An Abelian Quotient of the Symplectic Derivation Lie Algebra of the Free Lie Algebra

ABSTRACT

We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of . More specifically, we show that the weight 12 part of the abelianization of is 1-dimensional for g ⩾ 8. The computation is done with the aid of computers.

KEYWORDS:

• symplectic derivations
• outer automorphism group
• free group

2000 AMS SUBJECT CLASSIFICATION:

• Primary 20F28
• Secondary 20J06, 17B40

Acknowledgment

The authors would like to thank Gwénaël Massuyeau for helpful comments on the first draft of this article.

Funding

The authors were partially supported by KAKENHI (No. 15H03618, No. 15H03619, No. 16H03931, and No. 16K05159), Japan Society for the Promotion of Science, Japan.