Grossman-Larson Hopf algebras built on rooted trees

In [8], Dirk Kreimer discovered the striking fact that the process of renormalization in quantum field theory may be described, in a conceptual manner, by means of certain Hopf algebras (which depend on the chosen renormalization scheme). A toy model was studied in detail by Alain Connes and Dirk Kreimer in [3]; the Hopf algebra which occurs, denoted by HR, is the polynomial algebra in an infinity of indeterminates, one for each rooted tree, but with a non-cocommutative comultiplication. Some operators, denoted by N and L, have been defined on HR. The first one is the natural growth operator, which acts as a derivation; it defines some elements δk, for k ≥ 1, which provide the link between HR and another Hopf algebra, introduced in [5] by Alain Connes and Henri Moscovici in a completely different context, namely in noncommutative geometry. The operator L is a solution of the “Hochschild equation”, and the pair (HR, L) is characterized as the solution of a universal problem in Hochschild cohomology. It was proved also in [3] that HR is in duality with the universal enveloping algebra of a certain Lie algebra L, which has a linear basis indexed by all (non-empty) rooted trees. Let us note that this Lie algebra L appeared also, very recently, in [2], in the context of pre-Lie algebras and the operad of rooted trees. In this note we would like to draw the attention to another Hopf algebra built on rooted trees, introduced ten years ago by Robert Grossman and Richard Larson in [6] (see also their survey [7]). This Hopf algebra (denoted by A in what follows) has a linear basis consisting of all (non-empty) rooted trees, a noncommutative product, and is a cocommutative graded connected Hopf algebra, hence, by the Milnor-Moore theorem, it is the universal enveloping algebra of the Lie algebra of its primitive elements, which may also be described explicitly: P (A) has a linear basis consisting of all rooted trees whose root has exactly one child. Using these properties of A, Grossman and Larson gave a Hopf algebraic proof of the classical result of Cayley on the number of rooted trees. The construction of the Hopf algebra A is motivated