REDUCTION OF JACOBI MANIFOLDS

A. Ibort*, M. de Leon**, G. Marmo***

* Depto. de Fısica Teorica, Univ. Complutense de Madrid, 28040 Madrid, Spain.

** Inst. de Matematicas y Fısica Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain.

*** Dipto. di Scienze Fisiche, Univ. di Napoli, Mostra d’Oltremare, Pad. 19, 80125 Napoli,Italy

Abstract

A reduction procedure for Jacobi manifolds is described in the algebraic setting of Jacobi alge-

bras. As applications, reduction by arbitrary submanifolds and the reduction of Jacobi manifolds

with symmetry are discussed. This generalized reduction procedure extends the well–known reduc-

tion procedures for symplectic, Poisson, contact and cosymplectic structures.

MS Classification: 58F05, 53C15,

Contents

1. Introduction.

2. Jacobi manifolds.

2.1. Some properties of Jacobi manifolds and Jacobi algebras.

2.2. Derivations of the Lie Algebra structure.

2.3. Particular cases.

2.4. An example: The Jacobi structure of SU(2).

3. Generalized Reduction of Jacobi manifolds.

3.1. The reduction mechanism.

3.2. Reduction of Jacobi manifolds.

3.3. Reduction by a submanifold.

3.4. Reduction by a distribution.

4. Examples and applications.

4.1. Reduction of Jacobi manifolds with symmetry.

4.2. Reduction of Poisson, symplectic, contact and cosymplectic manifolds.

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1 Introduction

Jacobi manifolds were introduced by A. Lichnerowicz as a rich geometrical notion extend-ing several important geometrical structures, including among others, symplectic, Poisson,contact and cosymplectic structures [Li78]. However it is true that conceptually Jacobimanifolds arise from the notion of local Lie algebras, i.e., Lie algebras on spaces of smoothsections of vector bundles with a locality property [Sh74], [Ki76], [Gu84].

In this paper we analyze a geometrical reduction procedure for Jacobi manifolds. Itis well–known that reduction is a natural geometrical process in the categories of Poisson,symplectic, contact and cosymplectic manifolds [Ma86], [Ma74], [EY93], [Al89] and also inthe category of Poisson supermanifolds [Ca90]. It is thus natural to ask if there is a naturalmechanism for the reduction of Jacobi manifolds. It so happens that the adequate wayto address such generalization is using an algebraic formulation as proposed in [Gr94] forPoisson manifolds and in [Ma96] for Nambu manifolds. The algebraic approach benefits fromthe natural duality between algebras of functions and sets. We will use such approach hereand show that there is a natural generalization of this algebraic reduction procedure to thecategory of Jacobi manifolds and Jacobi algebras [Gr92].

This mechanism allows us to construct new Jacobi manifolds out of simple ones very muchlike symplectic reduction offers a simple way to construct complicated (from the topologicalpoint of view) symplectic manifolds from “simple” symplectic manifolds. As a byproduct ofthese ideas we will be able to offer another interpretation of several theorems concerning theexistence of Jacobi structures on certain submanifolds and projections of Jacobi manifolds[Ju84], [Da91].

There is a striking similarity between Jacobi manifolds and certain algebraic structuresthat have arose recently in Quantum Field Theories called BV–algebras and some of itsgeneralizations [Wi90], [Ge94], [Sc95]. In a sense that will be discussed in forthcomingpapers, the Jacobi algebras are one instance of a classical limit of a generalized notion ofBV– algebras. This remark raises the problem of quantizing Jacobi manifolds as a questionof physical interest.

It is worth pointing out that the algebraic approach to reduction of geometric structuresfocuses the attention on the way to reproduce the initial geometrical structures on algebrasconstructed from the initial one using simple algebraic constructions such us quotienting outby ideals and restricting to suitable subalgebras. These constitute in fact the two elementarysteps in the reduction process. This reminds very much the procedure of “localization” inalgebraic geometry even though the outcome is very different.

