simple proof of the duality theorem for Lie bialgebroid and the fact that the base of a Lie bialgebroid has a Poisson structure. Jacobi manifolds are generalization of Poisson manifolds. In [6], Iglesias and Marrero introduced a notion of generalized Lie bialgebroid (genaraliza- tion of Lie bialgebroid) in such a way that a Jacobi manifold associates a 199 200 a. das canonical generalized Lie bialgebroid structure. More precisely, a general- ized Lie bialgebroid ( (A,ϕ0),(A ∗,X0) ) over M is a Lie algebroid A over M together with 1-cocycle ϕ0 ∈ ΓA∗ and such that the dual bundle A∗ also car- ries a Lie algebroid structure with X0 ∈ ΓA be a 1-cocycle of it and satisfy some compatibility conditions like a Lie bialgebroid. Given a Jacobi mani- fold (M,Λ,E), if we consider the Lie algebroid TM × R → M with 1-cocycle (0,1) ∈ Γ(T∗M × R) = Γ(T∗M) ⊕ C∞(M) and the Lie algebroid on the dual 1-jet bundle T∗M × R → M with 1-cocycle (−E,0) ∈ Γ(TM × R) = Γ(TM) ⊕ C∞(M), then the pair ( (TM × R,(0,1)),(T∗M × R,(−E,0)) ) is a generalized Lie bialgebroid over M [6]. Moreover it is shown in [6] that if( (A,ϕ0),(A ∗,X0) ) is a generalized Lie bialgebroid over M, the base manifold M induces a Jacobi structure. We remark that, Lie algebroid structures on a vector bundle A → M are in one-to-one correspondence with the linear Poisson structures on the dual bundle A∗ [10]. This correspondence has been extended in [5] to the Jacobi setup by introducing a notion of linear Jacobi structure on a vector bundle. More precisely, they showed that given a Lie algebroid A → M with a 1- cocycle ϕ ∈ ΓA∗, the dual bundle A∗ carries a linear Jacobi structure and conversely, given a linear Jacobi structure on the dual A∗ of a vector bundle A induces a Lie algebroid structure on A together with a 1-cocycle ϕ ∈ ΓA∗. The notion of morphism between Lie bialgebroids were introduced in [11]. In this paper, we define a notion of morphism between generalized Lie bial- gebroids over a same base. If all the cocycles are zero, then it reduces to the morphism between Lie bialgebroids. We also showed that, if there is a morphism between two generalized Lie bialgebroids over a same base, then the induced Jacobi structures on the base coming from two generalized Lie bialgebroids are same (Section 3). Poisson groupoids were introduced by Weinstein [15] as a unification of Poisson Lie groups and symplectic groupoids. In [11], the authors gave an equivalent definition of a Poisson groupoid as a Lie groupoid G ⇒ M with a Poisson structure Λ on the total space G such that the bundle map Λ♯ : T∗G → TG is a Lie groupoid morphism from the cotangent Lie groupoid T∗G → A∗G to the tangent Lie groupoid TG → TM, where AG → M is the Lie algebroid of G. This motivates them to give an equivalent characterization of its infinitesimal object, that is Lie bialgebroid. They showed that a pair of Lie algebroids (A,A∗) in duality over M is a Lie bialgebroid if and only if the bundle map Λ ♯ A ◦ RA : T ∗A∗ → T∗A → TA is a Lie algebroid morphism from the cotangent Lie algebroid T∗A∗ → A∗ of the linear Poisson manifold on generalized lie bialgebroids and jacobi groupoids 201 A∗, to the tangent Lie algebroid TA → TM of A, where RA : T∗A∗ → T∗A is the canonical isomorphism defined in [11] and ΛA being the induced linear Poisson structure on A. This description is rather complicated but useful to show that Lie bialgebroids are infinitesimal of Poisson groupoids. Jacobi groupoids were introduced by Iglesias and Marrero [9] as a general- ization of both Poisson groupois and contact groupoids. More precisely, a Ja- cobi groupoid is a Lie groupoid G ⇒ M together with a Jacobi structure (Λ,E) on G and a multiplicative function σ on G, such that the bundle map (Λ,E)♯ : T∗G×R → TG×R defined by (Λ,E)♯(ωg,γ) = (Λ♯(ωg)+γE(g),−⟨ωg,E(g)⟩), is a Lie groupoid morphism between the twisted cotangent Lie groupoid T∗G × R ⇒ A∗G to the twisted tangent Lie groupoid TG × R ⇒ TM × R (cf. Definition 4.7). These twisting has been done by using the multiplicative function σ. Thus it is possible to give an equivalent characterization of a gen- eralized Lie bialgebroid in terms of some Lie algebroid morphism. More pre- cisely, we show that a pair of Lie algebroids with 1-cocycles ( (A,ϕ0),(A ∗,X0) ) in duality over M is a generalized Lie bialgebroid over M if and only if the map (ΛA,EA) ♯ ◦ (RA,−id) : T∗A∗ × R → T∗A × R → TA × R is a Lie al- gebroid morphism from the 1-jet Lie algebroid T∗A∗ × R → A∗ of the linear Jacobi manifold A∗, to the twisted tangent Lie algebroid TA × R → TM × R of (A,ϕ0), where (ΛA,EA) be the linear Jacobi structure on A (cf. Theorem 4.2). Note that an another characterization of a generalized Lie bialgebroid in terms of some Jacobi algebroid morphism was given in the thesis of D. Iglesias Ponte [4]. Then the Theorem 4.2 is being used to show that the infinitesimal form of Jacobi groupoids are generalized Lie bialgebroids (cf. Theorem 4.9) (Section 4). Coisotropic subgroupoids of a Poisson groupoid were introduced in [16] as a generalization of coisotropic subgroups of a Poisson Lie group. The Lie alge- broids of coisotropic subgroupoids of a Poisson groupoid are coisotropic sub- algebroids of the corresponding Lie bialgebroid. Here we introduce coisotropic subgroupoids of a Jacobi groupoid and show that the Lie algebroid of a coisotropic subgroupoid appears as some notion of coisotropic subalgebroid of the corresponding generalized Lie bialgebroid (Section 5). Notations. Given a Lie groupoid G ⇒ M, by α,β : G → M, we denote the source and target maps and ϵ : M → G the unit map. Two elements g,h ∈ G are composable if α(g) = β(h), and denote by G(2) ⊂ G × G the set of composable pairs. A morphism between two Lie groupoids G1 ⇒ M1 and G2 ⇒ M2 is a smooth map F : G1 → G2 over f : M1 → M2 which commutes with all structure maps. 202 a. das Given a vector bundle A → M, there is a cononical isomorphism RA : T∗A∗ → T∗A which is defined as follows. Suppose the vector bundle A is locally A ∣∣ U = U × V , where U ⊆ M is open, then the map RA is locally defined by RA(χ,ψ,Y ) = (−χ,Y,ψ), where χ ∈ T∗M, ψ ∈ V ∗, Y ∈ V (see [11] for more details). 