Gauge theories with graded differential Lie algebras

We present a mathematical framework of gauge theories that is based upon a skew-adjoint Lie algebra and a generalized Dirac operator, both acting on a Hilbert space.Comment: 10 pages, LaTeX2e, extended version (references and comments on the construction of physical models and on the relation to the axioms of noncommutative geometry added

On Kreimer's Hopf algebra structure of Feynman graphs

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements.Comment: 21 pages, LaTeX2e, requires feynmf package to draw Feynman graphs (see log file for additional information). Following an idea by Dirk Kreimer we introduced in the revised version a primitivator which maps overlapping divergences to primitive elements and which provides the link to the Hopf algebra of Kreimer (q-alg/9707029, hep-th/9808042). v4: error in eq (29) corrected and references updated; to appear in Eur.Phys.J.