The paper is organized as follows. We will review succintly some of the main featuresof Jacobi manifolds that will be used in the paper in §2, in particular we will review thecharacterization of Jacobi algebras and the role of derivations of the algebra of smoothfunctions. In section 3 we will describe the algebraic reduction theorem for associativecommutative algebras and for associative commutative algebras with a Lie algebra structure(but not necessarily Poisson algebras). Then, we will discuss the reduction of Jacobi algebras,i.e. algebras of functions of Jacobi manifolds with respect to an ideal, a submanifold and adistribution, and in Section 4 we will compare these results with the well–known reductionprocedures in symplectic, Poisson etc. manifolds.

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2 Jacobi manifolds and Jacobi algebras

2.1 Geometric and algebraic characterization of Jacobi manifolds

Definition 1 A Jacobi manifold [Li78] is a triple (M, Λ, X) where M is a smooth manifold,Λ is a bivector and X is a vector field defined on M , such that they satisfy the relations:

i.- [Λ, Λ] = 2X ∧ Λ,

ii.- [X, Λ] = 0.

In the previous definition [., .] denotes the Schouten bracket in the exterior algebra ofmultivectors on the manifold M [Sc54], [Tu74].

We will denote by F = F(M) the algebra of smooth functions on the manifold M . A Liealgebra structure [., .] on F will be said to be local if the linear operator Df on F defined by

Df (g) = [f, g] ∀g ∈ F , (1)

is local for all f ∈ F . It is an important result that Jacobi manifolds are singularized bythe following fact [Sh74], [Ki76], [Gu84].

Theorem 1 Any local Lie algebra structure [., .] on the algebra F of smooth functions of amanifold M is associated with a pair of tensors Λ, X satisfying the conditions i,ii in Def. 1,and such that the bracket of two functions f, g has the form

[f, g] = f, gΛ + fLXg − gLXf, (2)

where f, gΛ denotes the bracket defined by the bivector Λ given by

f, gΛ = Λ(df, dg),

and LXf denotes the Lie derivative of f along the vector field X.

The same result holds true for local Nambu structures [Mr96] replacing Λ by a N -multivector and X by a N − 1 multivector satisfying similar compatibility conditions toi, ii in Def. 1. It is worth pointing out that (F , [., .], ·) is not a Poisson algebra since

[f, gh] = g[f, h] + [f, g]h + gh(LXf), (3)

i.e., the bracket [., .] is not a derivation on each factor because of the last term in the r.h.s. ofequation (3). However, the bracket [., .] is not far from being a derivation because it definesa differential operator of order 1. Let us recall that a differential operator of order r on anassociative commutative algebra A is defined recursively as a linear map D on A such thatδ(x)D is a differential operator of order r−1 for all x ∈ A, where δ(x)D(y) = D(xy)−xD(y).Differential operators of order zero are defined as multiplication by elements on the algebra.It is not hard to see that a linear operator D on A is a differential operator of order r iffδ(x)r+1D = 0 for all x ∈ A. In this sense the linear operator Df on F defined by eq. (1) isa differential operator of order 1 for all f ∈ F . In fact, a simple computation using eq. (3)shows that δ(g)2Df = 0 for all g ∈ F . The skew–symmetry of the bracket [., .] guaranteesthat it is also a differential operator on the first argument too. Then, we will say that thebracket [., .] defines a skewsymmetric bidifferential operator on the associative commutativealgebra F . Consequently we will define abstractly algebras of smooth functions on Jacobimanifolds as follows [Gr92].

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Definition 2 A Jacobi algebra is an associative commutative algebra A with unit equippedwith a skew–symmetric bidifferential operator P which defines a Lie algebra structure on A.

It can be shown that if A contains no nontrivial nilpotent elements then the differentialoperator is of order ≤ 1 and then we have the following alternative formulation of Thm. 1,[Gr92].

Theorem 2 Let A be an associative commutative algebra with unit containing no nonzeronilpotent elements, then a skew–symmetric differential operator P defines a Jacobi structureon A iff there exists a bivector Λ and a vector field X verifying i and ii in Def. 1, such thatP (f, g) = [f, g] where [., .] is the bracket defined by eq. (2).

