A homotopy in the usual cochain complex of free Lie algebras (Q1366291)

The comparison of the standard cohomology theories of groups and associative algebras with the cotriple cohomology was studied by \textit{M. Barr} and \textit{J. Beck} [Acyclic models and triples, in Proc. Conf. Categorical Algebra, La Jolla 1965, 336-343 (1966; Zbl 0201.35403)]. The case of Lie algebras was examined by the author in [Sur les algèbres de Lie libres, PhD thesis, Univ. of Bourgogne, Dijon, 1992]. To implement the method of acyclic models (using the proofs of the above-mentioned results) in the case of Lie algebras, the natural contracting homotopy in the standard cochain complex of a free Lie algebra is constructed. In this article the author defines this functorial operator by an equivalent form which -- as he himself says -- has the advantage of simplifying the proofs. The aim of this note is to use the contracting homotopy to establish that the triple cohomology of Lie algebras coincides with a slightly different form of the standard cohomology theory.