C. R. Acad. Sci. Paris, t. 333, Série I, p. 861–866, 2001Géométrie différentielle/Differential Geometry

Matched pairs of Leibniz algebroids, Nambu–Jacobistructures and modular classRaúl IBAÑEZ a, Belén LOPEZ b, Juan C. MARRERO c, Edith PADRON c

a Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spainb Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas,

Gran Canaria, Spainc Departamento de Matemática Fundamental, Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain

E-mail: mtpibtor@lg.ehu.es; blopez@dma.ulpgc.es; jcmarrer@ull.es; mepadron@ull.es

(Reçu le 23 mai 2001, accepté après révision le 19 septembre 2001)

Abstract. The notion of a matched pair of Leibniz algebroids is introduced and it is shown that aNambu–Jacobi structure of ordern, n > 2, over a manifoldM defines a matched pair ofLeibniz algebroids. As a consequence, one deduces that the vector bundle

∧n−1(T∗M)⊕∧n−2

(T∗M) → M is a Leibniz algebroid. Finally, ifM is orientable, the modular classof M is defined as a cohomology class of order1 with respect to this Leibniz algebroid. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Algébroïdes de Leibniz croisés, structures de Nambu–Jacobi et classemodulaire

Résumé. La notion d’algébroïdes de Leibniz croisés est introduite et on démontre qu’une structurede Nambu–Jacobi d’ordren, n > 2, sur une variétéM définit deux algébroïdes deLeibniz croisés. Comme conséquence, on déduit que le fibré vectoriel

∧n−1(T∗M) ⊕∧n−2

(T∗M) →M est un algébroïde de Leibniz. Finalement, siM est orientable, la classemodulaire deM est définie comme une classe de cohomologie d’ordre1 par rapport à cetalgébroïde de Leibniz. 2001 Académie des sciences/Éditions scientifiques et médicalesElsevier SAS

Version française abrégéeLa notion d’algébroïde de Leibniz(à gauche) a été introduite récemment dans [4], comme une version

non commutative des algébroïdes de Lie. En plus des algèbres de Leibniz à gauche (au sens de [7]), unexemple typique d’algébroïde de Leibniz est le fibré vectoriel

∧n−1(T∗M) → M , où M est une variétémunie d’unestructure de Nambu–PoissonΛ d’ordre n, n > 2. Si L est la dérivée de Lie surM , lastructure d’algébroïde de Leibniz([[ , ]]Λ,#Λ) sur

∧n−1(T∗M) → M est décrite par#Λ(α) = i(α)Λet [[α,α′]]Λ = L#Λ(α)α

′ + (−1)n(i(dα)Λ)α′, pour toutα, α′ ∈Ωn−1(M) (voir [4]).Dans la première partie de cette Note, on introduit la notion d’algébroïdes de Leibniz croisés. Deux

algébroïdes de LeibnizA1 et A2 sur la même base sont dits croisés si la somme de Whitney defibrés vectorielsA = A1 ⊕ A2 a une structure d’algébroïde de Leibniz telle queA1 et A2 soientdes sous-algébroïdes de Leibniz deA. En utilisant une définition appropriée de représentation d’unalgébroïde de Leibniz sur un fibré vectoriel, on obtient des conditions nécessaires et suffisantes pourque deux algébroïdes de Leibniz soient croisés. On montre alors qu’une structure de Nambu–Jacobi sur

Note présentée par Charles-Michel MARLE.

S0764-4442(01)02150-4/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés 861

R. Ibañez et al.

une variétéM définit deux algébroïdes de Leibniz croisés. On rappelle qu’une structure de Nambu–Jacobi d’ordren sur M est unn-crochet de fonctions·, . . . , · anti-symétrique, qui agit comme unopérateur différentiel du premier ordre sur chacun de ses arguments et qui satisfait l’identité fondamentale(voir [10]). Le crochet de Nambu–Jacobi·, . . . , · induit un n-vecteurΛ et un (n − 1)-vecteursur M caractérisés par(df1, . . . ,dfn−1) = 1, f1, . . . , fn−1 et Λ(df1, . . . ,dfn) = f1, . . . , fn −∑n

i=1(−1)ifi1, f1, . . . , fi, . . . , fn. Les vecteursΛ et définissent des structures de Nambu–Poissond’ordre n et n − 1, respectivement [10] et l’on peut ainsi considérer les algébroïdes de Leibniz(∧n−1(T∗M), [[ , ]]Λ,#Λ