Even if Jacobi algebras are not Poisson algebras in general, we can define Hamiltonianvector fiels as follows:

Proposition 1 The map Φ:F(M) → X(M) defined by

f 7→ Φ(f) = Xf = Λ(df) + fX, ∀f ∈ F(M),

is a Lie algebra homomorphism, i.e.,

[Xf , Xg] = X[f,g].

The vector fields Xf will be called Hamiltonian vector fields and f will be called theHamiltonian of Xf .

We shall denote by FX the subalgebra of functions invariant under X, i.e., FX = f ∈F | LXf = 0 whose elements will be called basic functions on M . If the vector field X isfibrating, i.e., the space of integral curves N defines a regular foliation of M such that thecanonical projection π: M → N is a submersion, it is obvious that FX = π∗(F(N)) [Ju84].The triple (M, Λ, X) has been called then a regular Jacobi manifold and Λ projects onto aPoisson bivector Λ on N [Ch96].

2.2 Derivations of the Lie Algebra structure

A vector field Y on M is a derivation of the Lie Algebra structure on F defined by a Jacobistructure (Λ, X) if by definition,

LY [f, g] = [LY f, g] + [f,LY g], ∀f, g ∈ F . (4)

Notice that in addition Y is a derivation of the associative structure of F . This is thegeometrical translation of the notion of a derivation of a Jacobi algebra. Such derivationscan be called Jacobi derivations. It is an immediate computation to check that X is a Jacobiderivation. Then, it follows,

Proposition 2 The subalgebra FX is a Lie subalgebra of (F(M), [., .]).

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If Y is a derivation of the Lie algebra [., .], by using f = constant in the previous formulaeq. (4) we get

LY [f, g] = [f,LY g],

but using the definition eq. (2), [f, g] = fLXg, then

LY (LXg) = LX(LY g), ∀g ∈ F ,

then[X, Y ] = 0.

By using now, f, g ∈ FX , we findLY Λ = 0.

Thus we have proved, that Y is an infinitesimal automorphism of the Jacobi structure (Λ, X).In fact, the converse is also true as can be checked directly and we get,

Proposition 3 The vector field Y is a derivation for [., .] iff LY X = 0 and LY Λ = 0.

Hence, Props. 2 and 3 imply,

Corollary 1 The Lie algebra structure [., .] induces a Poisson structure on FX.

Thus, the formal quotient manifold whose algebra of functions is FX is a Poisson manifold.If X is fibrating then the quotient space N is actually a Poisson manifold as it was indicatedearlier. In this sense we can say that Jacobi manifolds are extensions of Poisson manifolds.These remarks provide a third way of characterizing Jacobi algebras. It is obvious that theconstant functions on M form an Abelian subalgebra of F . Moreover, it is clear that thecentralizer of the Abelian subalgebra of constant functions is FX . Thus a Jacobi algebracan be characterized as an unital algebra A such that the operator X = D1A

is a derivationand the centralizer of the Abelian subalgebra defined by multiples of the unit element 1A isa Poisson algebra and it coincides with the invariant subalgebra of X.

Hamiltonian vector fields are not Jacobi derivations in general, however we have,

Proposition 4 Hamiltonian vector fields are (inner) derivations iff LXh = 0, i.e., h ∈ FX.

Definition 3 A function C on M will be said to be a Casimir function if [C, f ] = 0 for allf ∈ F .

Notice that [C, f ] = 0 implies that

LXC = 0,

because by definition, eq. (2),

[C, f ] = Λ(dC, df) + CLXf − fLXC,

hence using f = constant, we obtain the conclussion above. Besides the arbitrariness of fimplies that

Λ(dC, .) = 0,

and then for any Hamiltonian vector field

LXfC = 0.

The converse follows the same argument, then we have,

Proposition 5 A function C is a Casimir iff LXfC = 0 for all f ∈ F .

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2.3 Particular cases

I. X = 0. Then (M, Λ) becomes a Poisson manifold. If Λ is nondegenerate we obtaina symplectic manifold with symplectic structure ω the inverse of the tensor Λ. The Liebracket [., .] becomes the ordinary Poisson bracket defined by the symplectic form ω.

