# The primitive filtration of the Leibniz complex

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 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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03358358
Contributor : Geoffrey Powell Connect in order to contact the contributor
Submitted on : Wednesday, September 29, 2021 - 12:22:31 PM
Last modification on : Saturday, January 29, 2022 - 3:34:43 AM

### Identifiers

• HAL Id : hal-03358358, version 1
• ARXIV : 2109.10766

### Citation

Geoffrey Powell. The primitive filtration of the Leibniz complex. 2021. ⟨hal-03358358⟩

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