a p-multivector eld (or simply a vector eld for p = 1) and a section 2 p (M) a p-dierential form (a Pfa form for p = 1). By convention, for p = 0, we

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1 The Schouten-Nijenhuis bracket and interior products Charles-Michel Marle Au Professeur Andre Lichnerowicz, en temoignage d'admiration et de respect. 1. Introduction The Schouten-Nijenhuis bracket was rst discovered by J. A. Schouten [24] [25] who, with A. Nijenhuis [21], establihed its main properties. A strong renewal of interest in that bracket occurred when A. Lichnerowicz began to consider generalizations of symplectic or contact structures which involve contravariant tensor elds rather than dierential forms. He dened Poisson and Jacobi structures [13, 14] and observed that the Schouten-Nijenhuis bracket allows to put under an intrinsic, coordinate free form the conditions under which a 2-multivector eld on a manifold denes a Poisson structure, as well as the conditions under which a pair (; E) of a 2-multivector eld and a vector eld E denes a Jacobi structure. Properties of the Schouten-Nijenhuis bracket were very actively investigated in recent years [1] [12] [20] [22], as well as its very numerous applications, in particular to Poisson geometry and Poisson cohomology [2] [3] [7] [8] [22] [26] [27] [28], bihamiltonian manifolds and integrable sytems [11] [19], Poisson-Lie groups [5] [16], Lie groupoids [6] [17] [18]. Generalizations of that bracket were considered [4] [9] [10] [15]. For some time, the sign conventions used in the denition of the Schouten- Nijenhuis bracket made by dierent authors were not always in agreement and very often led to rather complicated formulas. J.-L. Koszul [12] introduced in its denition new sign conventions much more natural than the original ones used by Schouten and Nijenhuis, leading to formulas easier to handle. We will use essentially these sign conventions (maybe with a slight change explicitely indicated in Remark 4.2). In this paper, we rst briey review the main already known properties of the Schouten-Nijenhuis bracket. We have tried to introduce these properties as simply as possible, and to state explicitely all the conventions made. Then we prove a formula (new, up to our knowledge) which relates the Schouten-Nijenhuis bracket with the right interior product of multivectors by 1-forms. That formula allows the recursive calculation of the Schouten-Nijenhuis bracket of multivector elds of any degree. 2. Right and left interior products on a manifold 2.1. The graded algebras of multivectors and forms. Let M be a real smooth (C 1 ) manifold of dimension m, T M and T M its tangent and cotangent bundles, respectively. For each integer p 1, we denote by A p (M) and p (M) the spaces of smooth sections, respectively of V p T M and of V p T M (the vector bundles p-th exterior powers of T M and T M, respectively). A section P 2 A p (M) will be called 1

2 a p-multivector eld (or simply a vector eld for p = 1) and a section 2 p (M) a p-dierential form (a Pfa form for p = 1). By convention, for p = 0, we set A 0 (M) = 0 (M) = C 1 (M; R), the algebra of smooth real functions on M. For p < 0, we set A p (M) = p (M) = f 0 g, the null module on C 1 (M; R). Of course, taking into account the skew-symmetry, we have also, for p > m, A p (M) = p (M) = f 0 g. Finally, we set A(M) = p2z A p (M), (M) = p2z p (M). We recall that a p-multivector P 2 A p (M) can be? considered as a map, p-multilinear on the ring C 1 (M; R)), alternate, dened on 1 (M) p, which takes its values in the ring C 1 (M; R). Explcitely, for 1 ; : : : ; p 2 1 (M) and x 2 M, P ( 1 ; : : : ; p )(x) = P (x)? 