2. Preliminaries In this section, we recall the definitions and basic facts about Jacobi man- ifolds, Lie algebroids and (generalized) Lie bialgebroids [3, 5, 6, 10, 11]. • Jacobi manifolds Definition 2.1. Let M be a smooth manifold. A Jacobi structure on M is a pair (Λ,E), where Λ is a 2-vector field and E is a vector field on M satisfying [Λ,Λ] = 2E ∧ Λ, LEΛ = [E,Λ] = 0, where [ , ] is the Schouten bracket on the space of multivector fields of M. The manifold M endowed with a Jacobi structure is called a Jacobi manifold. If (M,Λ,E) is a Jacobi manifold, then one can define a bilinear, skew- symmetric bracket on the space of smooth functions by the following {f,g} = Λ(δf,δg) + fE(g) − gE(f) for all f,g ∈ C∞(M). The bracket { , } satisfies the Jacobi identity and the property of a first order differential operator on each arguments, that is, {f,gh} = g{f,h} + h{f,g} − gh{f,1} for all f,g,h ∈ C∞(M). Conversely, any bilinear, skew-symmetric bracket on C∞(M) which satisfies Jacobi identity and also first order differential operator on each argument defines a Jacobi structure on M. Note that, if E = 0, then Λ defines a Poisson structure on M [14]. Apart from Poisson and symplectic manifolds, contact and locally conformal sym- plectic (l.c.s) manifolds are also examples of Jacobi manifolds [3, 9]. Given a Jacobi bracket { , } on M and any nowhere zero function a ∈ C∞(M), one can define a new Jacobi bracket { , }a by the following {f,g}a = 1 a {af,ag} ∀f,g ∈ C∞(M). on generalized lie bialgebroids and jacobi groupoids 203 A smooth map Φ : M → N between two Jacobi manifolds is called a Jacobi map if {h ◦ Φ,h′ ◦ Φ}M = {h,h′}N ◦ Φ, for any h,h′ ∈ C∞(N). For a nowhere zero function a ∈ C∞(M), the pair (Φ,a) is called a conformal Jacobi map if Φ is a Jacobi map between (M,{ , }aM) and (N,{ , }N). Remark 2.2. (i) Given a Jacobi manifold (M,Λ,E), there is a vector bun- dle morphism (Λ,E)♯ : T∗M × R → TM × R given by (Λ,E)♯(ωm,γ) = ( Λ♯(ωm) + γE(m),−⟨ωm,E(m)⟩ ) for (ωm,γ) ∈ T∗mM × R, m ∈ M. (ii) Let (Λ,E) be a Jacobi structure on M. Then the product manifold M ×R carries a Poisson structure Λ̃ = e−t ( Λ + ∂ ∂t ∧ E ) , where t is the usual coordinate on R. The manifold M × R together with the Poisson structure Λ̃ is called the Poissonization of the Jacobi manifold (M,Λ,E). The notion of coisotropic submanifolds of a Poisson manifold [15] is ex- tended to the context of Jacobi manifolds. Definition 2.3. ([3]) Let (M,Λ,E) be a Jacobi manifold. Then a sub- manifold S ↪→ M is called a coisotropic submanifold of M if Λ♯(TxS) 0 ⊆ TxS for all x ∈ S, where Λ♯ : T∗M → TM is the bundle map induced by the bivector field Λ, and (TxS) 0 = {α ∈ T∗xM| α(v) = 0,∀v ∈ TxS}. Similar to the Poisson case, one can prove the following result. Proposition 2.4. Let (M,Λ,E) be a Jacobi manifold with corresponding Jacobi bracket { , }, and C ↪→ M be a closed submanifold. Then the followings are equivalent: (i) C is a coisotropic submanifold; (ii) the vanishing ideal I(C) = {f ∈ C∞(M) ∣∣f|C ≡ 0} is a Lie subalgebra of (C∞(M),{ , }). 204 a. das • Lie algebroids Definition 2.5. A Lie algebroid (A, [ , ],ρ) over a manifold M is a smooth vector bundle A over M together with a Lie bracket [ , ] on the space ΓA of smooth sections of A and a bundle map ρ : A → TM, called the anchor, such that (i) the induced map ρ : ΓA → X(M) is a Lie algebra homomorphism; (ii) for any f ∈ C∞(M) and X,Y ∈ ΓA, the following condition holds [X,fY ] = f[X,Y ] + (ρ(X)f)Y. Given a Lie groupoid G ⇒ M, its Lie algebroid consist of a vector bundle AG → M whose fiber at x ∈ M consist of the tangent space Tϵ(x)(β−1(x)). Then the space of sections ΓAG can be identified with the left invariant vector fields on G. Since the space of left invariant vector fields on G is closed under the Lie bracket, thus it induces a Lie bracket on ΓAG. The anchor is defined to be the differential of α restricted to AG. Any Lie algebra is a Lie algebroid over a point, and the tangent bundle of any smooth manifold is a Lie algebroid with the usual Lie bracket on vector fields and identity as anchor. Given any Jacobi manifold, there is a canonical Lie algebroid associated to it, given by the following example. Example 2.6. ([6, 9]) Let (M,Λ,E) be a Jacobi manifold, then the 1-jet bundle T∗M × R has a Lie algebroid structure (T∗M × R, [ , ](Λ,E),ρ(Λ,E)) over M, where the bracket and anchor are given by [(α,f),(β,g)](Λ,E) = ( LΛ♯(α)β − LΛ♯(β)α − δ(Λ(α,β)) + fLEβ − gLEα − ιE(α ∧ β), Λ(β,α) + Λ♯(α)(g) − Λ♯(β)(f) + fE(g) − gE(f) ) and ρ(Λ,E)(α,f) = Λ ♯(α) + fE for all (α,f),(β,g) ∈ Γ(T∗M × R) = Ω1(M) ⊕ C∞(M). When E = 0 ( that is, when Λ is a Poisson structure on M), one recover to the cotangent Lie algebroid of M by projecting onto the first factor [14]. Given a Lie algebroid (A, [ , ],ρ), the exterior algebra Γ( ∧• A) of multi- sections of A together with the generalized Schouten bracket, forms a Ger- stenhaber algebra [10]. Moreover Γ( ∧• A∗) together with the Lie algebroid on generalized lie bialgebroids and jacobi groupoids 205 differential