Cocommutative Hopf Algebras of Permutations and Trees

Consider the coradical filtration of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We show 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. We also show that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra. This implies that the Hopf algebras of ordered trees and heap-ordered trees are tensor Hopf algebras. Introduction In the late 1980’s, Grossman and Larson constructed several cocommutative Hopf algebras from different families of trees (rooted, ordered, heap-ordered), 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 Connes-Kreimer Hopf algebra in renormalization theory [5] and the Loday-Ronco Hopf algebra in the theory of associativity breaking [17, 18]. The latter is closely related to other important Hopf algebras in algebraic combinatorics, including the Malvenuto-Reutenauer Hopf algebra [21] and the Hopf algebra of quasi-symmetric functions [20, 25, 29]. This universe of Hopf algebras of trees is summarized below. Family of trees Hopf algebra rooted trees Grossman-Larson ordered trees non-commutative, 89-90 heap-ordered cocommutative trees Loday-Ronco planar binary non-commutative, 98 trees non-cocommutative Connes-Kreimer rooted trees commutative, 98 non-cocommutative Date: March 1, 2004. 2000 Mathematics Subject Classification. Primary 16W30, 05C05; Secondary 05E05.