1 (x); : : : ; p (x) : Similarly, a section 2 p (M) can be considered as a map, p-multilinear on the ring C 1 (M; R), alternate, dened on? A 1 (M) p, with values in C 1 (M; R). Explicitely, for X 1 ; : : : ; X p 2 1 (M) and x 2 M, (X 1 ; : : : ; X p )(x) = (x)? X 1 (x); : : : ; X p (x) : We recall also that A(M) and (M) are Z-graded associative algebras on C 1 (M; R), with the wedge product (denoted by ^) as composition law. For example, the wedge product P ^ Q of P 2 A p (M) and Q 2 A q (M) is dened by the formula, in which the i, 1 i p + q, are Pfa forms, P ^ Q( 1 ; : : : ; p+q ) = X 2S(p;q) ()P ( (1) ; : : : ; (p) ) Q( (p+1) ; : : : ; (p+q) ) : We have denoted by S(p; q) the set of \shue" permutations of f 1; 2; : : : ; p + q g, i.e., permutations which satisfy (1) < (2) < < (p) and (p + 1) < (p + 2) < < (p + q) : In the above formula () is equal to 1 if the permutation is even and to?1 if that permutation is odd The pairing. There is a natural C 1 (M; R)-bilinear map of (M) A(M) into C 1 (M; R), called the pairing, denoted by (; P ) 7! h; P i. Let us recall its denition. If 2 q (M) and P 2 A p (M) with p 6= q, then h; P i = 0. If f 2 0 (M) = C 1 (M; R) and g 2 A 0 (M) = C 1 (M; R), then hf; gi is the ordinary product fg. If 1 ^ ^ p 2 p (M) is a decomposable p-form and X 1 ^ ^ X p 2 A p (M) a decomposable p-multivector eld, (therefore the i are Pfa forms and the X i are vector elds), then h 1 ^ ^ p ; X 1 ^ ^ X p i = det? h i ; X j i : The denition of the pairing extends then to (M)A(M) by bilinearity, in a unique way. For 2 p (M) and X 1 ; : : : ; X p 2 A 1 (M), we have h; X 1 ^ ^ X p i = (X 1 ; : : : ; X p ) : 2

3 Similarly, for P 2 A p (M) and 1 ; : : : ; p 2 1 (M), h 1 ^ ^ p ; P i = P ( 1 ; : : : ; p ) : The pairing is nondegenerate; therefore, a multivector eld (resp., a dierential form) is determined when one knows its pairing with any dierential form (resp., with any multivector eld) Interior products. Let X 2 A 1 (M) be a vector eld. The well known interior product by X, denoted by i(x), is the unique graded C 1 (M; R)-linear endomorphism of (M), of degree?1, such that, for 2 q (M), i(x) is the element of q?1 (M) dened by the formula, in which X 1 ; : : : ; X q?1 2 A 1 (M) are vector elds, i(x)(x 1 ; : : : ; X q?1 ) = (X; X 1 ; : : : ; X q?1 ) : (1) The interior product i(x) by a vector eld X is a derivation of degree?1 of the exterior algebra (M). It means that, for 2 q (M) and 2 (M), More generally, let P 2 A p (M). i(x)( ^ ) = i(x) ^ + (?1) q ^ i(x) : (2) We dene the right interior product by P as the unique C 1 (M; R)-linear endomorphism of (M), of degree?p, such that, for 2 q (M), with q p, i(p ) is the unique element of q?p (M) such that, for each R 2 A q?p (M), i(p ); R = h; P ^ Ri : (3) For f 2 A 0 (M) = C 1 (M; R), i(f) is just the ordinary product by f. For P 2 A p (M), p > 1, i(p ) in general is no more a derivation of (M). For P 2 A p (M) and Q 2 A q (M), we have i(p )i(q) = i(q ^ P ) : (4) Similarly, let 2 1 (M). The left interior product by, denoted by j(), is the unique graded C 1 (M; R)-linear endomorphism of A(M), of degree?1, such that, for Q 2 A q (M), j()q is the unique element of A q?1 (M) dened by the formula, in which 1 ; : : : ; q?1 2 1 (M) are Pfa forms, j()q( 1 ; : : : ; q?1 ) = Q( 1 ; : : : ; q?1 ; ) : (5) Observe that while X appears at the rst place on the right hand side of (1), appears at the last place on the right hand side of (5). The left interior product j() by a Pfa form is less often used in dierential geometry than the (right) interior product by a vector eld, probably because multivector elds are less often used than dierential forms. However, its properties are similar. In particular, it is a \derivation on the right" of degree?1 of the exterior algebra A(M). It means that, for P 2 A(M) and Q 2 A q (M), it satises the formula, which should be compared with (2), j()(p ^ Q) = P ^ j()q + (?1) q j()p ^ Q : (6) 3

