A homotopy in the usual cochain complex of free Lie algebras
type: Item
instance of: scholarly article
title: A homotopy in the usual cochain complex of free Lie algebras
zbMATH Open document ID: 0919.17016
DOI: 10.1023/A:1007312205995
author: Alexandros Patsourakos
published in: Letters in Mathematical Physics
publication date: 1997-09-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1059648
zbMATH Keywords: cochain complex
zbMATH Keywords: free Lie algebra
zbMATH Keywords: contracting homotopy
zbMATH Keywords: triple cohomology of Lie algebras
reviewed by: Jan Kubarski
MaRDI profile type: Publication
full work available at URL: a:1007312205995
OpenAlex ID: W71922890
description: scientific article; zbMATH DE number 1059648