Finite dimensional comodules over the Hopf algebra of rooted trees

الأصل · EN

The aim of this paper is an algebraic study of the Hopf algebra Hᵣ of rooted trees, which was introduced in Kreimer1,Connes,Broadhurst,Kreimer2. We first construct comodules over Hᵣ 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. In the next section, we give a formula about primitive elements of the subalgebra of ladders, and construct a projection operator on the space of primitive elements. It allows us to obtain a basis of the primitive elements by an inductive process, which answers one of the questions of Kreimer. In the last sections, we classify the Hopf algebra endomorphisms and the coalgebras endomorphisms, using the graded Hopf algebra gr(Hᵣ) associated to the filtration by degₚ of Kreimer. We then prove that Hᵣ is isomorphic to gr(Hᵣ), and deduce a decomposition of the group of the Hopf algebra automorphisms of Hᵣ as a semi-direct product.