(an observable) we attach an operator - Real Academia de Ciencias

RAC Rev. R. Acad. Cien. Serie A. Mat. VOL . 95 (1), 2001, pp. 65–83

Geometr´ıa y Topolog´ıa / Geometry and Topology

On the geometric prequantization of brackets ´ J. C. Marrero and E. Padron ´ M. de Leon,

Abstract. In this paper we consider a general setting for geometric prequantization of a manifold endowed with a non-necessarily Jacobi bracket. The existence of a generalized foliation permits to define a notion of prequantization bundle. A second approach is given assuming the existence of a Lie algebroid on the manifold. Both approaches are related, and the results for Poisson and Jacobi manifolds are recovered.

´ geometrica ´ Precuantizacion de corchetes Resumen. En este art´ıculo se considera un marco general para la precuantización geométrica de una variedad provista de un corchete que no es necesariamente de Jacobi. La existencia de una foliación generalizada permite definir una noción de fibrado de precuantización. Se estudia una aproximación alternativa suponiendo la existencia de un algebroide de Lie sobre la variedad. Se relacionan ambos enfoques y se recuperan los resultados conocidos para variedades de Poisson y Jacobi.

1. Introduction Since the seminal results by Kostant and Souriau [14, 26] a lot of work has been done in order to develop a geometric theory of quantization. The inspiration behind these ideas was to develop a method to quantize a classical system and to obtain the quantum system reproducing the Dirac scheme for canonical quantization. The procedure starts with a phase space which in the most favourable case is a symplectic manifold . Then, we associate to a Hilbert space, which at the first step is the space of sections of a complex line bundle over . Thus, to each function (an observable) we attach an operator , , where is the Hamiltonian vector field defined by , and is a covariant derivative on . is said to be a prequantization bundle of if , that is, the commutator of the operators corresponds to the Poisson bracket of the observables. This condition can be translated as the existence of a covariant derivative such that its curvature is . The condition is just fullfilled for integral symplectic manifolds. In constrained Hamiltonian systems and other physical instances, there appear more general phase spaces, endowed with a non-symplectic Poisson bracket, or even, a Jacobi bracket. An approach to this problem is the use of symplectic and contact groupoids, and there is an extensive list of results due mainly to Karasev, Maslov, Weinstein, Dazord, Hector and others (see [4], [5], [6], [11] and [31]; see also [29] and the references therein).

' "! #$ &%('*),+- #/.1 0-243 # 576 89 : :7 : >?

'

Presentado por Pedro Luis Garc´ıa Pérez Recibido: 3 de Noviembre 1999. Aceptado: 8 de Marzo 2000. Palabras clave / Keywords: Jacobi manifolds, Poisson manifolds, Lie algebroids, H-Chevalley-Eilenberg cohomology, Lichnerowicz-Jacobi cohomology, Lichnerowicz-Poisson cohomology, foliated cohomology, foliated covariant derivatives, geometric prequantization. Mathematics Subject Classifications: 53D50, 53D17, 17B66 c 2001 Real Academia de Ciencias, España.

