From left modules to algebras over an operad: application to combinatorial Hopf algebras
The purpose of this paper is two fold: we study the behaviour of the forgetful functor from -modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras. Let denote the forgetful functor from -modules to graded vector spaces. Left modules over an operad are treated as -algebras in the category of -modules. We generalize the...