  We focus in this text on the adaptation to the study of shuffles of the main
combinatorial tool in the theory of free Lie algebras, namely the existence of
a universal algebra of endomorphisms for tensor and other cocommutative Hopf
algebras: the family of Solomon's descent algebras of type A. We show that
there exists similarly a natural endomorphism algebra for commutative shuffle
algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra
of permutations, or algebra of free quasi-symmetric functions. We study this
new algebra for its own, establish freeness properties, study its generators,
bases, and also feature its relations to the internal structure of shuffle
algebras.
