LIU Ling,SU Nong

(School of Applied Science,Beijing Information Science and Technology University,Beijing 100192, China)

On Quasi-Jacobi Bialgebroid and Its Dirac-Jacobi Structure

LIU Ling,SU Nong

(School of Applied Science,Beijing Information Science and Technology University,Beijing 100192, China)

Notions of quasi-Jacobibialgebroid and its Dirac-Jacobistructure are introduced. The necessary and suffi cient conditions for a maximal isotropic subbundle L to be a Dirac-Jacobi structure are proved.Meanwhile several special examples are presented.

quasi-Jacobi bialgebroid;Jacobi-quasi bialgebroid;Dirac-Jacobi structure;triangular Jacobi bialgebroid

§1. Introduction

A Dirac structure on a diff erentiable manifold M is a subbundle L?T M⊕T?M that is maximal isotropic with respect to the symmetric bilinear form on T M ⊕T?M and satisfies an integrability condition.Then this notion was extended to subbundles L?E,where E is a vector bundle over M[7].Moreover,Liu characterized a Dirac structure in terms of its characteristic pair(D,π)[8].

Dirac structures were also used to study a simple interpretation of Jacobi manifolds and Jacobi reduction,such as E∞(M)-Dirac structure associated to a generalized Lie bialgebroid (T M×?,T?M×?)[23];the notions of generalized Courant algebroid and Dirac-Jacobi structure in[12],where several connections between Dirac structures for generalized Courant algebroids and Jacobi manifolds were also given.In[9],the authors gave two kinds of reductionrelating Jacobimanifolds through Dirac theories,without assuming the existence of momentum mappings or introducing admissible functions.

The notions of quasi-Jacobi and Jacobi-quasi bialgebroid are generalizations of quasi-Lie bialgebroid and Lie-quasi bialgebroid which were introduced in[6,14-16].In this paper,we study Dirac structures for quasi-Jacobiand Jacobi-quasibialgebroid and related properties.In Section 2,we review the notion of Dirac structure on vector space or Jacobi bialgebroid and some properties.In Section 3,we get the conditions for a subbundle L of the double of a quasi-Jacobi bialgebroid to be a Dirac structure and present some examples.

§2. Basic Defi nitions and Results

Let V be a vector space and V?its dual space.There exists a natural nondegenerate symmetric bilinear form(·,·)+on V⊕V?defined by

A subbundle L?V⊕V?which is maximalisotropic with respect to(·,·)+is called a Dirac subspace of V⊕V?.Set D=L∩(V+0)?L∩V.It is easy to prove that there exists a unique linear map

?πcan be seen as a bivector on V/D.Let p:V→V/D be the projection andπa bivector on V such that p(π)=?π,then

whereπ#:V?→V is the bundle map associate toπ.Therefore L=D⊕graph(π#|D⊥).(D,π) is called a characteristic pair[8]of L.

Let’s consider Dirac structures for Jacobi bialgebroids.It is well known that with each Lie algebroid(A,[·,·],a)a differential d on the graded space of sections of∧A?is associated, where A?is the dual vector bundle of A.Also the Lie bracket onΓ(A)can be extended to the algebra of sections of∧A,Γ(∧A)=⊕k∈ZΓ(∧kA).Let(A,[·,·],a)be a Lie algebroid over M andφ∈Γ(A?)a 1-cocycle for the Lie algebroid cohomology complex with trivial coeffi cients, i.e.,for all X,Y∈Γ(A),

Using the 1-cocycleφ,one can define a new representation aφof the Lie algebra(Γ(A),[·,·]) on C∞(M,?),by setting

Therefore,we obtain a new cohomology complex,whose diff erential cohomology operator is given by

Also,for any X∈Γ(A),the new Lie derivative operator with respect to X is given by

where LXis the usual Lie derivative LX=d?iX+iX?d.Usingφ,it is also possible to modify the Schouten bracket[·,·]onΓ(∧A)to theφ-Schouten bracket onΓ(∧A)defined,for any P∈Γ(ΛpA),Q∈Γ(ΛqA),by

where iφQ and iφP can be interpreted as the usual contraction of a multivector field with a 1-form.It is easy to see that when p=q=1,[P,Q]φ=[P,Q].

Defi nition 2.1[1]A Jacobialgebroid on a vector bundle A→M is a pair(A,φ),where (A,[·,·],a)is a Lie algebroid over M andφ∈Γ(A?)is a 1-cocycle.

