Lie theory for some quotients of the affine group represented by the Hopf Shuffle algebra

I develop a Lie theory for certain unipotent algebraic quotients of the affine group scheme G represented by the Hopf Shuffle algebra.;I prove that the Lie algebra of the Hopf Shuffle algebra is an algebra of Lie Series and that the Lie algebras of some unipotent algebraic quotients of G are quotients of the free Lie algebra.;I show that the affine group scheme of upper triangular unipotent matrices Un, for all n∈N, is a quotient of the affine group scheme represented by the Hopf shuffle algebra by injecting the representing algebra of Un into the Hopf Shuffle algebra as a sub Hopf algebra.;As an application of the Lie theory that I constructed, I give a simple proof that the Hopf shuffle algebra is a free commutative algebra, a result first proved by Radford [Rad79] in a rather involved way using combinatorics.