)et

(∧n−2(T∗M), [[ , ]],#). On démontre que ces algébroïdes sont croisés

et que la structure d’algébroïde de Leibniz([[ , ]],#) sur le fibré vectoriel∧n−1(T∗M)⊕

∧n−2(T∗M)→M est donnée par#(α,β) = #Λ(α) + #(β) et

[[(α,β), (α′, β′)]] =([[α,α′]]Λ + (−1)n−1 d

(i(α)

)∧ β′ + (−1)n−1

(i(dβ)

)α′ + L#(β)α

′

[[β,β′]] + L#Λ(α)β′ + (−1)n

(i(dα)Λ

)β′).

On introduit finalement laclasse modulaire d’une variété de Nambu–Jacobi(M,Λ,) orientable d’ordre

n, n > 2. Pour ce faire, on considère une forme volumeν surM et l’applicationM(Λ,)ν : C∞(M,R) ×

· · ·(n−1 · · ·×C∞(M,R) →C∞(M,R) caractérisée par la relationLXf1···fn−1ν = M

(Λ,)ν (f1, . . . , fn−1)ν,

où Xf1...fn−1 est le champ hamiltonien dansM des fonctionsf1, . . . , fn−1. On démontre queM(Λ,)ν

définit un1-cocycle dans le complexe de cohomologie associé à l’algébroïde de Leibniz(∧n−1(T∗M)⊕∧n−2(T∗M), [[ , ]],#

)et que la classe de cohomologie correspondante, la classe modulaire de(M,Λ,),

ne dépend pas de la forme volume considérée. Cette construction est une extension de celle réalisée pourdes variétés de Nambu–Poisson dans [4].

1. Matched pairs of Leibniz algebroids

A left Leibniz algebrais a real vector spaceg endowed with aR-bilinear map , : g× g→ g satisfyingthe left Leibniz identity, that is,

a1,a2, a3

−

a1, a2, a3

−

a2,a1, a3

= 0, for a1, a2, a3 ∈ g

(see[7]). Since the skew-symmetry condition is not required, a left Leibniz algebra can be considered as anon-commutative version of a Lie algebra. Note that

a1, a2+ a2, a1, a3

= 0, for a1, a2, a3 ∈ g. (1)

A representation of the left Leibniz algebra, (g, , ) over a real vector spaceV is a pair ofR-bilinearmappingsϕ1 : g× V → V andϕ2 : V × g→ V satisfying the following properties (see[7])

ϕ1

(a1, a2,m

)= ϕ1

(a1, ϕ1(a2,m)

)− ϕ1

(a2, ϕ1(a1,m)

),

ϕ2

(m,a1, a2

)= ϕ2

(ϕ2(m,a1), a2

)+ ϕ1

(a1, ϕ2(m,a2)

),

ϕ2

(ϕ2(m,a1), a2

)+ ϕ2

(ϕ1(a1,m), a2

)= 0.

A Leibniz algebroid structureon a differentiable vector bundleπ : A → M is a pair that consists of aleft Leibniz algebra structure[[ , ]] on the spaceΓ(A) of global cross sections ofπ : A → M and a vectorbundle morphismρ : A → TM , called theanchor map, such that the induced mapρ : Γ(A) → Γ(TM) =X(M) satisfies the following relations: (i)ρ : (Γ(A), [[ , ]]) → (X(M), [ , ]) is a left Leibniz algebrahomomorphism, and (ii)[[s1, fs2]] = f [[s1, s2]] + ρ(s1)(f)s2, for all s1, s2 ∈ Γ(A) andf ∈ C∞(M,R).In this case,(A, [[ , ]], ρ) is called aLeibniz algebroid overM (see[4]). Note that(A, [[ , ]], ρ) is a Liealgebroid if and only if the Leibniz bracket[[ , ]] is skew-symmetric. Moreover, if(g, , ) is a left Leibnizalgebra, then(g, , , ρ≡ 0) is a Leibniz algebroid over a point.