II. X = 0. Let (M, Ω, η) be an almost cosymplectic manifold, say, Ω is a closed 2–form andη is a closed 1–form on M such that Ωn ∧ η 6= 0, with dimM = 2n + 1. There exists a Reebvector field ξ defined by the equations iξΩ = 0, iξη = 1, but M carries a Poisson structure(see [Ca92], [Le93], [Ch96]).

A Jacobi manifold of constant rank k is defined by the condition X∧Λk 6= 0 and Λk+1 = 0.Jacobi manifolds of maximal rank are such that X ∧ Λn 6= 0 with dimM = 2n + 1, thenthere exists a 1–form θ such that iθΛ = X and θ is a contact 1–form.

III. X 6= 0. Let M be an exact contact manifold with contact form θ. Then let X bethe Reeb field of (M, θ). If the quotient space N of M by the Reeb field is a manifold, itinherits a symplectic structure ω and θ defines a connection on the fibration M → N . Welift horizontally the inverse of the symplectic structure on N by using the connection θ. Theresulting structure is a Jacobi manifold. The bracket [., .] becomes the Lagrange bracket forcontact structures [Lb58] (see the example of §2.4).

IV. Λ = 0. In this case we get a de Witt algebra. For instance on M = S1, we find aVirasoro algebra as a central extension of it [Gr96].

2.4 An example: The Jacobi structure on SU(2)

We will consider the manifold SU(2). We realize SU(2) in terms of matrices g of the form

g =

(

α β−β α

)

; α, β ∈ C,

satisfying |α|2 + |β|2 = 1. We define the left invariant vector fields XL1 , XL

2 , XL3 to be given

by the equations,iXL

aΘL = iσa,

where σ1, σ2, σ3 are Pauli matrices and ΘL the left–invariant Maurer–Cartan form g−1dg.

We set Λ = XL1 ∧ XL

2 and X = XL3 . Clearly X defines the Hopf fibration of π: S3 → S2

and FX = π∗F(S2). It is trivial to check that

[Λ, Λ] = 2X ∧ Λ; [X, Λ] = 0,

using the commutation properties of the vector field XLa given by

[XLa , XL

b ] = ǫabcXLc .

Then the previous tensors Λ, X define a Jacobi structure on the manifold S3.

It is simply to show that the previous Jacobi structure when restricted to S2 gives thestandard Poisson structure on S2,

xa, xb = ǫabcxc.

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Indeed, Λ is invariant under left translations on the group SU(2).

Because X = XL3 is a generator of the right action, the left action projects onto an action

on S2. The Poisson bracket induced on FX is invariant under this action. On S2 all SO(3)invariant bivector fields are multiple of each other.

3 Generalized Reduction of Jacobi manifolds

3.1 Reduction of commutative associative algebras

The generalized reduction process can be described better in the algebraic setting of com-mutative associative algebras. This way of thinking exploits the duality among topologicalspaces and the correspondings algebras of functions defined on them.

The idea is to obtain a reduced algebra out of a given one A combining two elementaryprocesses:

- First, “choosing a submanifold”, by definition consists in chosing a quotient B of thealgebra A. The projection map π: A → B defines an ideal I as ker π = I and A/I = B.Thus, “choosing a submanifold” is equivalent to fixing an ideal I of A.

- The second process, “quotienting out an equivalence relation”, by definition consists inchoosing a subalgebra AI of A/I. We will call the algebra AI a reduction of the algebra A.

Notice that there are two choices involved in the definition of the reduced algebra AI , thechoice of I and of the subalgebra AI itself. This can be made more explicit as follows. Theinverse image of the subalgebra AI defines a subalgebra A′ = π−1(AI) of A. Thus, the orderon which the two processes are performed can be reversed realizing that AI = A′/A′ ∩ I.Then, we can first select a subalgebra A′ of A and then an ideal I ′ of A′ and define AI = A′/I ′.The relation between the ideal I and I ′ is that I is the ideal generated by I ′, i.e., I = AI ′.We see in this way the equivalence among the two ways of constructing the reduced algebraAI and the dependence of AI in the choice of a subalgebra A′.

