Cohomologies of Lie Algebras of Vector Fields with Coefficients In Adjoint Representations Foliated Case

Abstract

Let M be a smooth manifold, and ^l(M) the infinite dimensional Lie algebra of all smooth vector fields on M. Let 91 be 9l(M) or a certain natural subalgebra of it. We arc interested in the cohomology #*(9l; V) of $1 with coefficients in some representation V9 which is an invariant of the Lie algebra 21. In 1968, I. M. Gel'fand and D. B. Fuks began to study the theory of cohomologies of Lie algebras of vector fields. First, they treated the case where 2I = 2l(M) and F=M (trivial coefficients). Since then, many mathematicians studied cohomologies in many cases, for instance [2], [4], [6] etc. They also treated the case of nontrivial coefficients, but restricted themselves to the representations induced from some finite dimensional ones. Their proofs were essentially based upon some finiteness of representations. Meanwhile, in 1973, F. Takens [7] proved that any derivations of 2I(M) is inner. It means that the first cohomology of 2I(M) with coefficients in its adjoint representation, a natural infinite dimensional representation, is trivial. In the present paper, the author treats a symplectic manifold (M, o>) and the subalgebra 210,(M) consisting of hamiltonian vector fields on M in this direction. Then he obtains the following results.