The primitive filtration of the Leibniz complex

TL;DR

This paper proves that the primitive filtration of the Leibniz complex forms an increasing, exhaustive filtration, confirming a conjecture by Loday, and explores the resulting spectral sequence and $L_$-structure.

Contribution

It generalizes Pirashvili's subcomplex result to the primitive filtration, establishing a conjecture by Loday and analyzing the homology and $L_$-structure of the complex.

Findings

The primitive filtration provides an increasing, exhaustive filtration of the Leibniz complex.

The spectral sequence confirms that the homology vanishes in all degrees except one for free Leibniz algebras.

The desuspension of the Pirashvili complex admits a natural $L_$-structure.

Abstract

Pirashvili exhibited a small subcomplex of the Leibniz complex $(T(s \mathfrak{g}), d_{\mathrm{Leib}})$ of a Leibniz algebra $\mathfrak{g}$. The main result of this paper generalizes this result to show that the primitive filtration of $T(s\mathfrak{g})$ provides an increasing, exhaustive filtration of the Leibniz complex by subcomplexes, thus establishing a conjecture due to Loday. The associated spectral sequence is used to give a new proof of Pirashvili's conjecture that, when $\mathfrak{g}$ is a free Leibniz algebra, the homology of the Pirashvili complex is zero except in degree one. This result is then used to show that the desuspension of the Pirashvili complex carries a natural $L_\infty$-structure that induces the natural Lie algebra structure on the homology of the complex in degree zero.