Towards cohomology of renormalization: Bigrading the combinatorial Hopf algebra of rooted trees

The renormalization of quantum field theory twists the antipode of a noncoc ommutative Hopf algebra of rooted trees, decorated by an infinite set of pr imitive divergences. The Hopf algebra of undecorated rooted trees, H-R, gen erated by a single primitive divergence, solves a universal problem in Hoch schild cohomology. It has two nontrivial closed Hopf subalgebras: the cocom mutative subalgebra H-ladder Of pure ladder diagrams and the Connes-Moscovi ci noncocommutative subalgebra H-CM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations o f the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainde r. The results for H-ladder are familiar from the theory of partitions, whi le those for H-CM involve novel transforms of partitions. Most beautiful is the bigrading of H-R, the largest of the three. Thanks to Sloane's superse eker, we discovered that it saturates all possible inequalities. We prove t his by using the universal Hochschild-closed one-cocycle B+, which plugs on e set of divergences into another, and by generalizing the concept of natur al growth beyond that entailed by the Connes-Moscovici case. We emphasize t he yet greater challenge of handling the infinite set of decorations of rea listic quantum field theory.