An antipode formula for the natural Hopf algebra of a set operad. (Q395254)

A commutative, non cocommutative Hopf algebra is attached to any symmetric set operad \(E\). It is called the natural Hopf algebra attached to \(E\). When \(E\) is left cancellative, it can be obtained as a quotient of the incidence Hopf algebra of a family of posets. A formula for the antipode is described with the help of Schröder trees. In the case of the operad of pointed sets, we obtain in this way the classical Lagrange inversion formula. Other cases are studied, as the operad of connected graphs or the NAP operad, which is related to the Connes-Kreimer Hopf algebra of trees. When \(E\) is not symmetric, a similar construction gives non commutative and non cocommutative Hopf algebras, generalizing the non commutative Hopf algebra of formal diffeomorphisms, and the Novelli-Thibon formula for the antipode.