The Hopf algebra of rooted trees, free Lie algebras, and Lie series

We present, based on our results in~[18] (where the Hopf algebra structure of the shuffle algebra is described in terms of the Hopf algebra of decorated rooted trees), a rewriting algorithm that provides a simple recursive way to compute the coproduct of the shuffle Hopf algebra in terms of a dual basis of the Poincar\'e-Birkhoff-Witt basis corresponding to a Hall basis of the free Lie algebra. This can be applied, for instance, to do computations in free Lie algebras and its enveloping associative algebra in terms of an arbitrary Hall basis. In addition, we show how to exploit our results to do algebraic manipulations with Lie series and exponentials of Lie series~[20], either using our rewriting algorithm, or working directly in terms of the Hopf algebra of decorated rooted trees. In particular, we give an explicit way to obtain the coefficients of the CBH formula (and also for its continuous version) read directly from the decorated rooted tree associated to each element of the Hall basis.