Cocommutative Hopf algebras of permutations and trees.
Journal of Algebraic Combinatorics, Tome 22 (2005) no. 4, pp. 451-470.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.
Mots-clés : keywords Hopf algebra, rooted tree, planar binary tree, symmetric group 
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     author = {Aguiar, Marcelo and Sottile, Frank},
     title = {Cocommutative {Hopf} algebras of permutations and trees.},
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Aguiar, Marcelo; Sottile, Frank. Cocommutative Hopf algebras of permutations and trees.. Journal of Algebraic Combinatorics, Tome 22 (2005) no. 4, pp. 451-470. https://geodesic-test.mathdoc.fr/item/JAC_2005__22_4_a1/