Cohomology of the Lie Algebra $\mathfrak{H}_2$: Experimental Results and Conjectures

The cohomology with trivial coefficients of the Lie algebra $\mathfrak{H}$ of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra $L_1\mathfrak{H}$ is considered. The cohomology $H^2(L_1\mathfrak{H})$ is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.