La cohomologie de Lichnerowicz-Jacobi pour les variétés de Jacobi

C. R. Acad. Sci. Paris, t. 324, !%rie I, p. 71-76, 1997 GbomCtrie diff~rentielle/Differential Geometry

Lichnerowicz-Jacobi of Jacobi manifolds

cohomology

Manuel de LEON, Juan C. MARRERO and Edith PADRON J. C. M. and E. P.: Departamento de Matemitica Fundamental, Facultad de Matemhticas, Universidad de La Laguna, 38200 La Laguna, Tenerife, Canary Islands, Spain E-mail: [email protected], [email protected] M. de L.: Instituto de Matemhticas Consejo Superior de Investigaciones E-mail: [email protected]

Abstract.

y Fisica Fundamental, Cientificas, Serrano 123, 28006 Madrid, Spain.

A cohomology for Jacobi manifolds which generalizes cohomology for Poisson manifolds is constructed.

La cohomologie pour

RCsumC.

the Lichnerowicz-Poisson

de Lichnerowicz-Jacobi

les vari&s

de Jacobi

On dtfjinit une cohomologie pour les vari.&s de Jacobi qui gPn&alise la cohomologie de Lichnerowicz-Poisson pour les vari&tTsde Poisson.

Version frangaise abr>!e Soit (M, A, E) une variete de Jacobi. On d&nit l’operateur differentiel u : Vp(A4) --f VP+‘(M), P]+pEAP, oii V’(M) est l’espace des r-vecteurs de M. On montre que L~og = CTO~E et a2(P) = -CEP A A. Alors, on obtient un complexe (Vi(M) = &V;(M), (~g), oti V;(M) est l’espace des p-vecteurs vCrifiant LEP = 0 et CT~est la restriction de cr. Les groupes de cohomologie de ce complexe sont appek les groupes de cohomologie de Lichnerowicz-Jacobi de la variete de Jacobi M. On denote le groupe d’ordre p par ITi, (M). s’etend en un L’homomorphisme # : O1(M) --f V1(M) defini par #((Y)(P) = A(o,p) + HiJM), homomorphisme # : RP(M) + V’(M) qui induit un homomorphisme #B : H;(M) oti iY$ (M) est le groupe de cohomologie basique. x Le couplage nature1 ( , ) : P(M) x VP(M) --$ C”(M) induit un couplage ( , ) : H,“““(M) H;,(M) + H,““” (M) entre la cohomologie de Lichnerowicz-Jacobi et l’homologie canonique de M introduite darts [4], [5]. (T(P) = -[A,

Note@sent& par Charles-Michel

MARLE.

0764~4442/97/0324007 1 0 AcadCmie des Sciences/Elsevier, Paris

71

M. de Leh,

J. C. Marrero and E. Padr6n

Finalement, nous Ctudions le cas particulier des vat-i&es de contact. Si (M, r)) est une vat%% de contact, nous montrons que i, 0 (T = -u 0 i,. Alors, nous obtenons un sous-complexe (V&(M) = = {P E V(M)/L

, oh V&JAI) et on,, est la restriction de (Tg. La cobomologie de ce sous-complexe est notee H&,(M). Ici, < est le champ de vecteurs de Reeb de la varitte de contact. On deduit des isomorphismes HI,(M) g HI,,(M) $ HI;:(M), HI,,(M) Z H:(M) et HIJ(M) Z H:(M) 63 HL-l(M). En utilisant un resultat dans [4], [5] on obtient aussi les isomorphismes HI,,(M) Z Hi&(M) et H;,(M) g H;:;,_,(M) e&t”_, (M), si 2n + 1 est la dimension de M.

1. Jacobi and Poisson manifolds A Jacobi structure on a m-dimensional manifold M is a pair (A, E) where A is a skew-symmetric tensor field of type (2,0) and E a vector field on M satisfying the following properties: [Ah] = 2E AA,

(I)

[E,A] =O.

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

(2)

if,

91= A(@,dg)+ fE(g) - g-W).

