An affirmative answer to a conjecture for Metoki class

In "The {G}el'fand-{K}alinin-{F}uks class and characteristic classes of transversely symplectic foliations" arXiv:0910.3414, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in \ds\HGF{7}{2}{}{8} is decomposed as a product $\eta\wedge \omega$ of some leaf cohomology class $\eta$ and a transverse symplectic class $\omega$. We show that the same formula holds for Metoki class, which is a non-trivial element in \ds \HGF{9}{2}{}{14}. The result was conjectured by Kotschick and Morita, where they studied characteristic classes of symplectic foliations due to Kontsevich. Our proof depends on Groebner Basis theory using computer calculations.Comment: 11 plain text files which are output of Maple calculations and also raw materials. These are stored subdirectory anc as ancillary files. You can see the file size on appendice

Another proof to Kotschick-Morita's Theorem of Kontsevich homomorphism

In \cite{KOT:MORITA}, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in \ds \HGF{7}{2}{}{8} is decomposed as a product $\eta\wedge \omega$ of some leaf cohomology class $\eta$ and a transverse symplectic class $\omega$. In other words, the Kontsevich homomorphism \ds\omega\wedge :\HGF{5}{2}{0}{10} \rightarrow\HGF{7}{2}{}{8} is isomorphic. In this paper, we give proof for the Kotschick and Morita's theorem by using the Gr\"obner Basis theory and computer symbol calculations

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

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.Comment: 133 page

Dual Lie algebras of Heisenberg Poisson Lie groups

In this note, we shall classify all the dual Lie algebra structures induced by multiplicative Poisson tensors on an arbirary dimensional Heisenberg Lie group