Let((A,[·,·],a),φ)and((A?,[·,·]?,a?),W)be Jacobi algebroids over M,in duality, with differentials d and d?,respectively.Then

Defi nition 2.2[4]The dual pair((A,φ),(A?,W))is said to be a Jacobi bialgebroid or a generalized Lie bialgebroid over M,if

Example 2.1 Let(M,Λ,E)be a Jacobi manifold.Then((T M×?,(0,1)),(T?M×?,(?E,0)))is a Jacobibialgebroid over M.

On the Whitney sum bundle A⊕A?,for any e1=X1+α1,e2=X2+α2∈Γ(A⊕A?),we consider two nondegenerate canonical bilinear forms(·,·)±

On the spaceΓ(A⊕A?)～=Γ(A)⊕Γ(A?)we have bracket[·,·]by setting

Letρ:A⊕A?→T M be the bundle map given byρ=a+a?,i.e.,

Defi nition 2.3 A Courant-Jacobi algebroid or a generalized Courant algebroid over a diff erential manifold M is a vector bundle E over M equipped with

(i)a nondegenerate symmetric bilinear form(·,·)+on the bundle;

(ii)a skew-symmetric bracket[·,·]onΓ(E);

(iii)a bundle mapρθ:E→T M×?and(iv)an E-1-formθsuch that,for any e1,e2,e3∈Γ(E),the condition〈θ,[e1,e2]〉=ρ(e1)〈θ,e2〉?ρ(e2)〈θ,e1〉holds,ρbeing the bundle map from E onto T M induced byρθ,satisfying the following relations

1)[[e1,e2],e3]+c.p.=DθT(e1,e2,e3),where T(e1,e2,e3)is the function on the base M defined byis the first-order differential operator defined by

2)ρθ([e1,e2])=[ρθ(e1),ρθ(e2)],where the bracket on the right-hand side is the Lie bracket (2.7)onΓ(T M×?);

3)ρθ(e)(e1,e2)+=([e,e1]+Dθ(e,e1),e2)++(e1,[e,e2]+Dθ(e,e2))+;

4)For any f,g∈C∞(M,?),(Dθf(wàn),Dθg)+=0.

Whenθ=0,we recover the definition of Courant algebroid,introduced in[8]or the equivalent version presented in[5].

Theorem 2.1[12]Let((A,φ),(A?,W))be a Jacobi bialgebroid over M,then the pair (E=A⊕A?,θ),withθ=φ+W∈Γ(E)is a Courant-Jacobialgebroid over M,equipped with ([·,·],(·,·)+,ρ),where the Lie bracket[·,·]onΓ(A⊕A?)is given by(2.8),the symmetric bilinear form(·,·)+is given by(2.7),the vector bundle mapρis given by(2.9)and the operator Dθis Dθ=(dφ+dW?)|C∞(M,?).

Defi nition 2.2 A Dirac-Jacobi structure for a Courant-Jacobi algebroid(E,θ)over M is a subbundle L?E which is maximal isotropic under(·,·)+and integrable,i.e.,Γ(E)is closed under the bracket[·,·].

Theorem 2.3[12]Let((A,φ),(A?,W))be a Jacobi bialgebroid and L?A⊕A?a maximal isotropic subbundle defined by a characteristic pair(D,π).Then L is a Dirac-Jacobi structure for((A,φ),(A?,W))if and only if

(1)Γ(D)is closed under the Lie algebroid bracket[·,·]of A;

(3)Γ(D⊥)is closed under the sum bracket[·,·]?+[·,·]π,where[,]πis the bracket defi ned, for anyα,β∈Γ(A?),by

In particular,when((A,φ),(A?,W))is a triangular Jacobibialgebroid,the above theorem takes the following form:

Corollary 2.1 Let((A,φ),(A?,W),P)be a triangular Jacobi bialgebroid and L?A⊕A?a maximal isotropic subbundle defined by a characteristic pair(D,π).Then L is a Dirac-Jacobi structure if and only if

(1)Γ(D)is closed under the Lie algebroid bracket[·,·]of A;

(2)[P+π,P+π]φ=0(mod D);

Example 2.2 In particular,when L=D⊕D⊥,the two conditions for L to be integrable are equivalent and both as follows:

(1)Γ(D)is closed under the Lie algebroid bracket[·,·]of A;

(2)Γ(D⊥)is closed under the Lie bracket[·,·]?of A?.

