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Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalencesof a fibration p:E\to B. We make a general study of this monoid, especially inrational homotopy theory. When E and B are simply connected CW complexes with Efinite, we identify the rational Samelson Lie algebra of the identity componentof Aut(p) as the homology of a certain DG Lie algebra of derivations arisingfrom the Koszul-Sullivan model of p. We obtain related identifications for therational homotopy groups of fibrewise mapping spaces and for therationalization of a natural nilpotent subgroup of Aut(p).

The rational homotopy type of the space of self-equivalences of a fibration