Finite dimensional comodules over the Hopf algebra of rooted trees

In [1, 3, 4, 5], a Hopf algebra of rooted trees HR was introduced. It was shown that the antipode of this algebra was the key of a problem of renormalization ([8]). HR is related to the Hopf algebra HCM introduced in [2]. Moreover, the dual algebra of HR is the enveloping algebra of the Lie algebra of rooted trees L. An important problem was to give an explicit construction of the primitive elements of HR. In [6], a bigradation allowed to compute the dimensions of the graded parts of the space of primitive elements. The aim of this paper is an algebraic study of HR. We first use the duality theorem of [3] to prove a result about the subcomodules of a finite dimensional comodule over the Hopf algebra of rooted trees. Then we use this result to construct comodules from finite families of primitive elements. Furthermore, we classify these comodules by restricting the possible families of primitive elements, and taking the quotient by the action of certain groups. We also show how the study of the whole algebra as a left-comodule leads to the bigrading of [6]. We then prove that L is a free Lie algebra. In the next section, we prove a formula about primitive elements of the Hopf algebra of ladders, which was already given in [6], and construct a projection operator on the space of primitive elements. This operator produces the operator S1 of [6]. Moreover, it allows to obtain a basis of the primitive elements by an inductive process, which answers one of the questions of [6]. The following sections give results about the endomorphisms of HR. First, we classify the Hopf algebra endomorphisms using the bilinear map related to the growth of trees. Then we study the coalgebra endomorphisms, using the graded Hopf algebra gr(HR) associated to the filtration by degp of [6]. We finally prove that HR ≈ gr(HR), and deduce a decomposition of the group of the Hopf algebra automorphisms of HR as a semi-direct product.