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Given $X$ a finite nilpotent simplicial set, consider the classifyingfibrations $$ X\to Baut_G^*(X)\to Baut_G(X),\qquad X\to Z\to Baut_{\pi}^*(X),$$  where $G$ and $\pi$ denote, respectively, subgroups of the free and pointedhomotopy classes of free and pointed self homotopy equivalences of $X$ whichact nilpotently on $H_*(X)$ and $\pi_*(X)$.  We give algebraic models, in terms of complete differential graded Liealgebras (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 $Der^{\pi} L$ of the Lie algebra $Der L$ of derivations of $L$such that the geometrical realization of the sequences of cdgl morphisms  $$  L\stackrel{ad}{\to} Der^G L\to Der^G L\widetilde\times sL,\qquad L\toL\widetilde\times Der^{\pi} L\to Der^{\pi} L  $$  have the rational homotopy type of the above classifying fibrations. Amongthe consequences we also describe in cdgl terms the Malcev $Q$-completion of$G$ and $\pi$ together with the rational homotopy type of the classifyingspaces $BG $ and $B\pi$.

Lie models of homotopy automorphism monoids and classifying fibrations