M. De Wilde

Local cohomology of the algebra of C∞ functions on a connected manifold

A multilinear version of Peetres theorem on local operators is the key to prove the equality between the local and differentiable Hochschild cohomology on the one hand, and on the other hand the equality between the second and third local Chevalley cohomology groups and their differentiable counterpart.

Formal Deformations of the Poisson Lie Algebra of a Symplectic Manifold and Star-Products. Existence, Equivalence, Derivations

The purpose of the paper is to present a proof of the existence theorem for star-products and formal deformations of the Poisson Lie algebra on a symplectic manifold. The algebraic formalism of formal deformations and the cohomological background needed are described. The method used in the proof of the existence theorem leads to the description of the algebra of derivations of a star-product and, for a non compact manifold, to that of a formal deformation of the Poisson Lie algebra. Graded cohomology is used to classify all 1-differential deformations of the Poisson Lie algebra.

Star-products on cotangent bundles

We prove the existence of a *-product on each cotangent bundle, by introducing homogeneity conditions which provide a way of getting rid of the cohomological obstructions usually encountered.

Some Characterizations of Differential Operators on Vector Bundles

Let A be a family of differential operators acting on a vector bundle E, and let T be a linear map of the space of smooth sections of E into itself. This paper describes conditions on A such that, if ad (T)T′ = T ° T′ − T′ ° T is a differential operator whenever T′ ∈ A, then T itself is a differential operator. The order of T is moreover related to the best upper bound of the order of ad (T)T′ (T′ ∈ A). Examples are given in the case of tensor bundles.

Invariant star-products on symplectic manifolds

Abstract Let (M,F) be a symplectic manifold and consider a Lie subalgebra G of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to G , extends to an invariant one-differentiable deformation of infinite order. If M admits a G -invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a G - -invariant linear connection, there always exists a G -invariant star-product.

Local Chevalley Cohomologies of the Dynamical Lie Algebra of a Symplectic Manifold

Let (M,F) be a connected symplectic manifold. We denote by N the space of all smooth functions of M, equipped with the Poisson bracket, by L (resp. L⋆) the space of all locally (resp. globally) hamiltonian vector fields on M, equipped with the Lie bracket. Recall that, if (G,[,]) is a Lie algebra and (F,ρ) a representation of G, the corresponding Chevalley cohomology H(G,ρ) is the cohomology of the complex

Existence of star-products revisited

... \to { \wedge^p}\left( {G,F} \right) \to { \wedge^{p + 1}}\left( {G,F} \right) \to ...

Derivations of the Lie algebra structure of the enveloping algebra of a semi-simple Lie algebra

where Λp(G,F) is the space of p-linear alternating maps from G ×...× G into F and