It is obvious that the reduction process can be repeated a number of times using eachtime an ideal Ia of the reduced algebra AIa−1

and a subalgebra AIaof the quotient algebra

AIa−1/Ia, a = 1, . . . , n, with AI0 = AI . It is obvious from the previous remark that we can

reorganize the ideals Ia and the subalgebras AIato obtain a unique ideal J such that AIn

isa subalgebra of A/J . Thus the reduction process can be always restricted to the two stepsdescribed above.

The previous ideas can be refined if we suppose that the algebra A carries some additionalstructure. For instance we can suppose that A carries a Lie algebra structure [., .] (notnecessarily compatible with the associative product). Then, if we are given an ideal I asbefore (with respect to the associative structure), we will like to choose the subalgebra AI ofA/I such that it will inherit a Lie algebra structure, reproducing in this way the structures inthe original algebra A. Let us assume that A′ is simultaneously a Lie subalgebra of (A, [., .])and a subalgebra of (A, ·). Now we will suppose that A′

I ∩ I is an ideal with respect tothe associative structure and an invariant Lie subalgebra of A′, then it is obvious that thequotient algebra AI = A′/A′ ∩ I is a subalgebra of A/I and carries a Lie algebra structure.Thus the reduced algebra AI can also be called a reduced Lie algebra for A.

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From the above considerations it is obvious that the conditions imposed on the sub-algebra A′ are very tight. This shows that the enormeous freedom we have to constructreduced algebras is only apparent. We will show in the coming section how to find adequatesubalgebras A′ for Jacobi algebras.

Before that we will briefly analyze the reduction of dynamics. By definition, dynamicalsystems on algebras are derivations of the algebra. Thus, it is obvious that if D is a derivationof an associative commutative algebra A and we want to reduce both the algebra and thedynamics, the subalgebra A′ and the ideal I must be invariant with respect to D, i.e.,D(I) ⊂ I, D(A′) ⊂ A′. In this case it is obvious that D induces a derivation DI on thereduced algebra AI that will be called the reduced dynamic induced by D on AI .

3.2 Reduction of Jacobi manifolds

We will apply the ideas in the previous section to the algebra of smooth functions on aJacobi manifold M . Thus, the algebra A above will be now F . We will choose an ideal ofthe associative commutative algebra F that will be denoted now as J , i.e., FJ ⊂ J andwe shall assume that LXJ ⊂ J , i.e., J is X–invariant.

With the ideal J we can associate as before the short exact sequence of associativecommutative algebras

0 → J → F → F/J → 0.

The ideal J allows us to choose directly the subalgebra A′ used to complete the reductionprocess. This subalgebra is the normalizer of J with respect to the Lie algebra structure[., .] on F which is defined as

NJ = f ∈ F | [f,J ] ⊂ J . (5)

The following propositions are devoted to show that NJ verifies all the properties requiredin the reduction process. We must remark the crucial role played by the invariance of Jwith respect to X in the proof of some of them. In particular this means that the reductionprocess as discussed here will not work for an arbitrary Lie algebra structure on an associativecommutative algebra.

Proposition 6 NJ is a Lie subalgebra of (F , [., .]).

Proof. Is an immediate consequence of the Jacobi identity for [., .], because

[[f, g],J ] = [[g,J ], f ] + [[J , f ], g]. (6)

The term inside the first bracket on the r.h.s. of the previous equation (6), is in J becauseg ∈ NJ . Then because f ∈ NJ we obtain that the first term is in J . The same argumentapplies to the second term in the r.h.s. of eq. (6). 2

Proposition 7 The subalgebra NJ is X–invariant, i.e., LXNJ ⊂ NJ .

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Proof. Let f be a function in NJ and h a function in J . Because X is a derivation of[., .], we have,

LX [f, h] = [LXf, h] + [f,LXh].

Then,[LXf, h] = LX [f, h] − [f,LXh],

is in J because [f, h] and LXh are in J . 2

Proposition 8 The subspace NJ is a subalgebra of F with respect to the associative com-mutative structure ·.