The space C”(M, R) of C” real-valued functions on M endowed with the Jacobi bracket is a local Lie algebra in the sense of Kirillov (see [9]). Conversely, a structure of local Lie algebra on Cm(M, W) defines a Jacobi structure on M (see [8], [9]). If the vector field E vanishes then { , } is a derivation in each argument and, therefore, { , } defines a Poisson bracket on M. In this case, (1) reduces to [A, A] = 0 and (M, A) is a Poisson manifold. We notice that Poisson and Jacobi manifolds were introduced by Lichnerowicz ([13], [14], see also [l], [12], [15]). Now, let (a, A) be a Poisson manifold. We can define the contravariant exterior derivative d : U”(A2) + VP+‘(A?) by a(P) = -[ A, P], where V-‘(M) denotes the space of p-vectors on ti. Since 8’ = 0, (T defines a cohomology on &f which is called the Lichnerowicz-Poisson (LP for simplicity) cohomology for the Poisson manifold i@ (see [ 131). The pth LP cohomology group is denoted by H:,(G).

2. Lichnerowicz-Jacobi

cohomology

Let (M, A, E) be a Jacobi manifold. Denote by VP(M) the space of p-vectors on M. We define the differential operator u : VP(M) t V*+l (M) as follows:

g(P) = --[A, P] + pE A P,

(3)

for

P E VP(M).

We remark that if M is a Poisson manifold, 0 is just the contravariant exterior derivative which defines the LP cohomology. From a straightforward computation, using (1) and (3), we conclude: ~OF’OSITION

2.1: L~Oo=~OL~,

(4) (5)

72

r?(P)

= -LEP

AA

for

P E V*(M),

Lichnerowicz-Jacobi cohomology of Jacobi manifolds

a(P A Q) = o(P) A Q + (-l)pP

(6)

A a(Q)

for

P E VP(M) and Q E VP(M),

where GE denotes the Lie derivative with respect to E. Denote by VL (M) the subspace of VP(M) given by V;(M)

= {P E VP(M)/LEP

= [E, P] = 0).

If Cg(M, R) = {f E C”(M, R)/E(f) = 0}, is the space of basic functions on M, i.e., then Vg (M) is a Cg (M, W)-module. From proposition 2.1, it follows COROLLARY 2.2. This

IfP E V;(M), then CT(P) E V:“(M) and c”(P) = 0. result allows us to introduce the differential complex ... -

V;-‘(M)

%

V;(M)

=+ V;+‘(M)

+

...

where cB = ‘TIV;;(M) and

the Lichnerowicz-Jacobi is then given by

Vi(M) = epVg(M). This complex defines a cohomology which is called (LJ for simplicity) cohomology of (M, A, E). The pth LJ cohomology group

Cr W =

ker{an : V:(M)

+ V:+‘(M)}

Im{aB : V;-‘(M)

+ V;(M)}

’

Notice that on(h) = (TB(E) = 0 and thus A and E define cohomology classes in Hi,(M) and I&(M), respectively. We also remark that (6) implies that A induces an associative product in K@f) = @P%(M). Let (M, A, E) be a Jacobi manifold. Define a mapping # : R1 (M) + X(M) from the space of l-forms R1( M) on M onto the Lie algebra X(M) of the vector fields on M as follows:

(#W> = A(a,P>,.

(7)

for CX,p E a’(M). The mapping # can be extended to a mapping, which we also denote by #, from the space of p-forms W(M) on M onto the space of p-vectors VP(M) by putting: (8)

#(.f) = f,

#(a)(%,

* * * 7Qp> =

(-1>“Q(#Q1,

*. . , #a,>,

for f E

C”(M,W), Q E W(M) and cxl,. . . ,ap E a’(M). Using (l), (3), (7) and (8) we have the following results (see [ll] for a proof): ~OFQSITION2.3.

- If (Y E W(M)

(9)

and d is the exterior diflerential, then

= #(‘b),

LE(#a)

~(#a) = -#da

(10)

+ #(iE~)

A

A.