Example 2.3 Let(A,φ)be a Jacobi algebroid and(A?,0)a trivial Jacobi algebroid. Then((A,φ),(A?,0))is a Jacobibialgebroid.Suppose that L?A⊕A?is a maximalisotropic subbundle with a characteristic pair(D,π).Then L is integrable if and only if

(1)Γ(D)is closed under[·,·];

(2)[π,π]φ=0(mod D);

(3)Γ(D⊥)is closed under[·,·]π.

§3. Generalization from Jacobi Bialgebroids to Quasi-Jacobi Bialgebroids

We know that a Jacobi algebroid structure on a vector bundle A→ M consists of a Lie algebroid structure([·,·],a)and a 1-cocycleφof A.While if[·,·]is not a Lie bracket, a:A→T M is not necessarily a homomorphism andφ∈Γ(A?)is not necessarily a cocycle, but allofthese conditions are quasisatisfied by a certain 3-form?∈Γ(∧3A?)which isφ-closed, we say that there is a quasi Jacobi structure on A.

Let(A,A?)be a pair of dual vector bundles over differential manifold M endowed with a 1-form φand W,respectively.The quasi Jacobi algebroid structure on(A,φ)consists of a bundle map a:A→T M,a skew-symmetric operation[·,·]onΓ(A)and a 3-form?of A satisfying,for X,Y,Z∈Γ(A)and f∈C∞(M,?),the following conditions:

Defi nition 3.1[11]A quasi-Jacobi bialgebroid structure on(A,A?)consists of structure ([·,·],a,φ,?)on A and a Jacobi algebroid structure([·,·]?,a?,W)on A?satisfying,for X,Y,Z∈Γ(A)and f∈C∞(M,?),the following conditions

2)[X,f Y]=f[X,Y]+(a(X)f)Y;

3)a([X,Y])=[a(X),a(Y)]?a?(?(X,Y,·)),where a?:A?→T M is a bundle map;

4)dφ=?(W,·,·),where d is the quasi-differential operator onΓ(∧A?)determined by the structure([·,·],a)on A;

5)dφ?=d?+φ∧?=0,where the quasi-differentialoperator d onΓ(∧A?)is extended to the graded spaceΓ(∧A?).

We denote the quasi-Jacobi bialgebroid by((A,φ),(A?,W),?).

By interchanging the roles of(A,φ)and(A?,W)in the above definition,we obtain the notion of Jacobi-quasi bialgebroid over a differential manifold M.

Proposition 3.1[11]If((A,φ),(A?,W),?)is a quasi-Jacobi bialgebroid over a differential manifold M,then((A?,W),(A,φ),?)is a Jacobi-quasi bialgebroid over M and conversely.

In the case when bothφ=0 and W=0,we get the notion ofquasi-Lie bialgebroid.On the other hand,if?=0,then((A,φ),(A?,W),0)=((A,φ),(A?,W))is a Jacobi bialgebroid over M.

Let((A,φ),(A?,W),?)be a quasi-Jacobibialgebroid over M.Lφand LW?the quasi-Lie and Lie derivative operators defined,respectively,by dφandOn the Whitney sum bundle A⊕A?,for any e1=X1+α1,e2=X2+α2∈Γ(A⊕A?),we consider the two nondegenerate canonical bilinear forms(·,·)±defined by(2.7)and on the spaceΓ(A⊕A?)～=Γ(A)⊕Γ(A?) the bracket[·,·]?by setting

where[·,·]is the bracket(2.8).

Theorem 3.1[13]Let((A,φ),(A?,W),?)be a quasi-Jacobibialgebroid over M.Then the vector bundle A⊕A?over M endowed with([·,·]?,(·,·)+,ρθ,Dθ),whereθ=φ+W ∈andis a Courant-Jacobibialgebroid over M.

Defi niton 3.2 A Dirac structure for a quasi-Jacobi bialgebroid((A,φ),(A?,W),?) is a subbundle L of A⊕A?which is closed under the bracket[·,·]?and is maximal isotropic with respect to the symmetric bilinear form(·,·)+.

If L is a Dirac structure,then(L,ρθ|L,[·,·]?|L)is a Lie algebroid over M.

Similar to those of Jacobi bialgebroid,any Dirac structure L of a quasi-Jacobi bialgebroid also has characteristic pair(D,π)given by

On spacesΓ(A)andΓ(A?)we define brackets[·,·]′andrespectively,by

and

where[·,·]πbeing the Koszulbracket defined by(2.10).