@

65

M. de León, J. C. Marrero and E. Padrón

On the other hand, in [10] Huebschmann extended the Kostant-Souriau geometric quantization procedure of symplectic manifolds to Poisson algebras and, particularly, to the Lie algebra of functions of a Poisson manifold. In [28] (see also [29]), Vaisman obtains essentially the same quantization of a Poisson manifold straightforwardly, without resorting to any special algebraic machinery. He introduced contravariant instead of covariant derivatives on , in order to take account that the Poisson bracket is given by a contravariant 2-vector. This permits to obtain similar results for Poisson manifolds. Here, the traditional de Rham cohomology has to be substitutted by the so-called Lichnerowicz-Poisson cohomology (LP-cohomology). The LP-cohomology has a clear lecture in the Chevalley-Eilenberg cohomology of the Lie algebra of functions on through the representation . The above results were recently extended for a Jacobi manifold (see [19]). In this case one has to consider the Lie algebroid associated to and the so-called Lichnerowicz-Jacobi cohomology (LJcohomology) of . The Lie algebroid (respectively, the LJ-cohomology) has been introduced in [12] (respectively, in [18, 19]). The LJ-cohomology is the cohomology of a subcomplex of the H-ChevalleyEilenberg complex. The cohomology of this last complex is just the cohomology of the Lie algebra of functions on relative to the representation defined by the Hamiltonian vector fields (for a detailed study of the cohomologies of another subcomplexes of the H-Chevalley-Eilenberg complex, we refer to [15, 16, 17]). The purpose of the present paper is to consider a more general setting which extends in some sense the precedent ones. We consider a manifold endowed with a skew-symmetric bracket of functions which satisfies the Jacobi identity, a rule that assigns a vector field to each function such that , and, in addition, we assume that the generalized distribution defined by the vector fields is in fact a generalized foliation. Note that does not satisfy necessarily any local property, so that it is not in principle a Jacobi bracket. Even in this general setting it is still possible to define the corresponding H-Chevalley-Eilenberg cohomology, and the de Rham and the -foliated cohomologies are related with it. By introducing a suitable definition of -foliated derivatives, we give a first definition of prequantization bundle, and obtain a characterization of a quantizable manifold (Theorem 3.9) in the setting of foliated forms. If the existence of a Lie algebroid is assumed, then we can define a new cohomology by using the representation of on and, under certain conditions, this cohomology is related with the precedent ones in a natural way. A second definition of prequantization bundle is given in this context making use of a convenient notion of -derivative, and the corresponding characterization of quantizable manifold is obtained (Theorem 4.8). It should be remarked that this second notion of prequantization bundle is more general that the precedent one. Finally, we discuss the cases of Poisson and Jacobi manifolds recovering the results previously obtained in [28] and [19], respectively.

A:B ,!C

9 A:D;

E

G 5 6 5IHKJ 5 % 6 75 L 6KM OH N

9 ";

E F% 5 6

P

P

S D9 !T";Q E

UR V R W&

R

2. Jacobi manifolds

X

YZ\[

G [`WY J %a*b

All the manifolds considered in this paper are assumed to be connected. A Jacobi structure on an -dimensional manifold is a pair vector field on satisfying the following properties:

GJ

P

9 ";

G YZ\Y J %]0-[^_Y

where

Y

0

is a -vector and

[

a (1)

Here denotes the Schouten-Nijenhuis bracket ([3, 29]). The manifold endowed with a Jacobi structure is called a Jacobi manifold. A bracket of functions (the Jacobi bracket) is defined by

9 : R

®R "! R ¢ ¬ RR ¬ & R 9 j; R h# " !¯#±° | ~ G J

V 9 # R #± °-; R % & 9 # R #±°-; . ¬ # i W #±°Qb R R R R R R R R 9 R WYc; R W [¬

S ³R ¢ % % £ ¢ ¢ £ £ ¶ µ´´· D!¸ ² U W& ²

R ~ h´ V & ib 9 ¹; ~ h´ VR W&

R 9 F ± :n ; % ºD»¼¾½h¼À¿QÁÂ 6wÃ Á:»¼Ä½h¼ÆÅQÁÂ H Ã Á YcFWf

R .I »¼Ä½h¼À¿}ÁÂ Á¤3 d:n ÇÈ»Z¼Ä½h¼ÉÅQÁÂ ÁfU3 f Ud \a}

% »j½h¼À¿QÁÊ»6wÃ½h¼ÆÅQÁÊ Ã d
follows that defines a generalized foliation, which is called the characteristic foliation in [7]. Moreover, the Jacobi structure of induces a Jacobi structure on each leaf. In fact, if is the leaf over a point of and then is a contact manifold with the induced Jacobi structure. If , is a l.c.s. manifold (for a more detailed study of the characteristic foliation of a Jacobi manifold we refer to [7] and [8]). If is a Poisson manifold then the characteristic foliation of is just the canonical symplectic foliation of (see [20] and [30]). Next, we will recall the definition of the Lie algebroid associated to a Jacobi manifold (see [12]). Let be a differentiable manifold. A Lie algebroid structure on a differentiable vector bundle is a pair that consists of a Lie algebra structure on the space of the global cross sections of and a vector bundle morphism ( the anchor map) such that: is a Lie algebra homomorphism.

