On the abelianization of the special derivation Lie algebras of free Lie algebras

TL;DR

The paper investigates the abelianization of the special derivation Lie algebra $\mathfrak{b}_n$ associated with the free Lie algebra and its relation to Morita trace maps and Johnson homomorphisms. It develops explicit decompositions and bases for $\mathfrak{b}_n(k)$ for $k\le4$, analyzes trace maps $\mathrm{MT}_k^B$ and Johnson cokernels, and proves that $H_1(\mathfrak{b}_n,\mathbb{Z})$ is not finitely generated by constructing infinite families detected by Morita traces. It also identifies nontrivial kernels of trace maps at degree $5$, establishes partial exact sequences for the abelianization, and shows that the natural Theta map is not injective on $\mathfrak{b}_n$. Overall, the work highlights deep structure and complexity in the abelianization, including infinite generation and delicate interactions with trace maps and braid-related Johnson homomorphisms.

Abstract

In this paper, we show that there are infinitely many linearly independent elements in the abelianization of the Lie algebra of special derivations of a free Lie algebra by using the Morita traces. Furthermore, we show that the abelianization contains non-trivial elements which are killed by the Morita traces.

Theorems & Definitions (35)

• Proposition 1: $=$ Propositions \ref{['T-John-1']} and \ref{['T-John-2']}
• Theorem 1: $=$ Theorem \ref{['T-im-mtr']}
• Theorem 2
• Definition 3.1
• Lemma 3.2
• Definition 3.3
• Theorem 4.1: Andreadakis And
• Theorem 5.1: McCool McC
• Lemma 5.2
• Lemma 7.1