Cohomologies of Lie Algebras of Vector Fields with Coefficients in Adjoint Representations Case of Classical Type

Abstract

Let M be a smooth manifold, and 2l(M) the Lie algebra of all smooth vector fields on M. Assume that M admits a volume form T, a symplectic form CD or a contact form 9. Then we have natural Lie subalgebras of 2l(M) as 21T(M), 2l;(M), 9IW(M), 2l;o(M), 210(M) (see §1.1). These Lie algebras including 2l(M) itself are called of classical type. Here we are interested in the cohomology ff*(2l; 21) of the Lie algebra 21 with coefficients in its adjoint representation. Calculations of them are not easy in general. But the first cohomology can be calculated rather easily since H *(2J; 21) is interpreted in terms of derivations of 21. From this point of view F. Takens [5] calculated ff ^(M); $l(M)) in 1973. Later A. Avez-A. LichnerowiczA. Diaz-Miranda [2] and the author [3] calculated H\Wm(M) 2IW(M)) of Lie algebra 2IW(M) of hamiltonian vector fields by different methods. In the present paper, we will calculate H(2l; 21) for all 21 of classical type. Our results can be summarized as follows.