Generalized bialgebras and triples of operads

We introduce the notion of generalized bialgebra, which includes
the classical notion of bialgebra (Hopf algebra) and
many others, like, for instance, the tensor algebra equipped
with the deconcatenation as coproduct. We prove that, under
some mild conditions, a connected generalized bialgebra
is completely determined by its primitive part. This structure
theorem extends the classical Poincaré-Birkhoff-Witt
theorem and Cartier-Milnor-Moore theorem, valid for co-commutative
bialgebras, to a large class of generalized bialgebras.
Technically we work in the theory of operads which
allows us to state our main results and permits us to give it
a conceptual proof. A generalized bialgebra type is determined
by two operads: one for the coalgebra structure C,
and one for the algebra structure A. There is also a compatibility
relation relating the two. Under some conditions,
the primitive part of such a generalized bialgebra is an algebra
over some sub-operad of A, denoted P. The structure
theorem gives conditions under which a connected generalized
bialgebra is cofree (as a connected C-coalgebra) and
can be re-constructed out of its primitive part by means of
an enveloping functor from P-algebras to A-algebras. The
classical case is (C, A, P) = (Com, As, Lie). This structure
theorem unifies several results, generalizing the PBW and
the CMM theorems, scattered in the literature. We treat
many explicit examples and suggest a few conjectures.