By Loday J.-L., Vallette B.

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By abuse of terminology we say that this augmented chain complex is acyclic when the augmentation map is a quasi-isomorphism. In other words the homology is 0 everywhere except in degree 0 where it is K. Let f and g be two chain maps between the two graded chain complexes (V, d) and (V , d ). A homotopy between f and g is a map h : V → V of degree +1 such that f − g = d h + hd =: ∂(h) . It is easy to check that the induced maps on homology are equal: f• = g• . Applied to Id and 0, it provides a way to prove that a chain complex is acyclic.

As an immediate consequence the functors cobar and bar are adjoint to each other. Then we investigate the twisting morphisms which give rise to quasi-isomorphisms under the aforementioned identifications. We call them Koszul morphisms. The main point is the following characterization of the Koszul morphisms. Any linear map α : C → A gives rise to a map dα : C ⊗A → C ⊗A, which is a differential if and only if α is a twisting morphism. Moreover, α is a Koszul morphism if and only if the chain complex (C ⊗ A, dα ) is acyclic.

The associated Lie bracket is given by [xp , xq ] = (p − q)xp+q , hence L−1 (also denoted W1 in the literature) is the Lie algebra of polynomial vector fields on the affine line K1 . Taking the universal algebra of the associated Lie algebra, we get a Hopf algebra U (L−1 ), which turns out to be the dual of the celebrated Fa` a di Bruno Hopf algebra [JR82]. When n ranges over Z we get a pre-Lie algebra whose associated Lie algebra is well-known. For K = R it is the Lie algebra of polynomial vector fields over the circle (Virasoro algebra without center).