Leibniz algebras are a non-(anti)commutative version of Lie algebras just satisfying an analogue of the Jacobi identity. In the paper under review, which considers right Leibniz algebras, this means that every right multiplication operator is a derivation, i.e., it satisfies the (Leibniz) product rule. In particular, every Lie algebra is a Leibniz algebra, and conversely, every Leibniz algebra \(\mathfrak{g}\) has a canonical homomorphic image \(\mathfrak{g}_{\mathrm{Lie}}\) which is a Lie algebra such that the Lie-ification functor \(\mathfrak{g}\mapsto\mathfrak{g}_{\mathrm{Lie}}\) is left adjoint to the inclusion functor from Lie algebras to Leibniz algebras. In particular, this induces a comparison map \(\mathrm{HL}_\bullet(\mathfrak{g},\Bbbk)\to\mathrm{H}_\bullet (\mathfrak{g}_{\mathrm{Lie}},\Bbbk)\) from the Leibniz homology of \(\mathfrak{g}\) to the Lie algebra homology of \(\mathfrak{g}_{ \mathrm{Lie}}\) with trivial coefficients.
Let \(\Bbbk\) be a field of characteristic zero, let \(V\) be a vector space over \(\Bbbk\), let \[T_\bullet(V):=\bigoplus_{n\ge 0}V^{\otimes n}\] be the tensor algebra on \(V\), and let \[L_\bullet(V):=\bigoplus_{n\ge 0}L_n(V)\] be the graded Lie subalgebra of \(T_\bullet(V)\) generated by \(V\), namely, the free graded Lie algebra generated by \(V\) and concentrated in degree 1. Around 2002, T. Pirashvili [“A subcomplex of Leibniz complex‘”, Preprint, arXiv:1904.00121] observed for a Leibniz algebra \(\mathfrak{g}\) that \(L_\bullet(\mathfrak{g})\) is a subcomplex of the Leibniz chain complex \(\mathrm{CL}_\bullet(\mathfrak{g},\Bbbk):= T_\bullet(\mathfrak{g})\). Finally, let \[ \mathrm{Lie}_\bullet(\mathfrak{g}):=\bigoplus_{n\ge 1}\mathrm{Lie}_n (\mathfrak{g}) \] be the homology of the complex \(L_\bullet(\mathfrak{g})\). Then Pirashvili proved that \(\mathrm{Lie}_1(\mathfrak{g})\cong\mathfrak{g}_{\mathrm{Lie}}\) (as \(\Bbbk\)-vector spaces) and conjectured that if \(\mathfrak{g}\) is a free Leibniz algebra, then \(\mathrm{Lie}_n(\mathfrak{g})=0\) for any integer \(n>1\). He also noted that this conjecture would imply that \(\mathrm{Lie}_n(\mathfrak{g})\) computes the Quillen derived functors of the Lie-ification functor \(\mathfrak{g} \mapsto\mathfrak{g}_{\mathrm{Lie}}\).
The main result of the paper under review is a proof of a conjecture of J.-L. Loday also contained in Pirashvili’s paper by defining an increasing and exhausting filtration of the Leibniz chain complex. The spectral sequence associated to this filtration implies then Pirashvili’s conjecture mentioned earlier. Previously, a proof of Pirashvili’s conjecture was given by J. Mostovoy [Int. Math. Res. Not. 2022, No. 1, 18927–18940 (2022; Zbl 1510.17028)] by different methods which are based on his interpretation of Leibniz algebras in terms of differential graded Lie algebras.
Pirashvili asked the author whether the complex \(\mathrm{Lie}_\bullet(\mathfrak{g})\) admits a natural \(L_\infty\)-structure that extends the Leibniz differential and which induces the Lie structure on \(\mathfrak{g}_{\mathrm{Lie}}\). The last section of the paper is devoted to a construction of such an \(L_\infty\)-structure that is unique up to isomorphism and was obtained independently by Mostovoy in the paper cited above.