4 More generally, let 2 q (M). We dene the left interior product by, denoted by j(), as the unique C 1 (M; R)-linear endomorphism of A(M), of degree?q, such that, for P 2 A p (M), with p q, j()p is the unique element of A p?q (M) which satises, for each 2 p?q (M), ; j()p = h ^ ; P i : (7) That formula should be compared with (3). For f 2 0 (M) = C 1 (M; R), j(f) is just the ordinary product by f. For 2 p (M), p > 1, j() is in general no more a \derivation on the right" of A(M). For 2 p (M) and 2 q (M), we have the formula, which should be compared with (4), j()j() = j( ^ ) : (8) The sign conventions used here in the denitions of the right and left interior products are in agreement with the widely used denition of the interior product by a vector eld. When the degrees of the multivector eld and of the form are equal, the interior products reduce to the pairing, since we have the very natural formulas, for P 2 A p (M) and 2 p (M), i(p ) = j()p = h; P i : (9) 3. The Schouten-Nijenhuis bracket The left and right interior products are \punctual" operations: the value of the right interior product of a dierential form by a multivector eld (resp., the left interior product of a multivector eld by a dierentail form) at a point x of the manifold M depends only the values at x of that dierential form and that multivector eld. Other operations, such that the exterior dierentiation d (for dierential forms), the Lie derivative with respect to a vector eld (for multivector elds as well as dierential forms), and the bracket of a vector eld with another vector eld (which is in fact a particular case of Lie derivative) are not punctual: their values at a point x 2 M depend on the 1-jets at x of the elds under consideration. The Schouten-Nijenhuis bracket is of that type. In fact, it is a natural extension of the Lie derivative of multivector elds with respect to a vector eld. The following proposition states some of its properties which can be used for its denition Proposition. Let M be a smooth real m-dimensional manifold and A(M) the exterior algebra of multivector elds on M. There exists a unique R-bilinear map, dened on A(M) A(M), with values in A(M), called the Schouten-Nijenhuis bracket and denoted by (P; Q) 7! [P; Q], which satises the following properties: Property 1. For f and g 2 A 0 (M) = C 1 (M; R), [f; g] = 0. Property 2. For a vector eld X 2 A 1 (M) and a multivector eld Q 2 A(M), [X; Q] is the Lie derivative L(X)Q of Q with respect to X. Property 3. For P 2 A p (M) and Q 2 A q (M), [P; Q] =?(?1) (p?1)(q?1) [Q; P ] : (10) 4

5 Property 4. For P 2 A p (M), Q 2 A q (M) and R 2 A(M), [P; Q ^ R] = [P; Q] ^ R + (?1) (p?1)q Q ^ [P; R] : (11) Proof. Several full proofs of that proposition can be found in the literature, for example in [27] (with dierent sign conventions), [22] or [12]. We will indicate here only the main ingredients of a very straightforward proof. Property 4 implies that the Schouten-Nijenhuis bracket [P; Q] is local: its values in an open subset of M depend only on the values of the multivector elds P and Q in that open subset. Therefore we may work in the domain of a chart, in which P and Q are nite sums of exterior products of vector elds (or maybe functions, if their degree is 0). Properties 1 to 4 allow to express [P; Q] in the domain of that chart in terms of a nite sum of exterior (or ordinary) products involving functions and Lie derivatives of functions or vector elds with respect to a vector eld. This ensures the unicity. Finally, to prove the existence we have just to prove that when the value of a Schouten-Nijenhuis bracket in the domain of a chart is calculated, using properties 1 to 4, in two dierent ways, the obtained result is the same. 3.2 Remarks. As an easy consequence of properties 1 to 4, the Schouten-Nijenhuis bracket satises the following additional property: Property 5. For P 2 A p (M) and Q 2 A q (M), we have [P; Q] 2 A p+q?1 (M). Properties 4 and 5 show that for a given P 2 A p (M), the map Q 7! [P; Q] is a derivation of degree p? 1 of the exterior algebra A(M). Using properties 3, 4 and 5, we see that the Schouten-Nijenhuis bracket satises: Property 6. For P 2 A(M), Q 2 A q (M) and R 2 A r (M), [P ^ R; Q] = P ^ [R; Q] + (?1) (q?1)r [P; Q] ^ R : (12) Properties 5 and 6 show that, for a given Q 2 A q (M), the map P 7! [P; Q] is a \derivation on the right" of A(M). The next proposition is a natural generalization of the well known fact that the space A 1 (M), with the bracket of vector elds as a composition law, is a Lie algebra Proposition. Let P 2 A p (M), Q 2 A q (M) and R 2 A r (M) be three homogeneous multivector elds on the manifold M. The Schouten-Nijenhuis bracket satises the following identity, called the graded Jacobi identity, (?1) (p?1)(r?1) P; [Q; R] + (?1) (q?1)(p?1) Q; [R; P ] + (?1) (r?1)(q?1) R; [P; Q] = 0 : (13) Proof. We will give only its main lines. First we observe that the formula is satised when the degrees p, q and r are equal to 0 or 1. Then the general result follows by induction on the degrees, using Properties 3 and 4 to replace, for example, P by P ^ X, with X 2 A 1 (M), and therefore p by p

6 3.4. Remarks. 1. Proposition 3.3, together with Properties 3 and 5, states that the graded vector space A(M), with the Schouten-Nijenhuis bracket as composition law, is a graded Lie algebra [23]. In order to have a simple rule for composing the degrees, one has to state that the \Lie degree" of a homogeneous multivector eld P 2 A p (M) (that means, its degree with respect to the Schouten-Nijenhuis bracket as a graded Lie algebra composition law) is p? 1. Of course, the Lie degree of P 2 A p (M) should not be confused with its ordinary degree (also called exterior degree), which is p. The space A 1 (M) of vector elds, which is a Lie algebra in the usual sense, is then the subspace of A(M) of Lie degree As observed by Grabowski [7], the graded Jacobi identity may be written under other forms, in which its meaning is clearer than under the form (13), and which still have a meaning for more general brackets which are not graded-skew-symmetric (i.e, which do not satisfy (10)). Let us set, for each P 2 A p (M) and Q 2 A(M), ad P Q = [P; Q] : (14) Then ad P is a graded linear endomorphism of degree p? 1 of A(M). Using (10), we see that the graded Jacobi identity (13) of Proposition 3.3 can be written as ad P? [Q; R] = [adp Q; R] + (?1) (p?1)(q?1) [Q; ad P R] ; (15) or as ad [P;Q] = ad P ad Q?(?1) (p?1)(q?1) ad Q ad P : (16) Equation (15) has a clear meaning: the graded endomorphism ad P, of degree p?1, is a derivation of the graded Lie algebra A(M) with the Schouten-Nijenhuis bracket as composition law (the degree of Q 2 A q (M) considered here being its Lie degree q? 1 rather than its exterior degree q). Equation (16) means that the endomorphism ad [P;Q] of A(M), of degree p + q? 2, is the graded commutator of the endomorphisms ad P (of degree p? 1) and ad Q (of degree q? 1). The denition of the graded commutator of two graded endomorphisms is recalled at the beginning of the next section for endomorphisms of (M), but the same denition holds for endomorphisms of any graded vector space, in particular for endomorphisms of A(M). 4. Interior product operations on a Schouten-Nijenhuis bracket Let us recall that if and are two graded linear endomorphisms of (M), of degrees ' and respectively, their graded commutator, denoted by [; ], is dened as [; ] =? (?1) ' : (17) The following proposition, due to Koszul [12], indicates a very nice and useful expression for the interior product by a Schouten-Nijenhuis bracket. A special case of that formula appears in the works of Lichnerowicz [13] [14]; in that special case, the interior product by a Schouten-Nijenhuis bracket [P; Q] (with P 2 A p (M) and Q 2 A q (M) is applied to a dierential form of degree p + q? 1, equal to the degree 6

