Realization of Lie algebras and classifying spaces of crossed modules

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/. 17B55, 55P62; 55U101 Background on Lie models A complete differential graded Lie algebra (henceforth cdgl) is a differential graded Lie algebra L equipped with a decreasing filtration of differential Lie ideals such that F 1 D L, OEF p L; F q L F pCq L and L D lim n L=F n L:If no filtration is specified, it is understood that we consider the lower central series.