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

TL;DR

This paper computes the relative Gel'fand-Kalinin-Fuks cohomology groups of formal Hamiltonian vector fields on R^4 for weights 2, 4, and 6, revealing specific Betti numbers and degrees.

Contribution

It extends previous work on R^2 to R^4, providing explicit calculations of cohomology groups for certain weights using computational methods.

Findings

Weight 2 cohomology has Betti number 1 at degree 2.

Weight 4 cohomology has Betti number 2 at degree 4.

Weight 6 cohomology is trivial across all degrees.

Abstract

In this paper, we investigate the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4. In the case of formal Hamiltonian vector fields on 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 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.