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Journal Articles Journal of Algebra Year : 2002

Finite dimensional comodules over the Hopf algebra of rooted trees

Loïc Foissy

Abstract

The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R 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 \cite{Kreimer}. In the last sections, we classify the Hopf algebra endomorphisms and the coalgebras endomorphisms, using the graded Hopf algebra gr(H_R) associated to the filtration by deg_p of \cite{Kreimer}. We then prove that H_R is isomorphic to gr(H_R), and deduce a decomposition of the group of the Hopf algebra automorphisms of H_R as a semi-direct product.
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Dates and versions

hal-00160836 , version 1 (09-07-2007)

Identifiers

  • HAL Id : hal-00160836 , version 1

Cite

Loïc Foissy. Finite dimensional comodules over the Hopf algebra of rooted trees. Journal of Algebra, 2002, 255 (1), pp.89-120. ⟨hal-00160836⟩

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