Lie models of homotopy automorphism monoids and classifying fibrations

Given $X$ a finite nilpotent simplicial set, consider the classifying fibrations $X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_π^*(X),$ where $G$ and $π$ denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of $X$ which act nilpotently on $H_*(X)$ and $π_*(X)$. We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if $L$ is a cdgl model of $X$, there are connected sub cdgl's $Der^G L$ and $Der^π L$ of the Lie algebra $Der L$ of derivations of $L$ such that the geometrical realization of the sequences of cdgl morphisms $L\stackrel{ad}{\to} Der^G L\to Der^G L\widetilde\times sL,\qquad L\to L\widetilde\times Der^π L\to Der^π L$ have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev $Q$-completion of $G$ and $π$ together with the rational homotopy type of the classifying spaces $BG$ and $Bπ$.

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