  The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a
noncommutative generalization of the Hopf algebra of quasi-symmetric functions,
can be endowed with an internal product that has several compatibility
properties with the other operations on $\WQSym$. This extends constructions
familiar and central in the theory of free Lie algebras, noncommutative
symmetric functions and their various applications fields, and allows to
interpret $\WQSym$ as a convolution algebra of linear endomorphisms of
quasi-shuffle algebras. We then use this interpretation to study the fine
structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In
particular, we compute their Adams operations and prove the existence of
generalized Eulerian idempotents, that is, of a canonical left-inverse to the
natural surjection map to their indecomposables, allowing for the combinatorial
construction of free polynomial generators for these algebras.