Let (A, [[ , ]], ρ) be a Leibniz algebroid over the manifoldM and letE be a vector bundle overM .Suppose that(ϕ1 : Γ(A)×Γ(E) → Γ(E), ϕ2 : Γ(E)×Γ(A) → Γ(E)) is a representation of the left Leibniz

862

Leibniz algebroids and Nambu–Jacobi structures

algebra(Γ(A), [[ , ]]) overΓ(E). Then, the pair(ϕ1, ϕ2) is said to be aLeibniz representation ofA overE iffor all s ∈ Γ(A), r ∈ Γ(E) andf ∈C∞(M,R), ϕ1(s, fr) = fϕ1(s, r) + ρ(s)(f)r, ϕ2(r, fs) = fϕ2(r, s).

If (A, [[ , ]], ρ) is a Lie algebroid overM and ϕ : Γ(A) × Γ(E) → Γ(E) is a representation ofAover E in the sense of [9] then the pair(ϕ, ϕ) defines a Leibniz representation ofA over E, whereϕ : Γ(E)× Γ(A) → Γ(E) is the map given byϕ(r, s) = −ϕ(s, r), for s ∈ Γ(A) andr ∈ Γ(E).

The following definition is an extension to Leibniz algebroids of the definition of matched pairs of Liealgebras (see[5,8]) and of matched pairs of Lie algebroids (see[11]).

DEFINITION 1. – Two Leibniz algebroids(A1, [[ , ]]1, ρ1) and(A2, [[ , ]]2, ρ2) overM form a matchedpair of Leibniz algebroidsif the Whitney sumA = A1 ⊕ A2 of vector bundles has a Leibniz algebroidstructure([[ , ]], ρ), such thatA1 andA2 are Leibniz subalgebroids ofA.

Under the identificationΓ(A1 ⊕ A2) ∼= Γ(A1) ⊕ Γ(A2), Definition 1 implies thatρ(s1, s2) = ρ1(s1) +ρ2(s2) and that[[si, s

′i]] = [[si, s

′i]]i, for all si, s

′i ∈ Γ(Ai), i ∈ 1,2. Moreover, if Πi : Γ(A1 ⊕ A2) ∼=

Γ(A1) ⊕ Γ(A2) → Γ(Ai) is the corresponding projection, then, from (1) and the Leibniz identity, wededuce that the mappingsϕj

ik : Γ(Ai)⊕Γ(Ak)→ Γ(Aj), with i, k, j ∈ 1,2, i = k, given byϕjik(si, sk) =

Πj([[si, sk]]), define a Leibniz representation(ϕ121, ϕ

112) of A2 over A1 and a Leibniz representation

(ϕ212, ϕ

221) of A1 over A2. In addition, using the fact thatρ : (Γ(A), [[ , ]]) → (X(M), [ , ]) is a Leibniz

algebra homomorphism, we obtain that:

[L1] ρi(ϕiik(si, sk)) + ρk(ϕk

ik(si, sk)) = [ρi(si), ρk(sk)], for all i, k ∈ 1,2, i = k.

On the other hand, the equality[[s, [[s′, s′′]]]] − [[[[s, s′]], s′′]] − [[s′, [[s, s′′]]]] = 0, for all s, s′, s′′ ∈ Γ(A),implies that:

[L2] [[si, ϕiik(s′i, sk)]]i − [[s′i, ϕ

iik(si, sk)]]i = ϕi

ik([[si, s′i]]i, sk)−ϕi

ik(si, ϕkik(s′i, sk))+ϕi

ik(s′i, ϕkik(si, sk)),

[L3] [[si, ϕiki(sk, s′i)]]i − [[ϕi

ik(si, sk), s′i]]i = ϕiki(sk, [[si, s

′i]]i)−ϕi

ik(si, ϕkki(sk, s′i))+ϕi

ki(ϕkik(si, sk), s′i),

[L4] [[ϕiki(sk, si), s′i]]i +[[si, ϕ

iki(sk, s′i)]]i = ϕi

ki(sk, [[si, s′i]]i)−ϕi

ik(si, ϕkki(sk, s′i))−ϕi

ki(ϕkki(sk, si), s′i),

for i, k ∈ 1,2, i = k. Note that,

[[(s1, s2), (s′1, s′2)]] =

([[s1, s

′1]]1 + ϕ1

12(s1, s′2) + ϕ1

21(s2, s′1), [[s2, s

′2]]2 + ϕ2

12(s1, s′2) + ϕ2

21(s2, s′1)

).

