The author proposes a framework in which to obtain the primitive elements in various combinatorial Hopf algebras over a field \(K\). If \(n=n_1+n_2+\cdots+n_r\) is a composition of \(n\), let \(Sh(n_1,n_2,\dots,n_r)\) denote the usual shuffles of \(\{1,2,\dots,n\}\) determined by the composition. A shuffle algebra is a vector space graded by the positive integers, with linear maps \(\cdot_g\colon A_n\otimes A_m\to A_{n+m}\) for each \(g\) in \(Sh(n,m)\) satisfying associative-type conditions. The reduced tensor algebra \(T(V)/K\) on a vector space \(V\) is a shuffle algebra. For a positively graded vector space \(V\), there is a free shuffle algebra \(Sh(V)\) on \(V\). A source of shuffle algebras is nonunital infinitesimal bialgebras, as introduced by J.-L. Loday and the author [J. Reine Angew. Math. 592, 123-155 (2006; Zbl 1096.16019)]. These are positively graded algebras with a coassociative coproduct \(c\) satisfying \(c(xy)=(xy_1)\otimes y_2+x_1\otimes (x_2)y+x\otimes y\) (Sweedler notation). They admit natural structures as shuffle algebras. A shuffle bialgebra is a positively graded shuffle algebra with a graded coassociative coproduct satisfying a rule for the coproduct of a product \(x \cdot_g y\). Free shuffle algebras on graded coalgebras are shuffle bialgebras. If \(A\) is a graded nonunital infinitesimal bialgebra, its shuffle algebra structure is a shuffle bialgebra (same coproduct). Primitive elements enter for \(H\) a conilpotent infinitesimal bialgebra, where conilpotent means that the union of the ascending filtration of \(H\) starting with \(\text{Prim}(H)\) is \(H\).
A principal theorem says that \(H\) is isomorphic to a certain enveloping-type algebra of its primitives. This theorem is proved in a more general context of preshuffle bialgebras and pre-Lie systems. All this is intertwined with and applied to various combinatorial Hopf algebras, including maps between finite sets, the Malvenuto-Reutenauer bialgebra of permutations, the bialgebra of surjective maps (of finite sets), the bialgebra of parking functions, and planar rooted trees. The primitive elements of some of these were computed earlier, but here this study is done in a very general framework.