d forms a differential graded algebra. When A = TM is the usual tangent bundle Lie algebroid, denote the differential of the Lie algebroid (that is, de-Rham differential of the manifold M) by δ. It is known that, Lie algebroid structures on a vector bundle A are in one- to-one correspondence with linear Poisson structures on the dual bundle A∗. This correspondence has been extended to the Jacobi set up by introducing the notion of linear Jacobi structure on a vector bundle [5]. We now describe these in details. Given a vector bundle, a Jacobi structure on the total space is called linear Jacobi structure, if the Jacobi bracket of two fibrewise linear functions is again linear, and the bracket of a linear function and the constant function 1 is a pull back function. Let q : A → M be a vector bundle over M with dual bundle q∗ : A∗ → M. Then any section X ∈ ΓA defines a fibrewise linear function lX on the dual bundle A ∗, namely lX(αx) = αx(Xx), where αx ∈ A∗x, x ∈ M. Conversely any fibrewise linear function on A∗ is of this form. Then the space of linear functions and pull back functions (that is, of the form f ◦ q∗, for f ∈ C∞(M)) generates C∞(A∗). Then we have the following theorem from [5]. Theorem 2.7. Let (A, [ , ],ρ) be a Lie algebroid over M and ϕ ∈ ΓA∗ be a 1-cocycle. Consider the bracket { , } on C∞(A∗) defined by {lX, lY } = l[X,Y ], {lX,f ◦ q∗} = (ρ(X)f + ϕ(X)f) ◦ q∗, {f ◦ q∗,g ◦ q∗} = 0, for X,Y ∈ ΓA and f,g ∈ C∞(M). Then { , } defines a linear Jacobi structure on A∗. Moreover, this Jacobi structure is given by Λ(A∗,ϕ) = ΛA∗ + ∆ ∧ ϕ v, E(A∗,ϕ) = −ϕ v, where ΛA∗ is the linear Poisson structure on A ∗ induced from the Lie algebroid structure on A, ∆ is the Liouville vector field on A∗ and ϕv ∈ X(A∗) is the vertical lift of ϕ ∈ ΓA∗. 206 a. das The converse part of the above theorem also holds true (see [5] for more details). A pair (A,ϕ) of a Lie algebroid A and a 1-cocycle ϕ ∈ ΓA∗ of it, is referred as a Jacobi algebroid. Let (A, [ , ],ρ) be a Lie algebroid over M and ϕ ∈ ΓA∗ be a 1-cocycle. Then the vector bundle à = A×R → M ×R carries a Lie algebroid structure ([ , ]˜ϕ, ρ̃ϕ). For any X̃, Ỹ ∈ Γà considered as time dependent sections of A, the Lie bracket and anchor are given by [X̃, Ỹ ]˜ϕ = [X̃, Ỹ ]˜ + ϕ(X̃) ∂Ỹ ∂t − ϕ(Ỹ ) ∂X̃ ∂t , ρ̃ϕ(X̃) = ρ̃(X̃) + ϕ(X̃) ∂ ∂t , where [X̃, Ỹ ]˜(x,t) = [X̃t, Ỹt](x) ρ̃(X̃)(x,t) = ρ(X̃t)(x), and ∂X̃ ∂t denotes the derivatie of X̃ with respect to time. Now let Ψ : à → à be the isomorphism of vector bundles over the identity on M × R defined by Ψ(v,t) = (etv,t), for (v,t) ∈ à = A × R. Then using Ψ and the Lie algebroid structure ([ , ]˜ϕ, ρ̃ϕ) on Ã, one can define a new Lie algebroid structure ([ , ]ˆϕ, ρ̂ϕ) on à such that the Lie algebroids (Ã, [ , ]˜ϕ, ρ̃ϕ) and (Ã, [ , ]ˆϕ, ρ̂ϕ) are isomorphic. Namely, we have [X̃, Ỹ ]ˆϕ = e−t ( [X̃, Ỹ ]˜ + ϕ(X̃) (∂Ỹ ∂t − Ỹ ) − ϕ(Ỹ ) (∂X̃ ∂t − X̃ )) , ρ̂ϕ(X̃) = e−t ( ρ̃(X̃) + ϕ(X̃) ∂ ∂t ) , for all X̃, Ỹ ∈ ΓÃ. Thus the total space of the dual bundle Ã∗ = A∗ × R → M × R carries a linear Poisson structure ΛA∗×R and this Poisson structure is the Poissonization of the linear Jacobi structure (Λ(A∗,ϕ),E(A∗,ϕ)) of A ∗ [5]. That is, ΛA∗×R = e −t ( Λ(A∗,ϕ) + ∂ ∂t ∧ E(A∗,ϕ) ) . The notion of morphism between Lie algebroids was introduced in [2]. Here we recall an alternative definition from [11]. Definition 2.8. Let A1 → M1 and A2 → M2 be two Lie algebroids. Then a vector bundle morphism F : A1 → A2 over f : M1 → M2 is called a on generalized lie bialgebroids and jacobi groupoids 207 Lie algebroid morphism if the graph of (F,f), that is, C := { (ϕ,ψ) ∈ A∗1 × A ∗ 2 | ⟨ϕ,X⟩ = ⟨ψ,F(X)⟩, for all X ∈ A1 compatible with ϕ } is a coisotropic submanifold of A∗1×A∗2, where A ∗ 1 and A ∗ 2 are equipped with the linear Poisson structures dual to the Lie algebroids A1 and A2, respectively. If M1 = M2 and f = identity, then F is a Lie algebroid morphism if and only if F preserves the Lie brackets and commute with anchors. • Lie bialgebroids Definition 2.9. ([11]) A Lie bialgebroid over M is a pair (A,A∗) of Lie algebroids in duality over M, such that the differential d∗ on Γ( ∧• A) defined by the Lie algebroid structure of A∗ and the Gerstenhaber bracket on Γ( ∧• A) defined by the Lie algebroid structure of A satisfies d∗[X,Y ] = [d∗X,Y ] + [X,d∗Y ], for all X,Y ∈ ΓA. Then there is a following characterization of a Lie bialgebroid [11]. Theorem 2.10. Let A be a Lie algebroid over M such that its dual bundle A∗ also carries a Lie algebroid structure. Consider the composition Π = Λ ♯ A ◦ RA, T∗A∗ −→ T∗A −→ TA, where ΛA being the linear Poisson structure on A coming from the Lie alge- broid A∗. Then (A,A∗) is a Lie bialgebroid if and only if T∗A∗ Π // �� TA �� A∗ ρ∗ // TM is a Lie algebroid morphism, where T∗A∗ → A∗ is the cotangent Lie algebroid of the linear Poisson structure on A∗ coming from the Lie algebroid A, and TA → TM is the tangent Lie algebroid of A. 208 a. das • Generalized Lie bialgebroids Given a Lie algebroid (A, [ , ],ρ) over M with 1-cocycle ϕ ∈ ΓA∗, there is an ϕ-deformed Lie algebroid representation ρϕ : ΓA × C∞(M) → C∞(M) (X,f) 7→ (ρϕ(X))f = ρ(X)f + ϕ(X)f. Thus one can define ϕ-deformed Lie algebroid differential dϕ which is given by dϕ : Γ( ∧• A∗) → Γ( ∧•+1 A∗) α 7→ dα + ϕ ∧ α. Then the ϕ-deformed Lie derivative is defined using Cartan formula LϕX : Γ( ∧• A∗) → Γ( ∧• A∗), α 7→ dϕιXα + ιXdϕα for X ∈ ΓA. One can also define ϕ-deformed Schouten bracket on the multi- sections of A by the formula [P,Q]ϕ = [P,Q] + (p − 1)P ∧ (ιϕQ) − (−1)p−1(q − 1)(ιϕP) ∧ Q, for P ∈ Γ( ∧p A),Q ∈ Γ( ∧q A) (see [1, 6]). Let (A, [ , ],ρ) be a Lie algebroid