In fact, a direct computation proves the following characterization:

THEOREM 2. – Two Leibniz algebroids(Ai, [[ , ]]i, ρi), i ∈ 1,2, form a matched pair of Leibnizalgebroids if and only if, there exist two Leibniz representations,(ϕ2

12, ϕ221) and (ϕ1

21, ϕ112) of A1 and

A2 on each other, such that the relations[L1], [L2], [L3] and [L4] hold.

2. The matched pair of Leibniz algebroids of a Nambu–Jacobi manifold

Let M be a differentiable manifold of dimensionm. A Nambu–Jacobi bracket of ordern (n m)on M is an n-linear mapping , . . . , : C∞(M,R) × · · ·(n . . . × C∞(M,R) → C∞(M,R) satisfyingthe following properties: (i) , . . . , is skew-symmetric; (ii) , . . . , acts as afirst-order differen-tial operator on each of its arguments: f1g1, f2, . . . , fn = f1g1, f2, . . . , fn + g1f1, f2, . . . , fn −f1g11, f2, . . . , fn; (iii) , . . . , satisfies thefundamental identity: f1, . . . , fn−1,g1, . . . , gn =∑n

i=1g1, . . . , gi−1,f1, . . . , fn−1, gi, gi+1, . . . , gn.Conditions (i) and (ii) imply that , . . . , defines a section of the vector bundle

∧n(TM) ⊕∧n−1(TM) → M , that is, ann-vectorΛ and an(n − 1)-vector on M characterized by the relations(df1, . . . ,dfn−1) = 1, f1, . . . , fn−1, Λ(df1, . . . ,dfn) = f1, . . . , fn+

∑ni=1(−1)i−1fi1, f1, . . . , fi,

. . . , fn, for all f1, . . . , fn ∈ C∞(M,R). The triple(M,Λ,) is called aNambu–Jacobi manifold[10] (seealso [3]). Note that if = 0, then(M,Λ) is aNambu–Poisson manifold of ordern (see[12]) and if n = 2,the pair(Λ,) is aJacobi structureon M [6] (seealso [1]). Moreover, if(M,Λ,) is a Nambu–Jacobimanifold of ordern, n > 2, thenΛ and are Nambu–Poisson structures onM of ordersn andn − 1. Inaddition, ifn = 3, is a Poisson structure onM of rank 2 (see[3,10]).

863

R. Ibañez et al.

Example1. – If M is an orientable manifold of dimensionn and if we choose a volume formν onM ,we have a Nambu–Poisson structureΛν of ordern on M given byΛν(df1, . . . ,dfn)ν = df1 ∧ . . .∧ dfn,for all f1, . . . , fn ∈ C∞(M,R). Now, assume thatθ is a closed1-form onM . Then,(M,Λν , i(θ)Λν) is aNambu–Jacobi manifold of ordern (see[3,10]).

The following theorem describes the local structure of a Nambu–Jacobi bracket of ordern, n > 2, arounda regular point.

THEOREM 3 ([10]). – Let (M,Λ,) be anm-dimensional Nambu–Jacobi manifold of ordern, n > 2.If x is a point ofM satisfyingΛ(x) = 0 and(x) = 0 then there exist local coordinates(x1, . . . , xn, xn+1,. . . , xm) aroundx such thatΛ = ∂

∂x1 ∧ . . .∧ ∂∂xn and = ∂

∂x1 ∧ . . .∧ ∂∂xn−1 .

Now, let (M,Λ) be a Nambu–Poisson manifold of ordern. Denote by#Λ :∧n−1(T∗M) → TM the

homomorphism of vector bundles defined by#Λ(α) = i(α)Λ. We also will denote by#Λ : Ωn−1(M) →X(M) the corresponding homomorphism ofC∞(M,R)-modules. If f1, . . . , fn−1 ∈ C∞(M,R), theHamiltonian vector fieldXΛ

f1···fn−1of (M,Λ) associated with the functionsf1, . . . , fn−1 is given by

XΛf1···fn−1

= #Λ(df1 ∧ · · · ∧ dfn−1). The fundamental identity implies thatLXΛf1 ···fn−1

Λ = 0, L being

the Lie derivative operator. Using this fact, we proved in [4] that, ifn > 2,

L#Λ(α)Λ = (−1)n(i(dα)Λ

)Λ, for all α ∈Ωn−1(M). (2)

