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Journal Articles Manuscripta mathematica Year : 2022

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.

Dates and versions

hal-03358358 , version 1 (29-09-2021)

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Geoffrey Powell. The primitive filtration of the Leibniz complex. Manuscripta mathematica, In press, 172 (3-4), pp.1009-1043. ⟨10.1007/s00229-022-01441-8⟩. ⟨hal-03358358⟩
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