7 of [P; Q]; that interior product is therefore simply the pairing ; [P; Q], and in the right hand side of Equation (18) below, the expression di(q)i(p ) vanishes, since i(q)i(p ) = i(p ^ Q) = 0, the multivector eld P ^ Q being of degree p + q and the dierential form of degree p + q? Proposition. Let P and Q be two multivector elds on the manifold M, and [P; Q] be their Schouten-Nijenhuis bracket. The interior product i? [P; Q]) is expressed, in terms of the exterior dierential d and the interior products i(p ) and i(q), by i? [P; Q] =? [i(q); d]; i(p ) ; (18) where the brackets which appear in the right hand side are graded commutators of graded endomorphisms of (M). Proof. We indicate only its main lines. We observe rst that Equation (18) is satised when P and Q are homogeneous of degrees 0 or 1. Then, by using (10) and (11), and by replacing P by P ^ X, where X is a vector eld, and therefore replacing p by p + 1, the result is obtained, by induction on the degrees, for homogeneous P and Q of all degrees. For P and Q eventually not homogeneous, the general result follows by bilinearity Remarks. 1. In [12], page 266, J.-L. Koszul writes, instead of (18), for a and b 2 A(M), [i(a); d]; i(b) = i? [a; b] : The sign dierences between that formula and (18) is probably due to dierent conventions about the interior product. 2. Let f 2 A 0 (M) = C 1 (M; R) be a smooth function, and P 2 A p (M). By using (18), we easily see that for any 2 p?1 (M), ; [P; f] = h ^ df; P i : (19) By a repeated application of that formula, we see that, for p smooth functions f 1 ; : : : ; f p 2 A 0 (M), and P 2 A p (M), h [P; f1 ]; f 2 ; : : : ; fp i = hdf p ^ df p?1 ^ ^ df 1 ; P i : (20) Let us now indicate a formula which relates the left interior product of a Schouten- Nijenhuis bracket by a Pfa form Proposition. Let P 2 A(M)be a multivector eld, Q 2 A q (M) an homogeneous multivector eld of degree q, and a Pfa form on the smooth manifold M. As in section 2.3, we denote by j() the left interior product by. We have j()[p; Q]? P; j()q? (?1) q?1 j()p; Q = (?1) q?2? j(d)(p ^ Q)? P ^ j(d)q? j(d)p ^ Q : 7 (21)

8 Proof. We may assume, without loss of generality, that P is homogeneous of degree p. Let 2 p+q?2 (M). We have ; j()[p; Q] = ^ ; [P; Q] = i? [P; Q] ( ^ ) : By using (18), we obtain ; j()[p; Q] =? d? i(p ) ; j()q Similarly, and + (?1) (p?1)(q?1) d? i(q) ; j()p? (?1) q?1 d? i? j()p ; Q + (?1) (p?1)q d? i? j()q ; P + (?1) p(q?1) d; j()(q ^ P + (?1) (p?1)q ; j(d)(q ^ P )? (?1) q ; P ^ j(d)q? (?1) (p?1)q ; Q ^ j(d)p : ; j()p; Q =(?1) p(q?1) d? i(q) ; j()p? d? i? j()p ; Q + (?1) (p?1)(q?1) d; Q ^ j()p ; ; P; j()q =? d? i(p ) ; j()q + (?1) (p?1)q d? i? j()q ; P + (?1) p(q?2) d; j()q ^ P : By putting together the three above expressions, we obtain Equation (21) Remarks. 1. When is a closed Pfa form, the right hand side of (21) vanishes, and that equation shows that the left interior product j() is a \derivation on the right" of the graded Lie algebra A(M), with the Schouten-Nijenhuis bracket as composition law. 2. Let f 2 0 (M) = C 1 (M; R) be a smooth function. Then j(f) is simply the ordinary product by f and, instead of (20), we have f[p; Q] = [fp; Q]? P ^ (fq) = [P; fq] + (fp ) ^ Q : (21) 3. In [12], page 264, Equation (2:4), J.-L. Koszul indicates the formula, where D is a dierential operator which generates the Schouten-Nijenhuis bracket (in a sense specied in that paper) and! a dierential form of degree p, Dj(!)? (?1) p j(!)d =?j(d!) ; 8

9 where we have denoted by j(!) the right interior product by!, in order to use the same notations as above (Koszul denotes that interior product by i(!)). That formula seems to be related with Equation (21) and may oer a way to generalize Proposition 4.3 for left interior products of Schouten-Nijenhuis brackets by dierential forms of degree higher than Formula (21) in Proposition 4.3 allows us to obtain an expression of [P; Q] in terms of exterior products and Schouten brackets of multivector elds of degree strictly lower than the degrees of P and Q. Therefore, a repeated use of that formula yields an expression of [P; Q] in terms of exterior products, Lie derivatives of functions with respect to vector elds and brackets of vector elds. References [1] Beltran, J. V., and