2001/05/25 by Loïc Foissy, Loic Foissy, Foissy, Loic · 1 citation
Mathematics · #05C05 (Secondary) #16W30 (Primary) 81T15 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:05C05 #msc:16W30 #msc:81T15
paper · pdf · doi:10.48550/arxiv.math/0105210
LaTeX, 26 pages
arxiv created 2001/05/25 · openalex publication_date 2001/05/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this paper is an algebraic study of the Hopf algebra HR of rooted trees, which was introduced in \citeKreimer1,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 \citeKreimer. 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 \citeKreimer. 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.