On the combinatorics of the graph-complex.
In [The Gelfand Mathematical Seminars 1990--1992 (ed. L.~Corwin et al.), Birkhäuser, 173--187 (1993; Zbl 0821.58018)], \textit{M. Kontsevich} introduced a certain graph-complex whose cohomology coincides with the cohomology of the Lie algebra of Hamiltonian vector fields (on the even space). This construction could be extended to the Lie (super)algebra of Hamiltonian vector fields on the odd space, yielding another graph-complex (it seems that the preprint of Kontsevich and Shoikhet the author is referring to, does not exist, at least at the time of writing of this review). In this paper, it is shown, in a purely combinatorical way, that the cohomology of these two graph-complexes coincide, which yieds coincidence of cohomology of corresponding Lie (super)algebras. It would be interesting to find a direct proof of the latter fact not appealing to graph-complexes, and, according to Feigin (quoted by the author), this seems to be a difficult question.
Lie algebra of Hamiltonian vector fields
scientific article; zbMATH DE number 1744390
scientific article