over M and ϕ0 ∈ ΓA∗ be a 1-cocyle. As- sume that the dual bundle A∗ also carries a Lie algebroid structure ([ , ]∗,ρ∗) and X0 ∈ ΓA be its 1-cocyle. Let the Lie derivative of the Lie algebroid A (resp. A∗) is denoted by L (resp. L∗). Definition 2.11. ([6]) The pair ( (A,ϕ0),(A ∗,X0) ) is said to be a gen- eralized Lie bialgebroid over M if the following conditions are hold dX0∗ [X,Y ] = [d X0 ∗ X,Y ] ϕ0 + [X,dX0∗ Y ] ϕ0, (1) LX0∗ϕ0P + L ϕ0 X0 P = 0 (2) for all X,Y ∈ ΓA and P ∈ Γ( ∧p A). Remark 2.12. (i) The condition (2) of the above definition is equivalent to ϕ0(X0) = 0, ρ(X0) = −ρ∗(ϕ0), L∗ϕ0X + [X0,X] = 0, (3) for all X ∈ ΓA. These follows from condition (2) by applying P = f ∈ C∞(M) and P = X ∈ ΓA. (ii) When ϕ0 = 0 and X0 = 0, one recover the definition of a Lie bialgebroid. on generalized lie bialgebroids and jacobi groupoids 209 Example 2.13. Given any smooth manifold M, the bundle TM×R → M has a Lie algebroid structure whose bracket and anchor are given by [(X,f),(Y,g)] = ( [X,Y ],X(g) − Y (f) ) , pr(X,f) = X, for (X,f),(Y,g) ∈ X(M) ⊕ C∞(M). Moreover (0,1) ∈ Ω1(M) ⊕ C∞(M) = Γ(T∗M × R) is a 1-cocycle of this Lie algebroid. If (M,Λ,E) is a Jacobi manifold, then the 1-jet bundle T∗M × R → M also carries a Lie algebroid structure (cf. Example 2.6) and one can check that, (−E,0) ∈ X(M) ⊕ C∞(M) = Γ(TM × R) is a 1-cocycle of it. For a Jacobi manifold (M,Λ,E), the pair ( (TM×R,(0,1)),(T∗M×R,(−E,0)) ) is a generalized Lie bialgebroid over M [6]. Another interesting class of examples of generalized Lie bialgebroids are provided by strict Jacobi-Nijenhuis manifolds [7]. The relation between Lie bialgebroid and generalized Lie bialgebroid is given by the following result. Proposition 2.14. ([6]) Let (A, [ , ],ρ) be a Lie algebroid over M with ϕ0 ∈ ΓA∗ be a 1-cocycle. Suppose the dual bundle A∗ → M also carries a Lie algebroid structure ([ , ]∗,ρ∗) and X0 ∈ ΓA be a 1-cocycle of it. Consider the Lie algebroids à = (A × R, [ , ]˜ϕ0, ρ̃ϕ0) and Ã∗ = (A∗ × R, [ , ]ˆX0∗ , ρ̂∗ X0) in duality over M × R. Then, (i) if ( (A,ϕ0),(A ∗,X0) ) is a generalized Lie bialgebroid over M, the pair (Ã,Ã∗) is a Lie bialgebroid over M × R; (ii) if (Ã,Ã∗) is a Lie bialgebroid over M × R, the pair ( (A,ϕ0),(A ∗,X0) ) is a generalized Lie bialgebroid over M. Thus using the duality of a Lie bialgebroid, one can conclude the duality of a generalized Lie bialgebroid. Proposition 2.15. ([6]) If ( (A,ϕ0),(A ∗,X0) ) is a generalized Lie bial- gebroid over M, then so is the pair ( (A∗,X0),(A,ϕ0) ) . Remark 2.16. One can also directly prove the duality of a generalized Lie bialgebroid following the proof of Kosmann-Schwarzbach [13] for Lie bialge- broids in the presence of cocycles, although these two proofs of the duality of a generalized Lie bialgebroid can be shown to be equivalent in the presence of the Proposition 2.14. 210 a. das Given a generalized Lie bialgebroid over M, it is proved in [6] that the base M carries a Jacobi structure. Let ( (A,ϕ0),(A ∗,X0) ) be a generalized Lie bialgebroid over M. Define a bracket { , } : C∞(M) × C∞(M) → C∞(M) by {f,g} = ⟨dϕ0f,dX0∗ g⟩. (4) Proposition 2.17. The bracket defined above satisfies the following prop- erties dϕ0{f,g} = [dϕ0f,dϕ0g]∗, (5) dX0∗ {f,g} = −[d X0 ∗ f,d X0 ∗ g]. (6) Proof. Since {f,g} = ⟨dϕ0f,dX0∗ g⟩ = ρX0∗ (dϕ0f)g = [dϕ0f,g]X0∗ , we have dϕ0{f,g} = [dϕ0f,dϕ0g]X0∗ = [d ϕ0f,dϕ0g]∗. Similarly, {f,g} = ρϕ0(dX0∗ g)f = [dX0∗ g,f]ϕ0, therefore dX0∗ {f,g} = [d X0 ∗ g,d X0 ∗ f] ϕ0 = −[dX0∗ f,d X0 ∗ g]. Theorem 2.18. ([6]) Let ( (A,ϕ0),(A ∗,X0) ) be a generalized Lie bialge- broid over M. Then the bracket above defines a Jacobi structure on M. Remark 2.19. (i) From (4), we have {f,g} = ⟨df,d∗g⟩+fρ∗(ϕ0)g+gρ(X0)f, therefore the induced Jacobi bivector field ΛM and the vector field EM is given by ΛM(δf,δg) = ⟨df,d∗g⟩ = ( ρ∗ ◦ ρ∗(δf) ) (g), EM = ρ∗(ϕ0) = −ρ(X0). (ii) Let (M,Λ,E) be a Jacobi manifold. If we consider the generalized Lie bialgebroid ( (TM × R,(0,1)),(T∗M × R,(−E,0)) ) given in Example 2.6, the induced Jacobi structure on M coincide with the original Jacobi structure. (iii) The dual generalized Lie bialgebroid ( (A∗,X0),(A,ϕ0) ) over M induces the opposite Jacobi structure of the above. on generalized lie bialgebroids and jacobi groupoids 211 3. Morphism between generalized Lie bialgebroids In this section, we introduce a notion of morphism between generalized Lie bialgebroids over a same base and prove that the induced Jacobi structure on the base of a generalized Lie bialgebroid is unique up to a morphism. • Jacobi algebroid maps Definition 3.1. Let (A,ϕ) and (B,ψ) be two Jacobi algebroids over M. Then a bundle map Φ : A → B over M is called a Jacobi algebroid map if Φ is a Lie algebroid map and Φ∗(ψ) = ϕ. Proposition 3.2. Let (A,ϕ) and (B,ψ) be two Jacobi algebroids over M. Then a bundle map Φ : A → B over M is a Jacobi algebroid map if and only if Φ∗ : B∗ → A∗ is a Jacobi map, under the dual linear Jacobi structures. Proof. Let Φ : A → B is a Jacobi algebroid map. Then for all X,Y ∈ ΓA, we have {lX, lY } ◦ Φ∗ = l[X,Y ] ◦ Φ ∗ = lΦ[X,Y ] = l[Φ(X),Φ(Y )] = {lΦ(X), lΦ(Y )} = {lX ◦ Φ∗, lY ◦ Φ∗}. For any f ∈ C∞(M), we also have {lX,f ◦ q∗A} ◦ Φ ∗ = ( ρA(X)f + ϕ(X)f ) ◦ q∗A ◦ Φ ∗ = ( ρB(Φ(X))f + (ψ,Φ(X))f ) ◦ q∗B = {lΦ(X),f ◦ q ∗ B} = {lX ◦ Φ∗,f ◦ q∗A ◦ Φ ∗}. Moreover for any f,g ∈ C∞(M), {f ◦ q∗A,g ◦ q ∗ A} ◦ Φ ∗ = {f ◦ q∗B,g ◦ q ∗ B} = {f ◦ q ∗ A ◦ Φ ∗,g ◦ q∗A ◦ Φ ∗}, since both sides are equals to zero. Therefore, Φ∗ : B∗ → A∗ is a Jacobi map. The converse part is similar. Proposition 3.3. Let (A,ϕ) and (B,ψ) be two Jacobi algebroids over M. Then a bundle map Φ : A → B over