The universal fibration with fibre X in rational homotopy theory

Let X be a simply connected space with finite-dimensional rational homotopy groups. Let \(p_\infty :UE \rightarrow B\mathrm {aut}_1(X)\) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map \( \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})\) expressed in terms of derivations of the relative Sullivan model of \(p_\infty \). We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space \(B\mathrm {aut}_1(X_\mathbb {Q})\) as a consequence. We also prove that \(\mathbb {C} P^n_\mathbb {Q}\) cannot be realized as \(B\mathrm {aut}_1(X_\mathbb {Q})\) for \(n \le 4\) and X with finite-dimensional rational homotopy groups.