On Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a non-commutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré–Birkhoff–Witt type theorems and collect information about the operads PrimP, e.g., in terms of characteristic functions.