Abstract : 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.