Given X a finite nilpotent simplicial set, consider the
classifying fibrations
X → B aut∗
G(X) → B autG(X) and X → Z → B aut∗
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 DerGL and DerΠL of the Lie algebra of
derivations of L such that the geometrical realizations of the
sequences of cdgl morphisms
L ad
→ DerGL → DerGL ̃×sL and L → L ̃×DerΠL → DerΠL
have the rational homotopy type of the above classifying
fibrations. Among the consequences we also describe in cdgl
*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 and of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphisms
have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev -completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.