Proof. Let h be a function on J and f1, f2 two arbitrary functions on NJ . Then,

[f1f2, h] = f1[f2, h] + f2[f1, h] − f1f2LXh.

The two first terms on the r.h.s. of the previous equation belong to J because fi are in NJ ,and the last term is also contained in J because J is X–invariant and is an ideal, hencef1f2 is in NJ . 2

Proposition 9 NJ ∩ J is an invariant Lie subalgebra of NJ , i.e.,

[NJ ∩ J ,NJ ] ⊂ NJ ∩ J .

Proof. Let f be in NJ ∩J and g in NJ . Then f is in NJ and then [f, g] ∈ NJ becauseof Prop. 6, but [f, g] is also in J because of the definition of the normalizer NJ , eq. (5).Then, [f, g] ∈ NJ ∩ J .

Now if h1, h2 are in NJ ∩ J , then h1, h2 are in NJ , then [h1, h2] is in NJ because ofProp. 6. On the other hand h2 ∈ J . Then, because h1 ∈ NJ , then [h1, h2] ∈ J . Hence[h1, h2] ∈ NJ ∩ J . 2

Proposition 10 The subspace NJ ∩ J is an ideal of the associative commutative algebraNJ .

Proof. Let f be in NJ ∩ J and g ∈ NJ . Then, f ∈ J and gf ∈ J because J is anideal. On the other hand because of Prop. 8, NJ is a subalgebra of F with respect to ·,then gf ∈ NJ and the conclussion follows. 2.

Thus we can state the following theorem,

Theorem 3 Let (M, Λ, X) be a Jacobi manifold and J an ideal of the associative commu-tative algebra of smooth functions F on M . Let us suppose that J is X–invariant. Then,the quotient space NJ /NJ ∩ J inherits a Jacobi algebra structure induced from that of F .Moreover, if there is a smooth manifold R such that F(R) = NJ /NJ ∩ J , then R inheritsthe structure of a Jacobi manifold, the bracket among functions given by the bracket of theLie algebra structure induced in NJ /NJ ∩ J .

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The algebra FJ = NJ /NJ ∩ J will be called the reduced algebra of the Jacobi algebraF with respect to the ideal J and the Lie bracket on it will be denoted by [., .]J . If theassociative commutative structure on FJ = NJ /NJ ∩ J defines a smooth manifold R, thereduced structure defined on this quotient manifold R will be called the reduction of theJacobi manifold M .

Proof. We see that because of Prop. 9, the Lie algebra (Prop. 6) NJ admits NJ ∩ Jas an invariant Lie subalgebra, therefore NJ /NJ ∩J is a Lie algebra. Moreover, because ofProp. 10, NJ ∩ J is an ideal of NJ with respect to its associative structure (Prop. 8) andthe quotient NJ /NJ ∩ J is an associative commutative algebra. Finally the derivation Xpasses to the quotient because of Prop. 7. Thus, we have on the reduced algebra NJ /NJ ∩Jan associative and a Lie algebra structures. It remains to show that they define a Jacobialgebra. This follows from the following Lemma.

Lemma 1 Let D be a differential operator of order r on the associative commutative algebraA. If I is an ideal such that D(I) ⊂ I then, the linear operator D induced by D on thequotient A/I is again a differential operator of order ≤ r.

Proof. The operator D(x + I) = D(x) + I defined on A/I verifies that

δ(x)D = δ(x)D,

for all x ∈ x = x + I. Then,

δ(x)r+1D = δ(x)r+1D = 0,

because δ(x)r+1D = 0 (D is of order r). 2

Then we conclude the proof of the main theorem, noting that the operator Df given byeq. (1), leaves invariant NJ for all f ∈ NJ (Prop. 6), then Df is a differential operator oforder 1 in NJ for all f ∈ NJ . Moreover J is invariant for each Df , f ∈ NJ . Thus, theoperator Df induces a differential operator Df of order 1 in the quotient algebra NJ /NJ ∩J .Moreover by definition of the Lie bracket in the reduced algebra, we have,

Df (g) = [f , g]J ,

Thus, the Lie bracket [., .]J defines a skew–bidifferential operator of order 1, hence a Jacobistructure. Moreover, if the reduced algebra is the algebra of functions of a smooth manifoldR, Thm. 2 implies that R is a Jacobi manifold. 2

3.3 Reduction by a submanifold

The canonical way of defining ideals in spaces of functions is by fixing subspaces. Let Σ bean embedded closed submanifold of M and we denote by JΣ the ideal of smooth functionsvanishing on Σ, JΣ = f ∈ F | f |Σ = 0 . Then F(Σ) ∼= F(M)/JΣ.

