Loday, Jean-Louis; Ronco, María Combinatorial Hopf algebras. (English) Zbl 1217.16033 Blanchard, Etienne (ed.) et al., Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29–April 6, 2007. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-5203-3/pbk). Clay Mathematics Proceedings 11, 347-383 (2010). Summary: We define a “combinatorial Hopf algebra” as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). We show that the choice of such an isomorphism implies the existence of a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.For the entire collection see [Zbl 1206.00042]. Cited in 1 ReviewCited in 31 Documents MSC: 16T30 Connections of Hopf algebras with combinatorics 18D50 Operads (MSC2010) Keywords:combinatorial Hopf algebras; free algebras; cofree coalgebras; pre-Lie algebras; Connes-Kreimer Hopf algebras; operads PDFBibTeX XMLCite \textit{J.-L. Loday} and \textit{M. Ronco}, Clay Math. Proc. 11, 347--383 (2010; Zbl 1217.16033) Full Text: arXiv