1. The induced map 2. For all

and for all

one has

(8)

A triple is called a Lie algebroid over (see [23], [25] and [29]). Now, let be a Jacobi manifold. In [12], the authors obtain a Lie algebroid structure on the jet bundle as follows. It is well-known that if is the cotangent bundle of , the space can be identified with the product manifold in such a sense that the space of the global cross sections of the vector bundle can be identified with Now, we consider on the bracket given by (see [12])

(9)

where

is the prolongation mapping defined by

(10)

We have (see [12])

Theorem 1 Let be a Jacobi manifold and the bracket on (9). Then, the triple is a Lie algebroid over where is the vector bundle morphism

defined by

(11)

for mapping

Moreover, if we consider on

the Jacobi bracket then the prolongation (12)

is a Lie algebra homomorphism. Remark 1 If vector bundle

then the canonical projection in such a sense that

defines a vector subbundle of the

for where (respectively, ) is the fibre of (respectively, ) over . Note that can be identified with the cotangent bundle and that, under this identification, the restriction of the anchor map to is just the vector bundle morphism 68

On the geometric prequantization of brackets

3. H-Chevalley-Eilenberg cohomology, foliated covariant derivatives and prequantization In this section we will assume that

is a differentiable manifold which satisfies the following conditions:

9 ; } W& h´ W&
1. There exists a bracket of functions

which is

EÎQ & h"! | ~ i

-bilinear, skew-symmetric and satisfies the Jacobi identity.

2. There exists a

= & ,!C E ,% 5 6 _| ~

G 5 6 5*HOJ % 57L 6KM HN ¢ P9 P % 5`6 ¨ W& ;Qb | P ib P 5 &P b

-linear map

and

,A : % V W& ib E 5`6 5P

for

(13)

is called the Hamiltonian vector field associated with .

If

is a point of

, we will denote by

the subspace of

A vector field on is said to be tangent to fields tangent to is denoted by

if

defined by

for all

SD! P r ¢

The space of the vector

3. The involutive generalized distribution

is completely integrable. Thus, teristic foliation.

P

defines a generalized foliation on

which is called the charac-

9 ;-

h´ hD! A:B ,!C 5 6 e:B b Ü ÝF£ Þ Ã ~

ÝF b ¼ ÞhÃ àOàKà~´ & D! &

? ´ O b O b ß V Â V V W&

± 9 ã ~
Ã Ã ç ã Ý ßä i wå àOàKà S% æçÀè ä Vvw 5`ç 6Aé ì ß ä wå àOàOà - ê ç àOàKà

ä . åæçënì AVv$ Â 9 ç ì ;} àOàKà êä ç àKàOà Òê ì àKàOà

ß ä wå ä

3.1.

H-Chevalley-Eilenberg cohomology

We consider the cohomology of the Lie algebra relative to the representation defined by the Hamiltonian vector fields, that is, to the representation given by

This cohomology is denoted by and it is called the H-Chevalley-Eilenberg cohomology associated to (see [16, 17, 18, 19] for the case of a Jacobi manifold). In fact, if is the real vector space of the -multilinear skew-symmetric mappings then Im

where

is the linear differential operator defined by

(14)

69


for

ßäs©`sQÝFä VÞ Ã ~ D$! å OVbKbO b ä &V W& b Y¹í % ã Ý © b YZí A:B ,% 5`6 e:B Ç 5 H Ç 9 :
$ # ,

%

Q # b 5 5 0 '7P b 2 « D ®!¯" !Õ b ' P 2 £ ÜP P b 0 P Ü ° P ib G J 2 ÚD!ª ' P

P

Definition 2 Let be a complex line bundle over on The curvature of is the mapping

If

is a point of

and

a

-foliated covariant derivative given by (19)

, we have that

where is the curvature of the covariant derivative on Thus, there exists a globally defined complex -foliated -form such that

. (20)

