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∞: UE→ Baut 1(X) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map ω: aut 1(Baut 1(XQ)) → Baut 1(XQ) expressed in terms of derivations of the relative Sullivan model of p∞. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space Baut 1(XQ) as a consequence. We also prove that CPQn cannot be realized as Baut 1(XQ) for n≤ 4 and X with finite-dimensional rational homotopy groups.