Let (M, A, E) be a Jacobi manifold. A p-form (Yon M is said to be basic if iEo = 0 and CE~ = 0. Next, we consider the subcomplex of the de Rham complex given by the basic forms ... -

S-lp,-‘(M) 2

R;(M)

++ n;+‘(M)

---) . . .

where 0%(M) is the space of basic p-forms, and d B = din;(M). Its cohomology is denoted by H;(M) and called the basic de Rham cohomology of (M, A, E) (see [4], [5]). From (9), we deduce 73

M. de Leh,

J. C. Marrero and E. Padrdn

that if a E R:(M) then #a: E V:(M). Thus, the mapping # : R*(M) --) V*(M) induces a homomorphism #B : R>(M) -+ Vi(M) of Cg(M, W)-modules. Moreover, using (lo), we obtain:

(11)

oB”#B=-#BOdB

and, therefore: THEOREM 2.4. - Let (M, A, E) be a Jacobi manifold. Then the mapping #B : R>(M) induces a homomorphism in cohomology

4

V&(M)

Let 6 : W(M) 4 nP-‘(M) be the differential operator given by the commutator of i(A) and the exterior differential, that is,

6 = i(A) o d - do i(h).

(13)

Notice that if M is a Poisson manifold, then S is just the Koszul operator (see [3], [lo]). Denote by SB the restriction of 6 to o:(M). It is proved (see [4], [5]) that fiB(fl%(kf)) & fiL-‘(h/i) and 6: = 0. These results allow us to introduce the differential complex (nk(M), SB), which is called the canonical complex of M. The homology of this complex is denoted by H,“&“(M), and it is called the canonical homology of 1M (we refer to [4], [5] for a more detailed study). We consider the natural pairing ( , ) : Rp(M) x VP(IM) 4 C” (M, R) defined by (a, P) = i(P)&

(14)

where i(P) denotes the contraction by P. Using (3), (13), (14) and the fact that [[i(P),d],i(Q)] F’ROFWSITION 2.5. - Ifa

E R;(M)

and P E V;(M),

= i([P, Q]) we deduce then (a, P) E Cg(M,

(a, OB(&)) - (fiBa> Q) =

R). Moreover,

-~B@)(-Y

and Q E VP-‘(M) B * 2.5 it follows

for every Q E flp (M)

From propositioi

THEOREM 2.6. - Let (M, A, E) be a Jacobi manifold. The mapping ( , ) defined in (14) induces a natural pairing ( ) ) : H,“““(M)

x H;,(M)

4

H,“““(M),

given by

3. Lichnerowicz-Jacobi

cohomology of contact manifolds

The basic examples of Jacobi manifolds that are not Poisson manifolds are the contact manifolds and the locally conformal symplectic manifolds [6]. In this last section we study the LJ cohomology of a contact manifold. Let it4 be a (2n+ 1)-dimensional manifold and 7 a l-form on M. We say that 71is a contact l-form if VIA(dq)” # 0 at every point. In such a case, (M, 7) is called a contact manifold (see, for example, [2]). 74

Lichnerowicz-Jacobi cohomology of Jacobi manifolds

A contact manifold (M, 11)is a Jacobi manifold. In fact, we define the skew-symmetric tensor field A of type GO) on M by putting

A(a, P) = W%),

(15)

b-1W,

for any cr,p E o’(M), where b : X(M) + R1 (M) is the isomorphism of C” (M, W)-modules given by b(X) = ixdv + q(X)q. The vector field E is just the Reeb vector field 6 = b-l(q) of (M,q). We remark that i~q = 1, icdq = 0. The mapping b can be extended to a mapping from the space VP(M) of p-vectors into the space of p-forms W(M) by putting b(X1 A.. . A X,) = b(X1) A . . . A b(X,). Thus, b is also an isomorphism of C”(M, R)-modules. Using (7), (8) and (15) we prove: PROFWITION3.1. - Let (M, 7) be a contact manifold with Reeb vector field <. Then for every a E W(M), we have #o = (-l)Pbwl(a) + 5 A #(ita). Notice that if (M,q) is a contact manifold and X E X(M), then i,L~xh = -itx17A = -#LxQ. On the other hand, b(X) = 12x77 - d(q(X)) +v(X)v. In particular, if q(X) = 0 then X = ~-‘(Lcx~]). Using these facts, (3) (6), (15) and proposition 3.1, we prove PROPOSITION 3.2. - Let (M,v) be a contact manifold. Then i, o u = -u o in. Proposition 3.2 allows us to consider the following subcomplex of the LJ complex (U;(M),

where V&(M)

on)

is the subspace of Vg (M) given by

U;,(M)

= {P E V;(M)/i,P

= 0} = {P E Vp(M)/L$’

= O&P = 0)

and on,, = CTIV$,(M). We denote the cohomology of this subcomplex by Hi,,(M) the 77-U cohomology of (M, r]).

which is called

THEOREM 3.3.-Let (M,Q) be a contact manifold. Then there is an isomorphism HI,(M)

= HZ,,(M)

@ HI;,:(M).