M is a Jacobi algebroid map if and only if Φ̃ = Φ × id : ( A × R, [ , ]˜ϕ, ρ̃A ϕ ) → ( B × R, [ , ]˜ψ, ρ̃B ψ ) is a Lie algebroid morphism over M × R. 212 a. das Proof. Suppose Φ is a Jacobi algebroid map. Then for any X̃, Ỹ ∈ Γ(Ã) = Γ(A × R), we have [ Φ̃(X̃),Φ̃(Ỹ ) ]˜ψ = [ Φ̃(X̃),Φ̃(Ỹ ) ]˜ + ψ ( Φ̃(X̃) )∂(Φ̃(Ỹ )) ∂t − ψ ( Φ̃(Ỹ ) )∂(Φ̃(X̃)) ∂t = Φ̃ [ X̃, Ỹ ]˜ + ( Φ̃∗(ψ),X̃ ) Φ̃ (∂Ỹ ∂t ) − ( Φ̃∗(ψ), Ỹ ) Φ̃ (∂X̃ ∂t ) = Φ̃ ( [X̃, Ỹ ]˜ϕ ) . Also for any X̃ ∈ Γ(Ã), (ρ̃B ψ ◦ Φ̃)(X̃) = ρ̃B ψ ( Φ̃(X̃) ) = ρ̃B ( Φ̃(X̃) ) + ψ ( Φ̃(X̃) ) ∂ ∂t = ρ̃B ◦ Φ(X̃) + ⟨ψ,Φ̃(X̃)⟩ ∂ ∂t = ρ̃A(X̃) + ⟨Φ∗ψ,X̃⟩ ∂ ∂t = ρ̃A(X̃) + ϕ(X̃) ∂ ∂t = ρ̃A ϕ (X̃). Hence Φ̃ defines a Lie algebroid morphism. The converse part is similar. Similarly Φ is a Jacobi algebroid map if and only if Φ̃ = Φ × id is a Lie algebroid map from (A × R, [ , ]ˆϕ, ρ̂A ϕ ) to (B × R, [ , ]ˆψ, ρ̂B ψ ) over M × R. It is known that given a Lie algebroid A, a section α ∈ ΓA∗ is a one cocycle of the Lie algebroid if and only if the image of α in A∗ is a coisotropic submanifold of A∗ with respect to linear Poisson structure [12]. Next we give an analogues of this result to Jacobi set up. Proposition 3.4. Let A be a Lie algebroid over M and ϕ ∈ ΓA∗ be a 1-cocycle. Then α ∈ ΓA∗ is a ϕ-deformed one cocycle of the Lie algebroid (that is, dϕα = 0) if and only if the image of α in A∗ is a coisotropic submanifold of A∗ with respect to linear Jacobi structure. Proof. Let α ∈ ΓA∗. Then for any X ∈ ΓA, the function lX − ⟨α,X⟩ ◦ q∗ on A∗ vanishes on the image of α. In fact, the space of functions on A∗ which vanishes on the image of α is generated by such kind of functions. Now for on generalized lie bialgebroids and jacobi groupoids 213 any X,Y ∈ ΓA, we have{ lX − ⟨α,X⟩ ◦ q∗, lY − ⟨α,Y ⟩ ◦ q∗ } = {lX, lY } − { ⟨α,X⟩ ◦ q∗, lY } − { lX,⟨α,Y ⟩ ◦ q∗ } + { ⟨α,X⟩ ◦ q∗,⟨α,Y ⟩ ◦ q∗ } = l[X,Y ] + ( ρ(Y )⟨α,X⟩ + ϕ(Y )⟨α,X⟩ ) ◦ q∗ − ( ρ(X)⟨α,Y ⟩ + ϕ(X)⟨α,Y ⟩ ) ◦ q∗. Therefore, { lX − ⟨α,X⟩ ◦ q∗, lY − ⟨α,Y ⟩ ◦ q∗ } (α(m)) = ⟨α, [X,Y ]⟩(m) + ( ρ(Y )⟨α,X⟩ ) (m) + (ϕ(Y ))(m)⟨α,X⟩(m) − ( ρ(X)⟨α,Y ⟩ ) (m) − (ϕ(X))(m)⟨α,Y ⟩(m) = − (dα)(X,Y )(m) − (ϕ ∧ α)(X,Y )(m) = − (dϕα)(X,Y )(m). Therefore, from Proposition 2.4, it follows that the image of α is a coiso- tropic submanifold of A∗ with respect to linear Jacobi structure if and only if dϕα = 0. • Generalized Lie bialgebroid morphisms Definition 3.5. A morphism between two generalized Lie bialgebroids( (A,ϕ0),(A ∗,X0) ) and ( (B,ψ0),(B ∗,Y0) ) over M is a map Φ : A → B of Lie algebroids such that the dual map Φ∗ : B∗ → A∗ is also a Lie algebroid map and they preserves the cocycles. That is, Φ(X0) = Y0, Φ ∗(ψ0) = ϕ0. Proposition 3.6. Let ( (A,ϕ0),(A ∗,X0) ) and ( (B,ψ0),(B ∗,Y0) ) be two generalized Lie bialgebroids over M. Then a bundle map Φ : A → B is a generalized Lie bialgebroid morphism if and only if the following conditions hold: (i) Φ : A → B is a Jacobi algebroid map from (A,ϕ0) to (B,ψ0); (ii) Φ : A → B is a Jacobi map under the linear Jacobi structures on A and B coming from the Jacobi algebroids (A∗,X0) and (B ∗,Y0) respectively. Proposition 3.7. Let ( (A,ϕ0),(A ∗,X0) ) and ( (B,ψ0),(B ∗,Y0) ) be two generalized Lie bialgebroids over M. Then a bundle map Φ : A → B is a generalized Lie bialgebroid morphism if and only if Φ̃ = Φ × id is a Lie bialgebroid morphism from (Ã,Ã∗) to (B̃,B̃∗) over M × R. 214 a. das Proof. Suppose Φ is a generalized Lie bialgebroid morphism. Therefore Φ is a Jacobi algebroid map from (A,ϕ0) to (B,ψ0). Hence from Proposition 3.3, we have Φ̃ = Φ × id is a Lie algebroid map from (A × R, [ , ]˜ϕ0, ρ̃A ϕ0) to (B × R, [ , ]˜ψ0, ρ̃B ψ0). Moreover Φ∗ is a Jacobi algebroid map (A∗,X0) to (B∗,Y0). Therefore Φ ∗ × id = Φ̃∗ is a Lie algebroid map from (A∗ × R, [ , ]ˆX0, ρ̂A X0) to (B∗ × R, [ , ]ˆY0, ρ̂B Y0). Hence Φ̃ is a Lie bialgebroid morphism from (Ã,Ã∗) to (B̃,B̃∗) over M × R. The converse part is similar. Theorem 3.8. Let ( (A,ϕ0),(A ∗,X0) ) be a generalized Lie bialgebroid over M and (ΛM,EM) denotes the induced Jacobi structure on M. Then the map ΦA : A → TM × R defined by ΦA(X) = (ρ(X),ϕ0(X)), for X ∈ ΓA, is a morphism between the generalized Lie bialgebroids ( (A,ϕ0),(A ∗,X0) ) and ( (TM ×R,(0,1)),(T∗M × R,(−EM,0)) ) , where ρ is the anchor of the Lie algebroid A. Moreover, if ( (A,ϕ0),(A ∗,X0) ) and ( (B,ψ0),(B ∗,Y0) ) are two generalized Lie bialgebroids over M and Ψ : A → B is a generalized Lie bialgebroid mor- phism, then the corresponding induced Jacobi structures on the base manifold M are same. Proof. The map ΦA is clearly a Lie algebroid map and ΦA(X0) = (ρ(X0),0) = (−EM,0). Note that the dual map Φ∗A : T ∗M × R → A∗ is such that Φ∗A(α,f)(X) = ⟨ (α,f),ΦA(X) ⟩ = ⟨ (α,f),(ρ(X),ϕ0(X)) ⟩ = α(ρ(X)) + fϕ0(X), for any X ∈ ΓA. Therefore, Φ∗A(α,f) = ρ ∗(α) + fϕ0. Hence, Φ ∗ A(0,1) = ϕ0. It is also a direct calculation to show that Φ∗A : ( T∗M × R, [ , ](ΛM,EM ),ρ(ΛM,EM ) ) → (A∗, [ , ]∗,ρ∗) preserves the Lie bracket. It also commutes with the anchors, as ρ∗ ◦ Φ∗A(α,f) = ρ∗(ρ ∗(α) + fϕ0) = ρ∗(ρ ∗(α)) + fρ∗(ϕ0) = Λ ♯ M(α) + fEM = ρ(ΛM,EM )(α,f), where ρ∗ is the anchor of the Lie algebroid A ∗. on generalized lie bialgebroids and jacobi groupoids 215 To prove the last part of the theorem, let the Lie algebroid differential of A and A∗ (resp. B and B∗) be denoted by dA and dA∗ (resp. dB and dB∗). Similarly the anchors are denoted by ρA and ρA∗ (resp. ρB and ρB∗). Then, {f,g}(A,A∗) = ⟨d ϕ0 A f,d X0 A∗g⟩ = ρ X0 A∗ (d ϕ0 A f)g = ρ X0 A∗ ( Φ∗A(δf,f) ) g = ρX0A∗ ( Ψ∗Φ∗B(δf,f) ) g = ρX0A∗ Ψ ∗(Φ∗B(δf,f))g = ρY0B∗(d ψ0 B f)g = {f,g}(B,B∗). Hence the proof. Therefore the induced Jacobi structure on the base of a generalized Lie bialgebroid is unique up to a morphism. 