Lie theory for Hopf operads

The present article takes advantage of the properties of algebras in the category of -modules (twisted algebras) to investigate further the fine algebraic structure of Hopf operads. We prove that any Hopf operad carries naturally the structure of a comonoid in the category of twisted -algebras. Many properties of Hopf algebraic structures are then shown to be encapsulated in this remarquable structure. In particular, various classical theorems of the theory of free Lie algebras are lifted to arbitrary Hopf operads.