 
Summary: COCOMMUTATIVE HOPF ALGEBRAS
OF PERMUTATIONS AND TREES
MARCELO AGUIAR AND FRANK SOTTILE
Abstract. 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 heapordered
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.
Introduction
In the late 1980's, Grossman and Larson constructed several cocommutative Hopf
algebras from different families of trees (rooted, ordered, heapordered), in connection
to the symbolic algebra of differential operators [10, 11]. Other Hopf algebras of trees
have arisen lately in a variety of contexts, including the ConnesKreimer Hopf algebra in
renormalization theory [5] and the LodayRonco Hopf algebra in the theory of associa
tivity breaking [18, 19]. The latter is closely related to other important Hopf algebras in
algebraic combinatorics, including the MalvenutoReutenauer Hopf algebra [22] and the
Hopf algebra of quasisymmetric functions [21, 27, 31].