4. Generalized Lie bialgebroids and Jacobi groupoids In this section, we give a characterization of generalized Lie bialgebroids and using it, we show that generalized Lie bialgebroids are infinitesimal form of Jacobi groupoids. We begin with an example of twisted version of tangent Lie algebroid. Given a Lie algebroid q : A → M, the bundle Tq : TA → TM carries a Lie algebroid structure, called tangent Lie algebroid of A. If ϕ ∈ ΓA∗ is a 1-cocycle of the Lie algebroid A, then one can define a twist on the tangent Lie algebroid. Example 4.1. Let A → M be a Lie algebroid and ϕ ∈ ΓA∗ be a 1-cocycle. Thus ϕ defines a linear function on A. Its complete lift defines a function on TA, which is linear with respect to the vector bundle structure TA → TM. Hence it defines a section ϕ of the dual bundle (TA)∗ → TM. Moreover ϕ ∈ Γ(TA∗) becomes a 1-cocycle of the tangent Lie algebroid TA → TM [4]. Thus there is a Lie algebroid structure on T̃A = TA× R over TM × R, whose sections are considered as dependent sections of the tangent Lie algebroid TA → TM. The Lie bracket and anchor of the Lie algebroid T̃A are given by [X̃, Ỹ ]˜ϕ = [X̃, Ỹ ]˜ + ϕ(X̃) ∂Ỹ ∂t − ϕ(Ỹ ) ∂X̃ ∂t , ρ̃ϕ(X̃) = ρ̃TA(X̃) + ϕ(X̃) ∂ ∂t , where [X̃, Ỹ ]˜(x,t) = [X̃t, Ỹt](x), ρ̃TA(X̃)(x,t) = ρTA(X̃t)(x), for X̃, Ỹ ∈ Γ(T̃A). 216 a. das The Lie algebroid T̃A = TA×R → TM ×R defined above is called the twisted tangent Lie algebroid of (A,ϕ). • An alternative characterization of generalized Lie bialgebroids Let (A, [ , ],ρ) be a Lie algebroid over M and ϕ0 ∈ ΓA∗ be a 1-cocycle of it. Suppose the dual bundle A∗ also carries a Lie algebroid structure ([ , ]∗,ρ∗) and X0 ∈ ΓA be its 1-cocycle. Let (ΛA,EA) be the linear Jacobi structure on A coming from the Lie algebroid A∗ and its 1-cocycle X0. Then we have the following characterization of a generalized Lie bialgebroid. Theorem 4.2. The pair ( (A,ϕ0),(A ∗,X0) ) is a generalized Lie bialge- broid over M if and only if the composition (ΛA,EA) ♯◦(RA,−id) : T∗A∗×R → T∗A × R → TA × R T∗(A∗) × R (ΛA,EA) ♯◦(RA,−id) // �� TA × R �� A∗ ( ρ∗( ),X0( ) ) // TM × R (7) is a Lie algebroid morphism, where the domain T∗(A∗)×R −→ A∗ is the 1-jet Lie algebroid of the linear Jacobi manifold A∗ coming from the pair (A,ϕ0), and the range TA × R → TM × R is the twisted tangent Lie algebroid of (A,ϕ0). Proof. Consider the pair of Lie algebroids ( (A × R, [ , ]˜ϕ0, ρ̃ϕ0),(A∗ × R, [ , ]ˆX0∗ , ρ̂∗ X0) ) in duality over M × R and consider T∗(A∗ × R) (πA×R) ♯◦RA×R // �� T(A × R) �� A∗ × R // T(M × R) (8) where the left hand side is the cotangent Lie algebroid of the linear Poisson structure of A∗ × R coming from the Lie algebroid (A × R, [ , ]˜ϕ0, ρ̃ϕ0), the right hand side is the tangent Lie algebroid of A × R, and πA×R is the linear Poisson structure on A × R coming from the Lie algebroid structure (A∗ × on generalized lie bialgebroids and jacobi groupoids 217 R, [ , ]ˆX0∗ , ρ̂∗ X0). Note that πA×R is the Poissonization of the linear Jacobi structure (ΛA,EA) of A. Thus (πA×R) ♯(wa + γdt|t) = e−t ( Λ ♯ A(wa) + γEA(a),−wa(Ea) ∂ ∂t ∣∣ t ) for wa + γdt|t ∈ T∗(a,t)(A × R), a ∈ A. Then,{( (X,λ),(Y,µ) )∣∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R} ⊆ (TA∗ × R) × (TA∗ × R) is in the graph of (7) if and only if for all (Φ,ζ) ∈ T∗ϕ(A ∗) × R, we have⟨ (Φ,ζ),(X,λ) ⟩ = ⟨ (ΛA,EA) ♯ ◦ (RA,−id)(Φ,ζ),(Y,µ) ⟩ = ⟨ (ΛA,EA) ♯(RA(Φ),−ζ),(Y,µ) ⟩ = ⟨( Λ ♯ A(RAΦ) − ζEA,−⟨RAΦ,EA⟩ ) ,(Y,µ) ⟩ = ⟨ Y,Λ ♯ A(RAΦ) − ζEA ⟩ − µ⟨RAΦ,EA⟩. On the other hand{(( X,λ d dt ∣∣ t ) , ( Z,ξ d dt ∣∣ t ))∣∣∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R } ⊆ T(A∗ × R) × T(A∗ × R) is in the graph of (8) if and only if for all (Φ,ζdt|t) ∈ T∗(ϕ,t)(A ∗ × R), we have⟨( Φ,ζdt|t ) , ( X,λ d dt ∣∣ t )⟩ = ⟨( πA×R )♯ ◦ RA×R(Φ,ζdt|t),(Z,ξ d dt ∣∣ t )⟩ = ⟨( πA×R )♯( RAΦ,−ζdt|t ) , ( Z,ξ d dt ∣∣ t )⟩ = ⟨ e−t ( Λ ♯ A(RAΦ) − ζEA,−⟨RAΦ,EA⟩ d dt ∣∣ t ) , ( Z,ξ d dt ∣∣ t )⟩ =e−t ⟨ Z,Λ ♯ A(RAΦ) − ζEA ⟩ − e−tξ⟨RAΦ,EA⟩. Therefore, {( (X,λ),(Y,µ) )∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R} is in the graph of (7) if and only if {( (X,λ d dt ∣∣ t ),(etY,etµ d dt ∣∣ t ) )∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R} is in the graph of (8). Moreover using the corresponding dual linear Poisson structures, one can show that{( (X,λ),(Y,µ) )∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R} ⊆ (TA∗ × R) × (TA∗ × R) 218 a. das is a coisotropic submanifold if and only if{(( X,λ d dt ∣∣ t ) , ( etY,etµ d dt ∣∣ t ))∣∣∣X ∈ TϕA∗,Y ∈ TψA∗;λ,µ ∈ R} ⊆ T(A∗ × R) × T(A∗ × R) is a coisotropic submanifold. Thus we have that (7) is a Lie algebroid morphism if and only if (8) is a Lie algebroid morphism. Hence the result follows from Proposition 2.14 and Theorem 2.10. • Jacobi groupoids and generalized Lie bialgebroids Given a Lie groupoid G ⇒ M, its tangent Lie groupoid is given by TG ⇒ TM, whose structure maps are Tα,Tβ,Tϵ and the composition is denoted by ⊕TG. Definition 4.3. Let G ⇒ M be a Lie groupoid, η ∈ Ω1(G) be a contact 1-form on G and σ : G → R be a multiplicative function on G. Then the triple (G ⇒ M,η,σ) is called a contact groupoid if ηgh(Xg ⊕TG Yh) = ηg(Xg) + eσ(g)ηh(Yh) for all (Xg,Yh) ∈ (TG)(2). Example 4.4. ([9]) Given a Lie groupoid G ⇒ M with Lie algebroid AG → M, it is known that the cotangent bundle T∗G has a Lie groupoid structure over A∗G. Denote the structure maps of this groupoid by α̃, β̃, ϵ̃ and the composition by ⊕T∗G. Now if σ : G → R be a multiplicative function on G, then one can twist the cotangent Lie groupoid and get a Lie groupoid structure on T∗G × R over A∗G whose structure maps are