On the other hand, ifn > 2, the triple(∧n−1(T∗M), [[ , ]]Λ,#Λ

)is a Leibniz algebroid overM , where

[[ , ]]Λ : Ωn−1(M)×Ωn−1(M)→Ωn−1(M) is the bracket defined by:

[[α,α′]]Λ = L#Λ(α)α′ + (−1)n

(i(dα)Λ

)α′, (3)

for all α, α′ ∈ Ωn−1(M) (see[4]). Moreover, the following equality holds (seerelation (3.10) in [4])

i(d([[α,α′]]Λ

))Λ = #Λ(α)

(i(dα′)Λ

)−#Λ(α′)

(i(dα)Λ

). (4)

Note that ifΛ is a Poisson structure onM of rank 2 then the triple(T∗M, [[ , ]]Λ,#Λ) is also a Leibnizalgebroid and relations (2) and (4) hold.

Next, suppose that(M,Λ,) is a Nambu–Jacobi manifold of ordern, n > 2. SinceΛ and are Nambu–Poisson structures, we can introduce theHamiltonian vector fieldXf1···fn−1 on (M,Λ,) associated withf1, . . . , fn−1 ∈C∞(M,R) as follows:

Xf1···fn−1 = XΛf1···fn−1

+n−1∑i=1

(−1)i−1fiXf1...fi...fn−1

. (5)

The Hamiltonian vector fields generate a generalized foliationF on M whose characteristic space at apointx ∈M is TxF = #Λ

(∧n−1(T∗xM)

)+ #

(∧n−2(T∗xM)

)[3]. In addition, [10],

LXf1 ···fn−2

Λ = 0, LXΛf1 ···fn−1

= (−1)n−1i(d((df1, . . . ,dfn−1)

))Λ. (6)

Now, from Theorem 3, it follows that:

#(β) ∧Λ = 0, #Λ(α) ∧ = (−1)n−1(i(α)

)Λ, (7)

for all α ∈Ωn−1(M) andβ ∈ Ωn−2(M). Thus, using (6) and (7), we deduce that:

L#Λ(α) = (−1)n−1(i(d(i(α)

))Λ−

(i(dα)Λ

)

)and L#(β)Λ = (−1)n−1

(i(dβ)

)Λ. (8)

On the other hand, from (8) and since[#Λ(α),#(β)

]= i(β)(L#Λ(α)) + #(L#Λ(α)β) and[

#(β), #Λ(α)]= i(α)(L#(β)Λ) + #Λ(L#(β)α), we deduce that[

#Λ(α),#(β)]= #Λ

(ϕ1

12(α,β))+ #

(ϕ2

12(α,β)),

[#(β),#Λ(α)

]= #Λ

(ϕ1

21(β,α)), (9)

864

Leibniz algebroids and Nambu–Jacobi structures

whereϕ112 : Ωn−1(M) × Ωn−2(M) → Ωn−1(M), ϕ1

21 : Ωn−2(M) × Ωn−1(M) → Ωn−1(M) andϕ212 :

Ωn−1(M)×Ωn−2(M)→ Ωn−2(M) are the maps given by:

ϕ121(β,α) = L#(β)α + (−1)n−1

(i(dβ)

)α, ϕ1

12(α,β) = (−1)n−1d(i(α)

)∧ β,

ϕ212(α,β) = L#Λ(α)β + (−1)n

(i(dα)Λ

)β.

(10)

PROPOSITION 4. – (i) The pair (ϕ121, ϕ

112) defines a representation of the Leibniz algebroid

(∧n−2

×(T∗M), [[ , ]],#) over the vector bundle

∧n−1(T∗M)→ M .

(ii) If ϕ221 : Ωn−2(M)× Ωn−1(M) → Ωn−2(M) is the zero map,(ϕ2

12, ϕ221) defines a representation of

the Leibniz algebroid(∧n−1(T∗M), [[ , ]]Λ,#Λ

)over the vector bundle

∧n−2(T∗M) →M .

Proof. –(i) From (3), (4) and (10), we obtain thatϕ121([[β,β′]], α) = ϕ1

21(β,ϕ121(β′, α)) − ϕ1

21(β′,ϕ1

21(β,α)) andϕ112(α, [[β,β′]]) = ϕ1

12(ϕ112(α,β), β′)+ϕ1

21(β′, ϕ1

12(α,β′)). On the other hand, using (10)and the Leibniz identity, we have thatϕ1

12(ϕ112(α,β), β′) + ϕ1

12(ϕ121(β,α), β′) = 0.

