The primitive filtration of the Leibniz complex

Abstract

Pirashvili exhibited a small subcomplex of the Leibniz complex $$(T(s {\mathfrak {g}}), d_{\textrm{Leib}})$$ 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.