  We study a noncommutative deformation of the commutative Hopf algebra of
rooted trees which was shown by Connes and Kreimer to be related to the
mathematical structure of renormalization in quantum field theories. The
requirement of the existence of an antipode for the noncommutative deformation
leads to a natural extension of the algebra. Noncommutative deformations of the
Connes-Kreimer algebra might be relevant for renormalization of field theories
on noncommutative spaces and there are indications that in this case the
extension of the algebra might be linked to a mixing of infrared and
ultraviolet divergences. We give also an argument that for a certain class of
noncommutative quantum field theories renormalization should be linked to a
noncommutative and noncocommutative self-dual Hopf algebra which can be seen as
a noncommutative counterpart of the Grothendieck- Teichmueller group.

Quantum Algebra