A homotopy in the usual cochain complex of free Lie algebras
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.
scientific article; zbMATH DE number 1059648