It is clear that JΣ is X–invariant iff Σ is invariant under the flow of X or equivalently,if X|Σ ∈ TΣ. The normalizer NΣ of JΣ consists of functions f such that the operator Df

leaves JΣ invariant. Is simple to check that this is equivalent to ask that the Hamiltonianvector field Xf is tangent to Σ.

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Some terminology is convenient now. Functions in JΣ will be called constraint functionsand functions in NΣ will be called first class functions. Functions in NΣ ∩ JΣ will becalled first class constraints and constraints which are not in NΣ ∩ JΣ will be called secondclass constraints. First class constraints define Hamiltonian vector fields which are tangentto Σ but because of Prop. 1, and Prop. 9 we have that the distribution DΣ generated byHamiltonian vector fields corresponding to first class constraints is an integrable distribution.It defines a foliation of M whose restriction to the submanifold Σ will be denoted by LΣ.

In general the reduced Jacobi algebra will not be the algebra of smooth functions on thequotient space Σ/LΣ unless some further condition are imposed on Σ. For instance if Σ issuch that JΣ ⊂ NΣ, we will say that Σ is first class or coisotropic. Then, the reduced Jacobialgebra is NΣ/JΣ. It is not hard to see that in this case F(Σ/LΣ) = NΣ/JΣ and the spaceof leaves of the foliation LΣ inherits a Jacobi structure. To conclude with we can state thatif the foliation is regular and Σ is coisotropic, then the reduced Jacobi algebra is the algebraof functions of the quotient manifold Σ/LΣ.

Simple cases of coisotropic submanifolds of Jacobi manifolds are provided by the followingexamples.

Let S be a regular level set of the Casimir functions on M . Then, it is clear that JS isthe ideal generated by the subalgebra of Casimirs on M . Then, NS = F because [f, C] = 0for all f ∈ F , and C an arbitrary Casimir. Hence, JS ⊂ NS = F and the submanifold S iscoisotropic. In this particular case, the reduced algebra is F/JS

∼= F(S) and the reducedmanifold is S itself with the natural Jacobi structure induced on it by Λ and X.

Another example is provided by submanifolds S such that JS + NS = F , then becauseNS/NS ∩JS

∼= NS +JS/JS, then, the reduced algebra will be again F/JS = F(S) and thereduced manifold will be the submanifold S again that will inherit a Jacobi structure. Inparticular we will obtain as corollaries of this situation most of results in [Da91], for instance:

Theorem 4 Let S be a submanifold of the Jacobi manifold M such that it satisfies,

TS + Λ♯(TS0) = TM,

where TxS0 = α ∈ T ∗

xM | α(v) = 0, ∀v ∈ TxS denotes the polar distribution to TSand Λ♯ is the bundle map T ∗M → TM defined by contraction with Λ. Then S inherits in anatural way the structure of a Jacobi manifold.

3.4 Reduction by a distribution

It is natural to select as a subalgebra in the reduction process not just the normalizer of anideal J but a subalgebra of it selected by imposing some invariance requirement. This isformulated using integrable distributions on F , i.e., Lie subalgebras D of the Lie algebra ofderivations of the algebra F . Any distribution defines an associative subalgebra of invariantfunctions, FD = f ∈ F | X(f) = 0, ∀X ∈ D . We will say that the integrable distributionD is compatible with the Jacobi algebra structure of F if FD is an X–invariant subalgebrawith respect to both structures, associative and Lie algebra, on F .

It is now obvious that if J is an ideal on F , the quotient algebra

FJ ,D = FD ∩NJ /FD ∩ NJ ∩ J ,

is a Jacobi algebra, that will be called the reduced Jacobi algebra of F with respect to Jand D.

