An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra

TL;DR

This paper constructs a specific abelian quotient of the symplectic derivation Lie algebra associated with a free Lie algebra, revealing a one-dimensional structure at weight 12 for sufficiently large genus, using computational methods.

Contribution

It explicitly identifies a one-dimensional abelian quotient of the symplectic derivation Lie algebra at weight 12 for genus at least 8, advancing understanding of its structure.

Findings

The weight 12 part of the abelianization is 1-dimensional for g ≥ 8.

Computational methods were used to verify the structure.

Provides new insights into the algebraic structure of symplectic derivations.

Abstract

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g \ge 8$. The computation is done with the aid of computers.