On a noncommutative deformation of the Connes-Kreimer algebra

Abstract

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.