## Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on $\mathbb R^4$

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 4, Page 699–716.

Mikami, Kentaro ; Nakae, Yasuharu
Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on $\mathbb R^4$
In this paper, we investigate the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on $\displaystyle \mathbb R^{4}$. In the case of formal Hamiltonian vector fields on $\displaystyle \mathbb R^{2}$, we computed the relative Gel'fand-Kalinin-Fuks cohomology groups of weight $<20$ in the paper by Mikami-Nakae-Kodama. The main strategy there was decomposing the Gel'fand-Fucks cochain complex into irreducible factors and picking up the trivial representations and their concrete bases, and ours is essentially the same. By computer calculation, we determine the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on $\displaystyle \mathbb R^4$ of weights $2$, $4$ and $6$. In the case of weight $2$, the Betti number of the cohomology group is equal to $1$ at degree $2$ and is $0$ at any other degree. In weight $4$, the Betti number is $2$ at degree $4$ and is $0$ at any other degree, and in weight $6$, the Betti number is $0$ at any degree.