Marc De Wilde

Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds

We prove the existence of star-products and of formal deformations of the Poisson Lie algebra of an arbitrary symplectic manifold. Moreover, all the obstructions encountered in the step-wise construction of formal deformations are vanishing.

On the local Chevalley cohomology of the dynamical Lie algebra of a symplectic manifold

We describe the relations between the local Chevalley cohomologies related to the adjoint representation of the Poisson Lie algebra of a symplectic manifold and the Lie algebras of all symplectic or globally Hamiltonian vector fields of the manifold. The proofs are based on the computation of the cohomology of the complex (E, ∂), where E is the space of multilinear local maps from a vector bundle of a manifold M into the space of forms on M and ∂L=d ∘ L.

Inductive limits and partitions of unity

Let E be the inductive limit of a family of locally convex topological vector spaces El. We introduce the concept of partitions of unity in E, which extends the usual partitions of unity in function spaces. Inductive limits admitting such a partition of unity have many interesting properties, similar to those of the strict inductive limits.They also have rather unusual properties concerning duals and spaces of mappings, which seem to be unknown, even in spaces likeD(Ω) andE(Ω).Some examples of such inductive limits are quoted at the end of the paper.

Barrelled spaces with a B-complete completion

Let (E, η) be a barrelled locally convex space. The aim of this paper is to describe the barrelled topologies on E weaker than η. When the completion Êη of (E, η) is B-complete, various properties of the barrelled topologies weaker than η are proved. Some examples are given to illustrate the possible situations.