Since the -foliated -form completely determines to the map , we will say also that curvature of Proceeding as in the case of a usual covariant derivative (see, for instance, [14]), we prove Theorem 4 Let derivative on Then:

1. The complex

be a complex line bundle over with curvature and that

-foliated -form

2. The cohomology class

is the

Suppose that is a -foliated covariant is the complex -foliated cohomology of

defines a cohomology class in

does not depend of the

3. If is a Hermitian metric on then is purely imaginary.

and

-foliated covariant derivative.

is a Hermitian

-foliated covariant derivative

ib 2 "!Õ 2 ê: ê: > ê A: W&

=ê [%$Ðd W

# & ,CD!T ê #w ,%?'*) + #h.¹0$243 #Q ' P 2 D!Õ2 D! b b Av'&Q

RE P ° P hDY í ! Ýh° Þ Ã 0 ~

2
D Ô ! P 3 RE P 0$2 ,% Yí b

3.3.

Prequantization

over we will denote by End For a complex line bundle endomorphisms of Then, we introduce the following definition. Definition 3 A complex line bundle

over

the space of the -linear

is said to be a prequantization bundle if

(21)

with

defined by

(22)

where is a -foliated covariant derivative on if there exists a prequantization bundle Let

The manifold

is said to be quantizable

over

be the -coboundary in the H-Chevalley-Eilenberg complex given by be the homomorphism defined by (18). From (13), (19), (20), (21), (22) and Definition 1, we deduce

and let

Lemma 1 The manifold is quantizable if and only if there exist a complex line bundle over and a -foliated covariant derivative on such that the curvature purely imaginary and

Next, we will obtain another characterization. For this purpose, we recall the following result. 72

of

is


) ( ' 2 Let

be a differentiable manifold as in Section 3. Moreover, we will assume that there is a Lie algebroid over with anchor map which satisfies the following conditions:

1. If phism

is the space of the cross sections of such that

then there exists a Lie algebra homomor-

2. For all where

3. For all

We will denote by

is the fibre over

we have that such that

the Lie bracket on

Furthermore, there is a vector subbundle

of

and by

the restriction to of the anchor map induced homomorphism between the cross sections of

4.1.

2 R *Ñ R
ÎD Ñ R s R

73


Ü £ R ,+ V & W b . R w´ bO bK b ¼ àOR àOà -´+ V R hDW!& V

Ü R -+ % áâ p 9±9±ã ã Q R -+ -+ " h! "! Â R - ,+ + \; ; g R R ã }sä R -+ V W& W hD! sä Â R -+ V W&

ç ã .sä # å àOàKà # ó% æçÄè ä AVv$ ¬ ç # ç ì i . ä # å àOàKà # ê ç àOàKà # W

R Rä R R R R . åæçëBì AVvw Â . ä 9 # R ç # R ì ; R # R å àOàKà # êR ä ç àOàKà # Rê ì àKàOà # R ä

# R å KbObKbO # R ä R b Ü £ R - + V i P Z ~

ù P ~ cb
P ¬Q UR
¬ R P P j
/ &

¬ R P i W i # R ObKbObK # R % ¬ # R KbObKbO ¬ # R

£#R KbO bObO- #+ R WR & b W ã ib ¬ R P

P ¬ R £ P iWdÒ Í P"!¶ ¬ R P Ü £ P R DV! Ü £ R - + V W& ib ¬ËR ~ sD! UR - + V W&

/ &

¬ R % ¬ R P > ù P b ¬R ¬R c£ ~ iWdÒ s!Ô £ R , + £ W& i ã

¬R Ü ñ£ ò ~
zRE QÜ ñ£ ò ~ hD! Ü ÝF£ Þ Ã ~

º QV
º R } R , + V W W ,! ÝF Þ Ã ~

W&

ºDR . ObKbOb ,% . ÀºD KbObKbOºD

ObK ¬ ' # R Î R P # \#±° b 2 D!Õ 2 3( ö #%6' 5¼ 3( ö Ái#}ô õ ö

# R Î R # _

R

2
2 D! b U

´

h

´ 2 8 7? R R ( ( i#w

# ± # ± °

$ #

~ % ( ( ( ( ( 7 R R 2 3 î > 2 3:9 2 3:9 > 2 3 î 2`L 3 î M 3 9 N # R # R ° & UR # & ib

4.2.