given by α̃σ(ωg,γ) = e −σ(g)α̃(ωg) β̃σ(νh,ζ) = β̃(νh) − ζ(δσ)Aβ(h)G (ωg,γ) ⊕T∗G×R (νh,ζ) = ( (ωg + e σ(g)ζ(δσ)g) ⊕T∗G (eσ(g)νh),γ + eσ(g)ζ ) . Then if we consider the canonical contact 1-form ηG on T ∗G × R and the multiplicative function σ◦pr, where pr : T∗G×R → G denotes the projection onto G, the triple (T∗G × R ⇒ A∗G,ηG,σ ◦ pr) is a contact groupoid. on generalized lie bialgebroids and jacobi groupoids 219 The Lie algebroid of a contact groupoid is given by the following [9]. Theorem 4.5. Let (G ⇒ M,η,σ) be a contact groupoid. Then the base M admits a unique Jacobi structure (Λ0,E0) such that (α,e σ) is a conformal Jacobi map and β is an anti-Jacobi map. Moreover the Lie algebroid of G ⇒ M is isomorphic to the 1-jet Lie algebroid of the Jacobi manifold (M,Λ0,E0). Let G ⇒ M be a Lie groupoid with Lie algebroid AG → M, and σ : G → R be a multiplicative function on G. Then σ induces a 1-cocycle ϕ0 ∈ Γ(A∗G) of the Lie algebroid AG, defined by ϕ0|x(Xx) = Xx(σ), for Xx ∈ AxG, x ∈ M. If we consider the contact groupoid (T∗G × R ⇒ A∗G,ηG,σ ◦ pr) over A∗G, then the induced Jacobi structure on A∗G is same as the linear Jacobi struc- ture on A∗G coming from the Lie algebroid AG and its 1-cocycle ϕ0 [9]. Thus the Lie algebroid of the twisted cotangent groupoid T∗G×R → A∗G is isomor- phic to the 1-jet Lie algebroid T∗(A∗G) × R → A∗G, where A∗G is the linear Jacobi manifold. Moreover this isomorphism s is given by (j′G) −1 followed by (RAG,−id), that is, s = (j′G) −1 ◦ (RAG,−id), where j′G is the isomorphism between the Lie algebroids A(T∗G × R) and T∗(AG) × R over A∗G. Given a multiplicative function on a Lie groupoid, one can also twist the tangent Lie groupoid and get a Lie groupoid structure on TG×R over TM×R. Example 4.6. ([9]) Let σ : G → R be a multiplicative function on G, then there is a Lie groupoid structure on TG×R over TM ×R whose structure maps are given by (Tα)σ(Xg,λ) = ( (Tα)(Xg),Xg(σ) + λ ) (Tβ)σ(Yh,µ) = ( (Tβ)(Yh),µ ) (Xg,λ) ⊕TG×R (Yh,µ) = (Xg ⊕TG Yh,λ). The Lie algebroid of this Lie groupoid TG × R ⇒ TM × R is given by TAG × R → TM × R, the twisted tangent Lie algebroid of (AG,ϕ0). Let jG denote the isomorphism between TAG × R and A(TG × R) over TM × R, generalizing the isomorphism between TAG and ATG over TM. Definition 4.7. ([9]) (Jacobi groupoid) Let G ⇒ M be a Lie groupoid with a Jacobi structure (Λ,E) on G and σ : G → R be a multiplicative 220 a. das function. Then (G ⇒ M,Λ,E,σ) is called a Jacobi groupoid if the bundle map (Λ,E) ♯ : T∗G × R → TG × R is a Lie groupoid morphism T∗G × R (Λ,E)♯ // �� �� TG × R �� �� A∗G // TM × R from the twisted cotangent Lie groupoid given in Example 4.4 to the twisted tangent Lie groupoid given in Example 4.6. Example 4.8. Contact groupoids are examples of Jacobi groupoids [9]. Jacobi Lie groups [8] are just Jacobi groupoids over a point [9]. Suppose (G ⇒ M,Λ,E,σ) be a Jacobi groupoid. Then the dual bundle A∗G ∼= (TM)0 also carries a Lie algebroid structure whose bracket and anchor are given by [α,β]∗(x) = ( π1[(α̃,0),(β̃,0)](Λ,E) ) (x), ρ∗(ωx) = Λ ♯(ωx), for x ∈ M, where α,β ∈ ΓA∗G ∼= Γ(TM)0; α̃, β̃ be their arbitrary extension to 1-forms on G, and T∗G×R → T∗G being the projection onto the first factor [9]. Moreover the vector field E induces a 1-cocycle X0 ∈ ΓAG of the Lie algebroid A∗G and is defined by X0|x(ωx) = −⟨ωx,E(x)⟩, for ωx ∈ A∗xG, x ∈ M. Let (ΛAG,EAG) be the linear Jacobi structure on AG coming from the Lie algebroid A∗G and its 1-cocycle X0. Theorem 4.9. Let (G ⇒ M,Λ,E,σ) be a Jacobi groupoid with Lie alge- broid AG. Then ( (AG,ϕ0),(A ∗G,X0) ) is a generalized Lie bialgebroid over M. Proof. Since (Λ,E) ♯ : T∗G×R −→ TG×R is a morphism of Lie groupoids, thus applying the Lie functor, we get a morphism A ( (Λ,E) ♯ ) : A(T∗G×R) −→ A(TG × R) A(T∗G × R) A ( (Λ,E)♯ ) // �� A(TG × R) �� A∗G // TM × R on generalized lie bialgebroids and jacobi groupoids 221 between corresponding Lie algebroids. Then by an argument similar to [11] shows that the diagram A(T∗G × R) A ( (Λ,E)♯ ) // j′ G �� A(TG × R) T∗(AG) × R (ΛAG,EAG) ♯ // T(AG) × R jG OO commutes, where jG : TAG×R → A(TG×R) is the isomorphism of Lie alge- broids over TM × R, and j′G : A(T ∗G× R) → T∗(AG) × R is the isomorphism of Lie algebroids over A∗G. Thus it follows that, A(T∗G × R) A ( (Λ,E)♯ ) // j′ G �� A(TG × R) T∗(AG) × R (ΛAG,EAG) ♯ // T(AG) × R jG OO T∗(A∗G) × R (RAG,−id) 44jjjjjjjjjjjjjjjj s 55 (ΛAG,EAG) ♯◦(RAG,−id) 55 also commutes, as A ( (Λ,E) ♯ ) ◦ s = A ( (Λ,E) ♯ ) ◦ (j′G) −1 ◦ (RAG,−id) = jG ◦ (ΛAG,EAG)♯ ◦ (RAG,−id). Note that, s = (j′G) −1 ◦ (RAG,−id) is an isomorphism of Lie algebroids over A∗G, and also jG is an isomorphism of Lie algebroids over TM ×R. From the diagram below, A(T∗G × R) xxqq qq qq qq qq A ( (Λ,E)♯ ) // A(TG × R) &&NN NNN NNN NNN A∗G ( ρ∗( ),X0( ) ) // TM × R T∗(A∗G) × R ffMMMMMMMMMM s OO (ΛAG,EAG) ♯◦(RAG,−id) // T(AG) × R 88ppppppppppp jG OO 222 a. das since the top row is a morphism between Lie algebroids over ( ρ∗( ),X0( ) ) : A∗G → TM × R, the bottom row is also a morphism between Lie algebroids over the map ( ρ∗( ),X0( ) ) : A∗G → TM × R. Thus the result follows from Theorem 4.2. 