Monterde, J., Graded Poisson structures on the algebra of dierential forms. Comment. Math. Helv. 70 (1995), 383{402. [2] Bhaskara, K. H., and Viswanath, K., Calculus on Poisson manifolds. Bull. London Math. Soc. 20 (1988), 68{72. [3] Bhaskara, K. H., and Viswanath, K., Poisson Algebras and Poisson Manifolds, Pitman Research Notes in Mathematics 174, Longman Sci., Harlow and New York, [4] Cabras, A., and Vinogradov, A. M., Extensions of the Poisson bracket to dierential forms and multi-vector elds. J. Geom. Phys. 9 (1992), 75{100. [5] Cahen, M., Gutt, S., Ohn, C., and Parker, M., Lie-Poisson groups: remarks and examples. Lett. Math. Phys. 19 (1990), 343{353. [6] Coste, A., Dazord, P., and Weinstein, A., Groupodes symplectiques. Publ. du Departement de Mathematiques de l'universite Claude Bernard (Lyon I) 2/A (1987), 1{65. [7] Grabowski, J., Z-graded extensions of Poisson brackets. Preprint, Mathematical Institute, Polish Academy of Sciences, [8] Huebschmann, J., Poisson cohomology and quantization. J. Reine Angew. Math. 408 (1990), 57{113. [9] Kosmann-Schwarzbach, Y., Grand crochet, crochets de Schouten et cohomologies d'algebres de Lie. C. R. Acad. Sc. Paris, serie I 312 (1991), 123{126. [10] Kosmann-Schwarzbach, Y., Exact Gerstenhaber algebras and Lie bialgebroids. Acta Applicandae Math. 41 (1995), 153{165. [11] Kosmann-Schwarzbach, Y. and Magri, F., Poisson-Nijenhuis structures. Ann. Inst. Henri Poincare, Phys. Theor. 53 (1990), 35{81. [12] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie. In Elie Cartan et les mathematiques d'aujourd'hui. Asterisque, numero hors serie (1985), [13] Lichnerowicz, A., Les varietes de Poisson et leurs algebres de Lie associees. J. Dierential Geometry 12 (1977), 253{300. [14] Lichnerowicz, A., Les varietes de Jacobi et leurs algebres de Lie associees. J. Math. Pures et appl. 57 (1978), 453{488. [15] Liu, Z.-J., Weinstein, A., and Xu, P., Manin triples for Lie bialgebroids. Preprint, Penn. State University, [16] Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations and 9

10 Bruhat decompositions. J. Dierential Geometry 31 (1990), 501{526. [17] Mackenzie, K. C. H., Lie groupoids and Lie algebroids in dierential geometry. London Math. Soc. Lecture Notes Ser. 124, Cambridge University Press, Cambridge, [18] Mackenzie, K. C. H. and Xu, P., Lie bialgebroids and Poisson groupoids. Duke Math. J. 73 (1994), 415{452. [19] Magri, F. and Morosi, C., A geometrical characterization of integrable hamiltonian systems through the theory of Poisson-Nijenhuis manifolds. Universita degli studi di Milano Quaderno S 19 (1984). [20] Michor, P. W., Remarks on the Schouten-Nijenhuis bracket. Suppl. Rend. Circ. Mat. di Palermo, Serie II 16 (1987), 207{215. [21] Nijenhuis, A., Jacobi-type identities for bilinear dierential concomitants of certain tensor elds I, Indagationes Math. 17 (1955), 390{403. [22] Ouzilou, R., Hamiltonian actions on Poisson manifolds, in Symplectic geometry (A. Crumeyrolle and J. Grifone, eds), Research notes in mathematics 80, Pitman, Boston, 1983, 172{183. [23] Roger, C., Algebres de Lie graduees et quantication. In Symplectic Geometry and Mathematical Physics (P. Donato et al., eds), Progress in Math. 99, Birkhauser, Boston, {421. [24] Schouten, J. A., Uber Dierentialkonkomitanten zweier kontravarianter Groen. Indag. Math. 2 (1940), 449{452. [25] Schouten, J. A., On the dierential operators of the rst order in tensor calculus. In Convegno Int. Geom. Di. Italia, Ed. Cremonese, Roma, 1954, 1{7. [26] Vaisman, I., Remarks on the Lichnerowicz-Poisson cohomology. Ann. Inst. Fourier Grenoble 40 (1990), 951{963. [27] Vaisman, I., Lectures on the geometry of Poisson manifolds. Birkhauser, Boston, [28] Xu, P., Poisson cohomology of regular Poisson manifolds. Ann. Inst. Fourier Grenoble 42 (1992), 967{988. Charles-Michel Marle Universite Pierre et Marie Curie, Institut de Mathematiques, 4, place Jussieu, Paris cedex 05, France. marle@math.jussieu.fr 10