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

TL;DR

The paper determines the second homology $H_2$ of the positive-weight part $\mathfrak{c}_g^{+}$ of Kontsevich's commutative symplectic derivation Lie algebra. It exploits the weight grading and $\mathrm{Sp}(2g;\mathbb{Q})$ representation theory, particularly the decomposition into highest-weight modules $V_\lambda$ and explicit detection of highest-weight vectors. The main result is the vanishing of weight-$w$ contributions to $H_2(\mathfrak{c}_g^{+})$ for $g,w\ge4$, while weights 1–3 are computed, yielding $H_2(\mathfrak{c}_g^{+})\cong [51]+[33]+[22]+[11]+[1]+[0]$ for $g\ge4$. This advances understanding of the stable homology of Kontsevich's commutative Lie algebra and its connection to commutative graph homology, with explicit boundary-construction techniques using $\omega_3\in\wedge^3\mathfrak{c}_g^{+}$.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 2.1
• Proposition 2.1
• Proof
• Theorem 3.1
• Lemma 3.1
• Proof
• Lemma 4.1
• Proof
• Lemma 4.2