5. Coisotropic subgroupoids of Jacobi groupoid In this section we introduce the notion of coisotropic subgroupoids of a Jacobi groupoid which is a straight forward generalization of coisotropic sub- groupoids of a Poisson groupoid [16]. We also study their infinitesimal coun- terpart. • Coisotropic subgroupoids Definition 5.1. Let (G ⇒ M,Λ,E,σ) be a Jacobi groupoid. Then a subgroupoid H ⇒ N of G ⇒ M is called a coisotropic subgroupoid if H is a coisotropic submanifold of G. Example 5.2. (i) Any Poisson groupoid (G ⇒ M,Λ) can be considered as a Jacobi groupoid with E = 0 and σ = 0. Then coisotropic subgroupoids of (G ⇒ M,Λ) are the coisotropic subgroupoids of (G ⇒ M,Λ,0,0). (ii) Let (G ⇒ M,Λ,E,σ) be a Jacobi groupoid. Then M is a coisotropic submanifold of G and carries an induced Jacobi structure [9]. If N ↪→ M be a coisotropic submanifold, the subgroupoid G|N := α−1(N) ∩ β−1(N) is a coisotropic subgroupoid of (G ⇒ M,Λ,E,σ). Note that, the infinitesimal object corresponding to a Jacobi groupoid (G ⇒ M,Λ,E,σ) is the generalized Lie bialgebroid ( (AG,ϕ0),(A ∗G,X0) ) . Therefore it is natural to ask how the Lie algebroid of a coisotropic sub- groupoid H ⇒ N is related to the generalized Lie bialgebroid ( (AG,ϕ0), (A∗G,X0) ) . To answer this question, we introduce coisotropic subalgebroids of a generalized Lie bialgebroid and show that infinitesimal form of coisotropic subgroupoids of a Jacobi groupoid appear as coisotropic subalgebroids of the corresponding generalized Lie bialgebroid ( (AG,ϕ0),(A ∗G,X0) ) . Definition 5.3. Let ( (A,ϕ0),(A ∗,X0) ) be a generalized Lie bialgebroid over M. Then a Lie subalgebroid B → N of A → M is called a coisotropic subalgebroid of ( (A,ϕ0),(A ∗,X0) ) if B ↪→ A is a coisotropic submanifold, where A is equipped with the linear Jacobi structure coming from (A∗,X0). on generalized lie bialgebroids and jacobi groupoids 223 Proposition 5.4. Let A → M be a Lie algebroid and ϕ0 ∈ ΓA∗ be a 1-cocycle. Then a subbundle B → N of A → M is a Lie subalgebroid (and hence ϕ0|N ∈ ΓB∗ is a 1-cocycle of it) if and only if B0 is a coisotropic submanifold of A∗, where A∗ is equipped with the linear Jacobi structure and B0x = {γ ∈ A∗x ∣∣ γ(v) = 0, ∀v ∈ Bx}, x ∈ N. Proof. First suppose that, B is a Lie subalgebroid of A → M. Note that for any X ∈ ΓA, the function lX is a linear function on A∗. Among the functions lX for X ∈ ΓA, those which vanishes on B0 are precisely those for which X ∣∣ N ∈ ΓB. Let X,Y ∈ ΓA be such that X ∣∣ N ,Y ∣∣ N ∈ ΓB. Then lX, lY are linear functions on A∗ vanishes on B0. We have the Jacobi bracket {lX, lY } = l[X,Y ]. Since B → N is a Lie subalgeboid of A → M, we have [X,Y ] ∣∣ N ∈ ΓB. Therefore the function {lX, lY } also vanishes on B0. Among the pull back functions on A∗, those which vanishes on B0 are of the form f ◦ q∗, for some f ∈ C∞(M) with f ∣∣ N ≡ 0. Therefore for any X ∈ ΓA and f ∈ C∞(M) with X ∣∣ N ∈ ΓB, f ∣∣ N ≡ 0, we have {lX,f ◦ q∗} = ( ρ(X)f + ϕ(X)f ) ◦ q∗. Since ρ(X) ∣∣ N ∈ TN and f ∣∣ N ≡ 0 ( that is, (δf)|N ∈ (TN)0 ) , the function ρ(X)f +ϕ(X)f = ⟨ρ(X),δf⟩+ϕ(X)f is vanishes on N, and hence, {lX,f ◦q∗} is vanishes on B0. Thus by Proposition 2.4, we have B0 is a coisotropic submanifold of A∗. Thus from the above proposition and since (B0)0 = B, we have the fol- lowing. Proposition 5.5. If B → N be a coisotropic subalgebroid of ( (A,ϕ0), (A∗,X0) ) , then B0 → N is a coisotropic subalgebroid of ( (A∗,X0),(A,ϕ0) ) . It is known that (cf. Proposition 2.18), the base of a generalized Lie bialgebroid carries an induced Jacobi structure. The next result shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced Jacobi structure. Proposition 5.6. Let ( (A,ϕ0),(A ∗,X0) ) be a generalized Lie bialgebroid over M and B → N be a coisotropic subalgebroid of ( (A,ϕ0),(A ∗,X0) ) . Then N is a coisotropic submanifold of M. 224 a. das Proof. If (ΛM,EM) denote the induced Jacobi structure on M, then from the Remark 2.19, we have Λ ♯ M = ρ∗ ◦ ρ ∗, where ρ and ρ∗ denote the anchors of the Lie algebroids A and A ∗ respec- tively. We first prove that ρ∗(TN)0 ⊆ B0. This is true because, ⟨ρ∗ξx,v⟩ = ⟨ξx,ρ(v)⟩ = 0, for ξx ∈ (TN)0x and v ∈ Bx. Thus we have Λ ♯ M(TN) 0 = ρ∗ ◦ ρ∗(TN)0 ⊆ ρ∗(B0) ⊆ TN, in the last inclusion we have used that B0 is a Lie subalgebroid of A∗. There- fore, N is a coisotropic submanifold of M. • Infinitesimal form of Coisotropic subgroupoids Proposition 5.7. Let (G ⇒ M,Λ,E,σ) be a Jacobi groupoid with gen- eralized Lie bialgebroid ( (AG,ϕ0),(A ∗G,X0) ) . Let H ⇒ N be a coisotropic subgroupoid with Lie algebroid AH → N. Then AH → N is a coisotropic subalgebroid of ( (AG,ϕ0),(A ∗G,X0) ) . Proof. Since H ⇒ N is a Lie subgroupoid of G ⇒ M, therefore AH → N is a Lie subalgebroid of AG → M. Next we claim that the anchor ρ∗ = Λ♯ ∣∣ (TM)0 of the Lie algebroid A∗G maps (AH)0 to TN. Observe that, for any x ∈ N, (AH)0x = (TM) 0 x ∩ (TH)0x and TxN = TxM ∩ TxH. Therefore, ρ∗(AH) 0 = Λ♯(TM)0 ∩ Λ♯(TH)0 ⊆ TM ∩ TH = TN, here we have used the fact that both M and H are coisotropic submanifolds of G. Let θ,ϑ ∈ ΓA∗G ∼= (TM)0 be such that θ ∣∣ N ,ϑ ∣∣ N ∈ (AH)0. Let θ̃, ϑ̃ be their respective extensions to 1-forms on G which are conormal to H. Then the 1-form π1[(θ̃,0),(ϑ̃,0)](Λ,E) on G is conormal to both M and H, as M and H are both coisotropic submanifolds of G. Therefore, ( π1[(θ̃,0),(ϑ̃,0)](Λ,E) )∣∣ N ∈ (TM)0 ∩ (TH)0 = (AH)0. Hence (AH)0 → N defines a Lie subalgebroid of A∗G → M. Therefore, by Proposition 5.4, it follows that AH ↪→ AG is a coisotropic submanifold. Thus, AH → N is a coisotropic subalgebroid of ( (AG,ϕ0),(A ∗G,X0) ) . on generalized lie bialgebroids and jacobi groupoids 225 Corollary 5.8. Let (G ⇒ M,Λ,E,σ) be a Jacobi groupoid and H ⇒ N be a coisotropic subgroupoid of it. Then N is a coisotropic submanifold of M. Acknowledgements The author would like to thank the referee for his comments on the earlier version of this manuscript. References [1] Y. Hagiwara, Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A: Math. Gen. 37 (26) (2004), 6713 – 6725. [2] P. J. Higgins, K. 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