We review a Laplace operator on the cochain complex of the Lie algebra of the formal Poisson vector fields to obtain an analogy of the Hodge theorem for its cohomology. We present intermediate results of the computation on the plane.
"The Cohomology of the Lie Algebras of Formal Poisson Vector Fields and Laplace Operators." Tokyo J. Math. 32 (1) 105 - 111, June 2009. https://doi.org/10.3836/tjm/1249648412