Lie algebroids and derivatives on complex line bundles

Let

by onto

maps of

be a complex line bundle over the fibre over

Definition 4 A

-derivative

Denote by and by Lin

on

the space of cross sections of the space of the -linear

is a map

Lin

which

satisfies the following conditions: 1. If

then

2. The map

Lin

.

Lin

3. For

is

and

4. If

is an open subset of then the map

for

is a

we have

and given by

for

If

is a

A

-derivative

, we obtain a (Hermitian)

Definition 5 Let The curvature of

on

is said to be (27)

, and is a (Hermitian) -foliated covariant derivative on

and

-differentiable local section of

-differentiable local section of

Let be a Hermitian metric on Hermitian if for

-linear.

and we put

-derivative.

be a complex line bundle over is the mapping

and

a

-derivative on given by

(28)

for

and

75


R hD! | ~

¬ &

7 V 7 # R #±R °± i#$ %Ø 7 #KR ° # R #$ b ì I´ W &

. 7 * R UR
Y R £ D!Õ . 7 R R V # ± # ± ° ,

Ì % . 7 R R . 7 # R #±R °± Ð. 3i . 7 °} # R #KR °w b . =FR ¼?7 2 > í 7 Á UR < ! R V

ã . 7 > . 7 % = R @¼ 7 > 7 Á b G . 7 > J % G . 2 D!(

.;7

2
2 D!Õ b . 7

Theorem 9 Let with curvature 1.

be a complex line bundle over Then:

Suppose that

is a

-derivative on

defines a cohomology class in

2. If

is another

-derivative on such that

there exists a

-linear mapping

In particular,

3. If is a Hermitian metric on then is purely imaginary.

and

is a Hermitian

2 Prequantization D!Õ 2 ®"!Õ UR

2 ì ¼ ÁÊ¼ ÁA#c%an # ib F

] % a 5 2

-derivative on

4.3.

be an arbitrary complex line bundle over . -derivative on In this section, we will assume that a conditions: Let

(C1) If 76

then

for all

always satisfies the following


(C2) If

5 %]y a # & b ' P

and there exists

#R Ñ R

for all Note that if

for

is a

#ìR ,% 5

3( ö #% 2 ¼ ÁÊ¼ ÁA#

such that

2

2 D!Õ 3( ö #%'65¼ 3( öKÁi#}ô õnö-

-foliated covariant derivative on

# R Î R # _ i

2 2

and

2

then

is the

R

-derivative defined by

and then satisfies the above conditions. Next, we will introduce a new definition of prequantization bundle for the manifold

2 ê: ê: > ê

Definition 6 A complex line bundle

with

= ê [%$Ðd W

over

b

is said to be prequantization bundle if

A: W&

(30)

# ,C
º 2 R 2 2 D!Õ Ø %

. i 3

° .%7 . V ã

RE % Yí . Ý ºDR = R % ã Ý © . ºDR = R b 5 i %a # R £ ÀºDAv$ i %aBb 5 i Í%~y a # R Z% 5 # R Ñ R # R £ ÀºDAv$ i ib given by

-derivative on where is a prequantization bundle

over

.

The manifold

(31)

is said to be quantizable if there exits a

It is clear that if is quantizable in the sense of Section 3.3 (see Definition 3) then is also quantizable in the above sense. However, in general, the converse is not true. Let be the -coboundary in the H-Chevalley-Eilenberg complex given by (15) and let

be the homomorphism defined by (26). Using (28), (29), (30), (31), Definition 4 and the fact that is a Lie algebra homomorphism, we deduce

Lemma 2 The manifold is quantizable if and only if there exist a complex line bundle over and a -derivative on with curvature such that

