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

TL;DR

This paper investigates the structure of the abelianization of the Lie algebra of special derivations of free Lie algebras, revealing infinite independent elements and non-trivial elements nullified by Morita traces.

Contribution

It demonstrates the existence of infinitely many linearly independent elements and non-trivial elements in the abelianization, using Morita traces.

Findings

Infinite linearly independent elements in the abelianization.

Existence of non-trivial elements killed by Morita traces.

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.