Realization of Lie algebras and classifying spaces of crossed modules Article Swipe
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· 2024
· Open Access
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· DOI: https://doi.org/10.2140/agt.2024.24.141
· OA: W3177445321
The category of complete differential graded Lie algebras provides nice\nalgebraic models for the rational homotopy types of non-simply connected\nspaces. In particular, there is a realization functor, $\\langle -\\rangle$, of\nany complete differential graded Lie algebra as a simplicial set. In a previous\narticle, we considered the particular case of a complete graded Lie algebra,\n$L_{0}$, concentrated in degree 0 and proved that $\\langle L_{0}\\rangle$ is\nisomorphic to the usual bar construction on the Malcev group associated to\n$L_{0}$.\n Here we consider the case of a complete differential graded Lie algebra,\n$L=L_{0}\\oplus L_{1}$, concentrated in degrees 0 and 1. We establish that the\ncategory of such two-stage Lie algebras is equivalent to explicit subcategories\nof crossed modules and Lie algebra crossed modules, extending the equivalence\nbetween pronilpotent Lie algebras and Malcev groups. In particular, there is a\ncrossed module $\\mathcal{C}(L)$ associated to $L$. We prove that\n$\\mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to\nthe simplicial pair $(\\langle L\\rangle, \\langle L_{0}\\rangle)$. Our main result\nis the identification of $\\langle L\\rangle$ with the classifying space of\n$\\mathcal{C}(L)$.\n