Proof. - We consider the homomorphism,

:

(%(M),dd + (G,(M),

-B&

Consequently, using theorem 3.3, we obtain: THEOREM 3.4. - Let (M, q) be a contact manifold. Then the mapping #n : 0:(M) + V&(M) induces an isomorphism between the r&J cohomology group HI,,(M) and the basic de Rham cohomology group H:(M), f or every p. Thus, there is an isomorphism H;,(M)

x H;(M)

~3H;-l(M). 75

M. de Leh,

J. C. Marrero and E. Padr6n

In [5] (see also [4]) the authors have proved that if (M, 17)is a (272+ 1)-dimensional contact manifold, then the canonical homology group H,“““(M) is isomorphic to the basic de Rham cohomology group HgVP(M). Using this fact and theorem 3.4 we deduce the following result: COROLLARY 3.5. - Let (M, 71) be a contact manifold of dimension 2n + 1. Then the pth r]-LJ cohomology group HI,,(M) is isomorphic to the (2n - p) th canonical homology group Hgzp (M) . Therefore, H;,(M)

= H;“,,_,(M)

CDH;&(M).

Acknowledgements. This work has been partially supported through grants DGICYT (Spain), Project PB94-0106 and Consejeria de Educaci6n de1 Gobiemo de Canarias-Spain.

Note remise le 9 septembre 1996, acceptke le 30 septembre 1996.

References [1] Bhaskara K. H. and VIswanath K., 1988. Poisson algebras and Poisson manifolds, Res. Notes Math., 174, Pitman, London. [2] Blair D. E., 1976. Contact manifolds in Riemannian geometry, Lecture Notes in Math., 509, Springer-Verlag, Berlin. [3] Brytinski J. L., 1988. A differential complex for Poisson manifolds, J. Difirenriul Geom., 28, pp. 93-114. [4] Chinea D., de Le6n M. and Marrero J. C., 1996. The canonical double complex for Jacobi manifolds, C. R. Acad. Sci. Paris, 322, Series I, pp. 637-642. [5] Chinea D., de L&n M. and Marrero J. C., 1996. A canonical differential complex for Jacobi manifolds, Preprint, IMAPP-CSIC. [6] Dazord P., Lichnerowicz A. and Marie Ch. M., 1991. Structure locale des varitietbsde Jacobi, J. Mark Pures AppZ., 70, pp. 101-152. [7] Femtidez M., Ibtiez R. and de Le6n M., 1996. Poisson cohomology and canonical homology of Poisson manifolds, Arch. Math. (Bmo), 32, 1, pp. 29-56. [8] Guedira F. and Lichnerowicz A., 1984. GBom&rie des algebres de Lie locales de Kirillov, J. Math. Pures AppZ., 63, pp. 407-484. [9] KlrIIIov A., 1976. Local Lie algebras, Russian Math. Surveys, 31, No. 4, pp. 55-75. [lo] Koszul J. L., 1985. Crochet de Schouten-Nijenhuis et cohomologie, in Elk Cartan et les Math. d’Aujour d’Hui, Asterisque hors-serie, pp. 25 l-27 1. [11] de Le6n M., Marrem J. C. and Padr6n E., 1996. A cohomology differential complex for Jacobi manifolds, Preprint, IMAFF-CSIC. [12] Libermann P. and Marle Ch. M., 1987. Symplectic Geometry nnd Analytical Mechanics, Kluwer, Dordrecht. [13] Lichnerowicz A., 1977. Les vari&% de Poisson et les algebres de Lie associees, J. Diferenfial Geom., 12, pp. 253-300. [14] Lichnerowicz A., 1978. Les vati&% de Jacobi et leurs algbbres de Lie associbes, J. Math. Pures AppZ., 57, pp. 453-488. [ 151 Vaisman I., 1994. Lectures on the Geometry of Poisson Manifolds, Progr. Math. 118, Birkhauser, Basel.

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