Now, if is the homomorphism given by (16) then, proceeding as in the proof of Theorem V.2 of [19] and using (23), Lemma 2, Theorems 5, 8 and 9 and the fact that is a Lie algebra homomorphism, we conclude Theorem 10 The manifold is quantizable if and only if there exist an integral closed -form and a cross section of the dual bundle such that: 1. If

is the

2. If

is a point of

and

3. If

is a point of

such that

-linear map induced by

with

on

then

then

and

then

#R £ i #R % 77


5. The particular cases: Jacobi and Poisson manifolds

\YZ\[ £ ~ ã 9 ; ÝhÞ Ã Ý Ycí :n ,%Ì ã Ý © A:B %]Ycd \d-:B : b vb ä ß ~& Ýh Þ Ã ~ ù v ~º Ö ~ ~

ç ~

v Ö ÝhÞ Ã ñ 6w6 ~ hD! Ýh Þ Ã ~

º = A ObKbObK ,% = d ObObKbO\d Ð. æ è Vvw B Â B AÍd KbObObKDd C B ObObKbWd b B ç º ~ ~ ,% Ýh Þ Ã ñ 6w6 ~

~ Ö ~

ÝFÞ Ã ñ ç 6±6 ~Ö Y¹í % ã Ý © % º ° YcWag b ã Ý =R v ÂÝF Þ ñ ç ~ i = Ýh Þ 9 Vñ ;ç ~

R & Ã 6w6 ç ã Ã 6±6 Ýh£ Þ Ã ñ 6w6 ~ Ý ô FÞ GIE HJ î Dÿ K é +\+ ¼ ¡ ÁW

ç ~

v £ Ü ÝhÞ Ã ñ 6w6 ã Ý Àº = A W % º Â Mõ L = A

Mõ L = A F%ØA G Yc = J . [Mõ ^ L = . Y=~)^ A* G Yc A J~È j D! vw A[] Â *^ A]~. G [` = J ~ b

£ ~Ö Ö £ ~ i Nõ L

õ ° L %aBÒb Ö M õÜ £ L ~

M º £ ~ ~ Ö £ ~ ~ õM ËL D ! Ýh Þ £ Ã ~ ç ~ i ã ô Ö ÝhÞ Ã ñ 6w6 Ý ÞOGIE H"J î ÿDK é +W+ ¼ ¡ Á

Mõ L aBOv$ ,%~YZ\a} b [Ì%ag

Mõ L = \a} ,%ÌA G YZ = J \a}

Â = ~ b Mõ P Z ~ UD! ~

Mõ P = %Ø G Yc = J

Let

be a Jacobi manifold and the Jacobi bracket. Suposse that is the H-Chevalley-Eilenberg complex associated to (6) and (15), we deduce that

. Using (2), (4), (32)

Next, we will recall the definition of the Lichnerowicz-Jacobi cohomology (see [18] and [19]). is said to be -differentiable if it is defined by a linear differential operator A -cochain of order If is the space of -vectors on then we can identify the space with the space of all -differentiable -cochains as follows: define the monomorphism given by (33)

Then

and

which implies that the spaces are isomorphic. Note that (see (32) and (33))

(34)

On the other hand, using that for

is a linear differential operator of order , we deduce that . Thus, we have the corresponding subcomplex

of the H-Chevalley-Eilenberg complex whose cohomology will be called the -differentiable H-Chevalley-Eilenberg cohomology of . Moreover, we obtain that (see [18, 19]) (35)

where

(36)

This last equation defines a mapping which is in fact a differential operator that verifies Therefore, we have a complex whose cohomology will be called the Lichnerowicz-Jacobi cohomology (LJ-cohomology) of and denoted by (see [18, 19]). Note that the mappings given by (33) induce an isomorphism between the complexes and and consequently the corresponding cohomologies are isomorphic. Furthermore, from (36), we obtain that (37)

Now, if

is a Poisson manifold

, it follows that (see (36))

(38)

for defined by

Thus, we deduce that the linear differential operator

(39)

78


£ ~ Mõ P ib õ ° P %anb N Ü õM£ P ~

\YZ\[

P ÓR ~ D! ~ Ö Â ~ Ó ~ s
R , %~ , O 3 Ã W

~ b R > d%~ Mõ L > R b ~£ Í! ~ K ~ õMLÖ ~

R ¯ ~

\ g d &

D ! ~ £ £ ò R ~ h"! Ü õM£ L ~ Ö R Ü ñ£ Ú º > R %ÌAVv$ RE ÎRE ~
ù P Z ~ Ë D!

