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

Abstract

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

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

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

$$ \partial C\left( {{u_o}...{u_p}} \right) = \sum\limits_{i \leqslant p} {{{\left( { - 1} \right)}^i}\rho \left( {{u_i}} \right)C\left( {{u_o},...{{\hat u}_i}...,{u_p}} \right) + \sum\limits_{i < j} {{{\left( { - 1} \right)}^{i + j}}C\left( {\left[ {{u_i},{u_j}} \right],{u_o},...{{\hat u}_i}...{{\hat u}_j}...,{u_p}} \right)} } $$

where $\hat X$ means that X is omitted. We restrict here our attention to the above mentionned Lie algebras and their adjoint representation. A further restriction consists in taking not all p-cochains C but only the local ones (By Peetre’ theorem (2), these are precisely the multilinear differential operators). The corresponding cohomology spaces are denoted H_loc (N), H_loc (L), H_loc (L^⋆).