Year: 2024
Type: article
Abstract: The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of nonsimply connected spaces.In particular, there is a realization functor, h i, of any complete differential graded Lie algebra as a simplicial set.In a previous article, we considered the particular case of a complete graded Lie algebra, L 0 , concentrated in degree 0 and proved that hL 0 i is isomorphic to the usual bar construction on the Maltsev group associated to L 0 .Here we consider the case of a complete differential graded Lie algebra, L D L 0 ˚L1 , concentrated in degrees 0 and 1.We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Maltsev groups.In particular, there is a crossed module Ꮿ.L/ associated to L. We prove that Ꮿ.L/ is isomorphic to the Whitehead crossed module associated to the simplicial pair .hLi;hL 0 i/.Our main result is the identification of hLi with the classifying space of Ꮿ.L/.
Source: Algebraic & Geometric Topology
Authors: Yves Félix, Daniel Tanré
Institutions: UCLouvain, Université de Lille
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: Agence Nationale de la Recherche
OpenAlex ID: W3177445321