Theorem 3.2 Let((A,φ),(A?,W),?)be a quasi-Jacobi bialgebroid and L?A⊕A?a subbundle defined by a characteristic pair(D,π).Then L is a Dirac structure if and only if

i)Γ(D)is closed under the bracket[·,·]′;

iii)Γ(D⊥)is closed under the sum bracket

iv)?≡0(mod D⊥).

Proof It is easy to check that L is a maximalisotropic subbundle of A⊕A?with respect to the symmetric bilinear form(·,·)+.So we only have to verify that the closedness of L under the bracket[·,·]?is equivalent to conditions i)-iv).

For any sections X+π#α+α,Y+π#β+βof L=D⊕graph(π#|D⊥),we have

Γ(L)is integrable if and only if terms(I)～(IV)are allinΓ(L).For term(I),

is inΓ(L)if and only if?(X,Y,·)∈Γ(D⊥)and[X,Y]?π#(?(X,Y,·))=[X,Y]′∈Γ(D). So term(I)is inΓ(L)if and only ifΓ(D)is closed under the bracket[·,·]′(condition i))and ?≡0(mod D⊥)(condition iv)).

We compute term(II)as following

For any Z∈Γ(D),

thus,LφXβ+?(X,π#β,·)∈Γ(D⊥)ifand only ifΓ(D)is closed under the bracket[·,·]′(condition i)).Consequently,from(3.4)we deduce that

Forξ∈Γ(D⊥),we compute

By formulae

and

we know that(3.6)holds if and only ifπ=(∧3π#)(?)(mod D)(condition ii)).

Example 3.1

1)Null Dirac structures:The subbundle L={X+α|X∈D,α∈D⊥}=D⊕D⊥?A⊕A?is a Dirac structure if and only if

i)Γ(D)is closed under the bracket[·,·];

ii)Γ(D⊥)is closed under the sum bracket[·,·]?;

iii)?≡0(mod D⊥).

2)When?=0,we obtain conditions for L to be a Dirac structure for Jacobi bialgebroid in Theorem 2.7.

3)Ifφ=0 and W=0,we recover the conclusions obtained in[17]which were called Dirac structures of proto bialgebroids.

4)Dirac structure for triangular quasi-Jacobibialgebroid

Let(A,[·,·],a,φ)be a Jacobi algebroid over a diff erential manifold M,P a section ofΛ2A andψa trivector on A such that

From[11],we know that the triple((A,[·,·],a,φ),(A?,[·,·]P,a?,W),ψ)is a Jacobi-quasi bialgebroid over M,which is called a triangular Jacobi-quasi bialgebroid,where[·,·]Pis the Koszulbracket(2.10),a?=a?P#:A?→T M,P#:Γ(∧kA?)→Γ(∧kA)being the bundle map associate to P and W=?P#(φ).

Ifψ=P#(?),with?a dφ-closed 3-form on A and the spacesΓ(A?)andΓ(A)are equipped, respectively,with the bracketsand[·,·]′.Then the triplea?,W),?)is a triangular quasi-Jacobi bialgebroid over M.

Theorem 3.3 For a triangular quasi-JacobibialgebroidW),?),L=D⊕graph(π|D⊥)?A⊕A?is a Dirac structure if and only if

i)Γ(D)is closed under the bracket[·,·]+[·,·](P+π)#?,where[·,·](P+π)#?is the bracket (2.10);

iv)?≡0(mod D⊥).

Proof Using Theorem 3.6,conditions i)and iv)are easy to check.We prove ii)and iii).

In the case of triangular quasi-Jacobibialgebroid,we have dW?π=[P,π]φand[P+π,P+ π]φ=[P,P]φ+2[P,π]φ+[π,π]φ.By condition ii)in Theorem 3.6,in case of modulo D,we have

By condition iii)in Theorem 3.6,for anyα,γ∈Γ(D⊥),

As a result,for any X∈Γ(D),we have

In particular,if?=0,then we get Corollary 2.8.

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tion:53D17,58H05,17B62,17B66

1002–0462(2014)01–0097–10

Chin.Quart.J.of Math. 2014,29(1):97—106

date:2012-10-09

Supported by the Scientifi c Research Common Program of Beijing Municipal Commission of Education(SQKM201211232017);Supported by the Beijing Excellent Training Grant(2012D005007000005); Supported by the Funding Program for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality(11530500015)

Biography:LIU Ling(1976-),female,native of Jinzhai,Anhui,an associate professor of Beijing Information Science and Technology University,Ph.D.,engages in Poisson geometry.

CLC number:O186.1 Document code:A