(ii) It follows from (3), (4) and (10). Next, we will prove the main result of this section.

THEOREM 5. –Let (M,Λ,) be a Nambu–Jacobi manifold of ordern, n > 2. Then, the Leibnizalgebroids

(∧n−1(T∗M), [[ , ]]Λ,#Λ

)and

(∧n−2(T∗M), [[ , ]],#)

associated with the Nambu–Poisson structuresΛ and , respectively, form a matched pair of Leibniz algebroids and if([[ , ]],#) isthe Leibniz algebroid structure on

∧n−1(T∗M)⊕∧n−2(T∗M)→M , we have

#(α,β) = #Λ(α) + #(β),

[[(α,β), (α′, β′)]] =([[α,α′]]Λ + (−1)n−1d

(i(α)

)∧ β′ + (−1)n−1

(i(dβ)

)α′

+L#(β)α′, [[β,β′]] + L#Λ(α)β

′ + (−1)n(i(dα)Λ

)β′).

(11)

Proof. –Using (9), we deduce that the Leibniz representations(ϕ121, ϕ

112) and (ϕ2

12, ϕ221) satisfy the

property [L1]. Now, from (3), (8) and (10), we obtain that:

ϕ112

([[α,α′]]Λ, β

)= (−1)n−1d

(#Λ(α)

(i(α′)

)− i(L#Λ(α)α

′))∧ β − d

((i(dα)Λ

)(i(α′)

))∧ β,

which implies that (see(3) and (10)),

[[α,ϕ112(α

′, β)]]Λ − [[α′, ϕ112(α,β)]]Λ = ϕ1

12

([[α,α′]]Λ, β

)− ϕ1

12

(α,ϕ2

12(α′, β)

)+ ϕ1

12

(α′, ϕ2

12(α,β)).

Therefore, sinceϕ221 is the zero map, it follows that the relation [L2] holds. In a similar way, using

again (3), (8) and (10), we prove that relations [L3] and [L4] also hold. Consequently, from Theorem 2 andProposition 4, we deduce the result.

Remark1. – If sf1...fn−1 is the pair(df1∧ · · · ∧dfn−1,

∑n−1i=1 (−1)i−1fidf1 ∧ · · ·∧ dfi ∧ · · · ∧dfn−1

),

then#(sf1···fn−1) = Xf1···fn−1 and[[sf1···fn−1 , sg1···gn−1 ]] =∑n−1

i=1 sg1···gi−1f1,...,fn−1,gigi+1···gn−1 .

3. The modular class of a Nambu–Jacobi manifold

Let (M,Λ,) be an orientable Nambu–Jacobi manifold of ordern, n > 2, and(∧n−1(T∗M) ⊕∧n−2(T∗M), [[ , ]],#

)be the associated Leibniz algebroid. Suppose thats is a section of the dual bundle

to∧n−1(T∗M) ⊕

∧n−2(T∗M) → M , that is, s is a pair (P,Q), whereP is an (n − 1)-vector andQ is an (n − 2)-vector onM or, equivalently,s : C∞(M,R) × · · ·(n−1 · · · × C∞(M,R) → C∞(M,R)is a skew-symmetric map which acts as a first-order differential operator on each of its arguments. Thesections is a1-cocycle (with respect to

(∧n−1(T∗M) ⊕∧n−2(T∗M), [[ , ]],#)

)if s[[(α,β), (α′, β′)]] =

865

R. Ibañez et al.

#(α,β)(s(α′, β′)) − #(α′, β′)(s(α,β)) and it is a1-coboundary if there existsf ∈ C∞(M,R) such thats(α,β) = #(α,β)(f), for all (α,β), (α′, β′) ∈ Ωn−1(M) ⊕ Ωn−2(M) (see[4]). Note that from (2), (3)and (8), we deduce that the pair(,0) is a1-cocycle.

