  Many features of classical Lie theory generalize to the broader context of
algebras over Hopf operads. However, this idea remains largely to be developed
systematically. Quasi-shuffle algebras provide for example an interesting
illustration of these phenomena, but have not been investigated from this point
of view.The notion of quasi-shuffle algebras can be traced back to the
beginings of the theory of Rota--Baxter algebras, but was developed
systematically only recently, starting essentially with Hoffman's work, that
was motivated by multizeta values (MZVs) and featured their bialgebra
structure. Many partial results on the fine structure of quasi-shuffle
bialgebras have been obtained since then but, besides the fact that each of
these articles features a particular point of view, they fail to develop
systematically a complete theory.This article builds on these various results
and develops the analog theory, for quasi-shuffle algebras, of the theory of
descent algebras and their relations to free Lie algebras for classical
enveloping algebras.
