Isomorphisms of the Jacobi and Poisson Brackets

We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as for example those given by Poisson or contact structures, but we consider degenerate structures as well. Introduction We shall admit different classes of smoothness, so by a manifold of class C, where C = C∞ , C = C , or C = H, and by the algebra C(M) of class C functions on M we shall mean a) a real paracompact finite-dimensional smooth manifold and the algebra C∞(M) of all real smooth functions on M , if C = C∞ ; b) a real-analytic paracompact finite-dimensional manifold and the algebra C(M) of all real-analytic functions on M , if C = C ; c) a complex finite-dimensional manifold for which each connected component is Stein and the (complex) algebra of all holomorphic functions on M , if C = H. A Jacobi structure on M is a pair (Ω, D) consisting of a vector field D ∈ Γ(TM) and an antisymmetric bi-vector field Ω ∈ Γ(ΛTM) of class C satisfying: i) [D,Ω] = 0; ii) [Ω,Ω] = 2D ∧ Ω , where [·, ·] stands for the Schouten bracket. If D = 0, then [Ω,Ω] = 0 and we call Ω the Poisson structure. Note that in the case of C = H vector fields are holomorphic and of type (1, 0) , since we shall understand vector fields as derivations of the algebra C(M). Every Jacobi structure induces a Lie bracket {·, ·} on the algebra C(M), called the Jacobi (resp. Poisson) bracket, by (∗) {f, g} = Ω(f, g) + fD(g)− gD(f). 1991 Mathematics Subject Classification. 17B65, 17B66, 58F05.