Now, let ν be a volume form onM and consider the skew-symmetric mapM(Λ,)ν : C∞(M,R) ×

· · ·(n−1 · · · × C∞(M,R) → C∞(M,R) characterized byLXf1 ···fn−1ν = M

(Λ,)ν (f1, . . . , fn−1)ν, for all

f1, . . . , fn−1 ∈ C∞(M,R). In the particular case whenM is a Nambu–Poisson manifold ( = 0), M(Λ,)ν

defines an(n − 1)-vector onM which is just themodular tensorMΛν of M with respect toν (see[2,

4]). MΛν is a1-cocycle with respect to the Leibniz algebroid

(∧n−1(T∗M), [[ , ]]Λ,#Λ

)and the resultant

cohomology class is themodular classof M (see[4]).WhenM is not Nambu–Poisson manifold,M

(Λ,)ν acts as a first-order differential operator on each of its

arguments. In fact, from (5), we obtain thatM(Λ,)ν defines the section(MΛ

ν ,Mν ) + (−1)n(n− 1)(,0)

of the vector bundle∧n−1(TM) ⊕

∧n−2(TM) → M . Moreover, from relation (4.7) in [4], we havethat L#(α,β)ν = [(MΛ

ν ,Mν )(α,β) + (−1)n−1(Λ,)(d(α,β))]ν, where d is the differential operator

given byd(α,β) = (dα,−dβ). On the other hand, (8) and (11) imply that(Λ,)(d([[(α,β)(α′, β′)]])) =#(α,β)((Λ,)(d(α′, β′)))−#(α′, β′)((Λ,)(d(α,β))). Consequently, using the above results and since(,0) is a cocycle, we conclude that

THEOREM 6. –Let (M,Λ,) be an orientable Nambu–Jacobi manifold of ordern, n > 2, andν be

a volume form. Then,M(Λ,)ν defines a1-cocycle in the cohomology complex associated with the Leibniz

algebroid(∧n−1(T∗M) ⊕

∧n−2(T∗M), [[ , ]],#). Moreover, the resultant cohomology class(which is

called the modular class of(M,Λ,)) does not depend on the chosen volume form.

Example2. – LetM be an orientable manifold,ν be a volume form onM andθ be a closed1-formon M . Then,MΛν

ν = 0 (seeExample 1 and [4]). Thus, it is easy to prove that the modular class of theNambu–Jacobi structure(Λν , i(θ)Λν) onM vanishes if and only ifθ is exact.

Acknowledgements. Research supported by DGICYT grant BFM2000-0808 and by 1/UPV/EHU00127-310-EA-7781/2000. BL is grateful for the financial support of the Consejería de Educación del Gobierno de Canarias.

References

[1] Dazord P., Lichnerowicz A., Marle Ch.-M., Structure locale des variétés de Jacobi, J. Math. Pures et Appl. 70(1991) 101–152.

[2] Dufour J.-P., Singularities of Poisson and Nambu structures, Banach Center Publ. 51, Inst. Math., Polish Acad.Sci. Warszawa (2000) 61–68.

[3] Ibáñez R., de León M., Marrero J.C., Padrón E., Nambu–Jacobi and generalized Jacobi manifolds, J. Phys. A:Math. Gen. 31 (1998) 1267–1286.

[4] Ibáñez R., de León M., Marrero J.C., Padrón E., Leibniz algebroid associated with a Nambu–Poisson structure,J. Phys. A: Math. Gen. 32 (1999) 8129–8144.

[5] Kosmann-Schwarzbach Y., Magri F., Poisson Lie groups and complete integrability. I. Drinfeld bialgebras, dualextensions and their canonical representations, Ann. Inst. Henri-Poincaré 49 (1988) 433–460.

[6] Lichnerowicz A., Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures et Appl. 57 (1978)453–488.

[7] Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, L’Enseign. Math. 39(1993) 269–293.

[8] Majid S., Matched pairs of Lie groups associated to solutions of the Yang–Baxter equations, PacificJ. Math. 141 (2) (1990) 311–332.

[9] Mackenzie K., Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge Univ. Press, 1987.[10] Marmo G., Vilasi G., Vinogradov A.M., The local structure ofn-Poisson andn-Jacobi manifolds, J. Geom.

Phys. 25 (1998) 141–182.[11] Mokri T., Matched pairs of Lie algebroids, Glasgow Math. J. 39 (1997) 167–181.[12] Takhtajan L., On foundations of the generalized Nambu mechanics, Commun. Math. Phys. 160 (1994) 295–315.

866