On the Hopf algebra of functional graphs and differential algebras Original Research Article

We develop the Hopf algebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of functional graphs and we introduce several kinds of brackets in accordance with the decomposition in connected components of the graph of a mapping of a finite set into itself, i.e. basins of attraction as in the frame of the discrete dynamical systems. We compute the antipode in a natural basis. The use of the noncommutative polynomials gives a symbolic calculus useful for differential algebras and algebras of differential operators.