A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not. The Levi-Malcev theorem asserts that, if L is a finite-dimensional Lie algebra over a field of characteristic 0, then L splits over its soluble radical and that all complements are conjugate. In this note, I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not. As for Lie algebras, the splitting theorem achieves some reduction of the investigation of representations of Leibniz algebras to problems of Leibniz representations of semi-simple Lie algebras and of representations of soluble Leibniz algebras. T 1. Let L be a finite-dimensional Leibniz algebra over a field of characteristic 0 and let R be its soluble radical. There exists a semi-simple subalgebra S of L such that S + R = L and S ∩ R = 0.

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Abstract.We show that the problem addressed by classical homological perturbation theory can be reformulated as a fixed point problem leading to new insights into the nature of its solutions. We show, under mild conditions, that the solution is essentially unique.