Cocommutative Hopf algebras of permutations and trees

Details Cocommutative Hopf algebras of permutations and trees Cocommutative Hopf algebras of permutations and trees

2004-03-04

arxiv.org/abs/math/0403101v4

Journal of Algebraic Combinatorics, Volume 22 (2005), Number 4, pp. 451--470.

Mathematics

Quantum Algebra

Revised and clarified main results

Scientific paper

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.

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