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 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.

21 pages: revised version, following suggestions of a referee (accepted). (v1: 18 pages.)