£ P \dg "!¯ £ ~ £ P ~ K Mõ L P R P P Ö

£ P iWdÒ R P ü SO T £ ~ Ö £ ~ iO õML

QR ù P Q Q S S R £ ~ \dg

P P ! ~

/

P i % P , ObObKbO\ ,%ÌAVv$ KbObObK W

=&V P ObObKbW ~ i R P i, %Ì P i iO P i3 Ã , ib

satisfies the condition This fact allows us to consider the differential complex The cohomology of this complex is the Lichnerowicz-Poisson cohomology (LP-cohomology) associated to and it is denoted by (see [20] and [29]). Next, we will obtain the relation between the -foliated cohomology of a Jacobi manifold and the LJ-cohomology. Denote by the map given by (4) and (5) and let be the homomorphism of -modules defined by (40)

for

In [19] (see also [18]), we prove that

(41)

This implies that the mappings induce a homomorphism of comand, thus, a homomorphism in cohomology plexes (see [18, 19]). In fact, using (4), (5), (6), (33) and (40), we have that (42)

being the map given by (16). On the other hand, if is the space of the -foliated -forms on and is the canonical homomorphism, it is clear that there exists a homomorphism of complexes such that the following diagram is commutative

In fact, if

is the homomorphism of

-modules defined by

(43)

for

,

and

then

(44)

Thus, from (6), (33), (43) and (44), we deduce that

RE P P hD! Ýh Þ Ã ~

º >= R P ,%ÌAVv$ RE P i

(45)

is defined by (18). where Therefore, using (35) and Theorem 3, we conclude that

R P > d%~ õML > R P b

Consequently, for the corresponding homomorphisms in cohomology, we obtain the following commutative diagram 79


P Q X X X ø Q Q X X X X Vvw RE P

X X X P QOU X R ù P

õÜ SOM L T ~ º ü X Ü Y ÝF Þ Ã ~

V V W S V V RS V V AVv$ RE S V V V V V V ñÜ ò ~

P P
R R 0 WYcW[I Z R F% Mõ L Z ib [ %a %~vb [ %®y a 7% [ `% Z ¤.vQb v 0 \Yj

Z %]YU. Mõ P Z %]Y » [Ycb

5.2.

Let be the Lie algebroid associated to a Jacobi manifold of the dual bundle Section 2) and thus a cross section in such a way that the corresponding is given by

Then, (see can be identified with a pair -linear map (46)

for

Therefore, if is the homomorphism defined by (26) then, is the linear differential operator of order . Using from (10), (33) and (46), we obtain that these facts, (34), (35), (37), (42) and Theorem 10, we deduce a result which has been proved in [19]. Theorem 12 [19] Let be a Jacobi manifold. Then, is quantizable if and only if there exist an integral closed -form , a vector field and a real differentiable function such that: 1.

2. If

is a point of

3. If

is a point of

and

and

then

is a -form at

such that

and

then

From Theorem 12, it follows a result of Vaisman [28]. Corollary 2 [28] Let integral closed -form

be a Poisson manifold. Then, and a vector field such that

is quantizable if and only if there exist an

Finally, for the transitive Jacobi manifolds we obtain the same results that in Section 5.1 (see [19] for more details). Acknowledgement. This work has been partially supported through grants DGICYT (Spain) (Project PB94-0106) and University of La Laguna (Spain). ML wishes to express his gratitude for the hospitality offered to him in the Departamento de Matemática Fundamental (University of La Laguna) where part of this work was conceived.

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M. de León Instituto de Matemáticas y F´ısica Fundamental Consejo Superior de Investigaciones Cient´ıficas 28006 Madrid, Spain [email protected]

J. C. Marrero, E. Padrón Departamento de Matemática Fundamental Universidad de La Laguna La Laguna, Tenerife, Canary Islands, Spain [email protected], [email protected]

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