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4 Examples and applications

4.1 Reduction of Jacobi manifolds with symmetry

Let G a Lie group and g its Lie algebra.

Definition 4 A Lie algebra homomorphism g → X(M), a 7→ ξa, defines an infinitesimalaction of G on the Jacobi manifold M by automorphisms if ξa is a derivation for the Liebracket on F . The action of g on M will be called Hamiltonian if ξa is Hamiltonian for alla ∈ g.

Notice that the infinitesimal action of g on M is Jacobian iff LξaΛ = 0, [ξa, X] = 0, for

all a ∈ g. We also notice that an action by infinitesimal automorphisms is Hamiltonian ifthe Hamiltonians fa associated to the elements of g are in FX . Such action will be calledJacobian.

We will consider in what follows a Jacobian action of a Lie group on a Jacobi manifoldM . Denoting as before the Hamiltonian defined by the element a ∈ g by fa we have,

ξa = Xfa.

Hence, we define a momentum map µ: M → g∗ by setting

〈µ, a〉 = fa, ∀a ∈ g, (7)

or equivalently,ξa = 〈µ, a〉X + Λ(d〈µ, a〉), ∀a ∈ g. (8)

If 0 is a regular value for the momentum map µ, then Σ = µ−1(0) is a submanifold ofM . Consider the ideal JΣ of functions vanishing on Σ. It is obvious that JΣ is the ideal onF generated by the functions fa, a ∈ g and consequently it will be denoted by J0.

We will denote by N0 the normalizer of J0. It is not hard to check that Σ is first class,i.e., J0 ⊂ N0. Moreover, N0 is the algebra of functions such that their restriction to Σare G-invariant, i.e., if D denotes the integrable distribution generated by ξa, a ∈ g then,N0 ⊂ FD. The reduced algebra will therefore be F0 = N0/J0 or equivalently the algebra offunctions on Σ invariant with respect to D.

Then if the quotient space µ−1(0)/G is a smooth manifold, the quotient algebra F0 can beidentified with the algebra of functions on Σ/G. Then, the Jacobi reduction theorem allowsus to conclude that the quotient manifold Σ/G inherits the structure of a Jacobi manifold.

4.2 Reduction of symplectic, Poisson, contact and cosymplectic

structures

I. Reduction of symplectic manifolds. X = 0, Λ nondegenerate, the previous procedureagrees with symplectic reduction as discussed in [Ma85], [Gr94]. Specializing the discussionin §4.1 to symplectic manifolds with symmetry we will obtain the well–known Marsden–Weinstein symplectic reduction theorem.

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II. Reduction of Poisson manifolds. X = 0, Λ arbitrary. The previous discussion leadsus to the construction of reduced Poisson manifolds as discussed for instance in [Gr94].

If we are given a submanifold Σ of a Poisson manifold M and a subbundle E of TM ,we can define the annihilator E0

Σ of E restricted to Σ, then we consider the distribution Dgenerated by Hamiltonian vector fields such that the differentials of their hamiltonians lie onE0

Σ. If E verifies the conditions stated on [Ma86], then the distribution D is compatible withthe Poisson structure and the reduced Poisson agrees with the Marsden-Ratiu reduction ofM by Σ and E.

III. Reduction of cosymplectic manifolds. X = 0. Particularizing the results above tocosymplectic manifolds we will obtain the reduction of cosymplectic manifolds with symme-try [Al89]. The reduction of cosymplectic manifolds for singular momentum maps discussedin [Le93] can be described in this setting with the obvious modifications.

IV. Reduction of contact manifolds. X 6= 0. Similarly, reduction of contact manifoldswith symmetry is included in our previous discussion. The reduction of contact manifoldshas been discussed recently in [EY93], [EY94].

Ackwnoledgements The author AI wishes to ackwnoledge the partial financial supportprovided by DGICYT under the programme PB92-0197 as well as the NATO collaborativeresearch grant 940195. The author ML acknowledges the partical finantial support providedby DGICYT project PB94-0106.

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