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

TL;DR

This paper computes the second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra, linking algebraic and graph homology theories.

Contribution

It provides the first detailed calculation of H_2 for the commutative symplectic derivation Lie algebra using representation theory.

Findings

Determined H_2 of the algebra for various g values.

Connected homology results to graph homology.

Enhanced understanding of the algebra's structure.

Abstract

The symplectic derivation Lie algebras defined by Kontsevich are related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them and its Chevalley-Eilenberg chain complex have a $\mathbb{Z}_{\geq 0}$-grading called weight. We consider one of them $\mathfrak{c}_g$, called the "commutative case", and its positive weight part $\mathfrak{c}_g^{+} \subset \mathfrak{c}_g$. The symplectic invariant homology of $\mathfrak{c}_g^{+}$ is closely related to the commutative graph homology, hence there are some computational results from the viewpoint of graph homology theory. However, the entire homology group $H_\bullet (\mathfrak{c}_g^{+})$ is not known well. We determined $H_2 (\mathfrak{c}_g^{+})$ by using classical representation theory of $\mathrm{Sp}(2g; \mathbb{Q})$ and the decomposition by weight.