Differential cohomology of C ∞ (a ∗)

Abstract

Let G be a finite dimensional real Lie algebra and G ∗ its dual. G ∗ is a Poisson manifold. Thus the space C ∞( G ∗) of C ∞ functions on G ∗ has an associative and a Lie algebra structure. The problem of formal deformations of such a structure needs the determination of some cohomology groups of C ∞( G ∗), considered as a module on itself for left multiplication or adjoint representation. We determine here these groups. The result is very similar to the case of C ∞(W), where W is a symplectic manifold except for the Lie algebras h r × R m, direct products of Heisenberg and abelian Lie algebras.