The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra

Abstract

In 1993, Kontsevich introduced the symplectic derivation Lie algebras related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them is a graded algebra, so that its Chevalley-Eilenberg chain complex has another $Z_{\ge 0}$-grading, called weight, than the usual homological degree. We focus on one of the Lie algebras $c_g$, called the “commutative case”, and its positive weight part $c_g^{+}\subset c_g$. The symplectic invariant homology of $c_g^{+}$ is closely related to the commutative graph homology, hence some computational results are obtained from the viewpoint of graph homology theory. On the other hand, the details of the entire homology group $H_{•}\left(c_g^{+}\right)$ are not completely known. We determine $H_2\left(c_g^{+}\right)$ by decomposing it by weight and using the classical representation theory of the symplectic groups.