Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Abstract

Abstract. The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra or a cocycle bialgebra.

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

“…(b) If X = ∅, then F(X, Ω) was studied by Foissy [12,13], in which a decorated noncommutative version of Connes-Kreimer Hopf algebra was constructed. (c) If Ω is a singleton set, then F(X, Ω) was introduced and studied in [40] to construct a cocycle Hopf algebra on decorated planar rooted forests. (d) The rooted forests in F(X, Ω) with leaf vertices decorated by elements of X and internal vertices decorated by elements of Ω were introduced in [22].…”

“…which is the usual 1-cocycle condition employed in [9,11,40]. Here the empty tree ½ is the unique group like element in the Connes-Kreimer Hopf algebra.…”

“…See also [6,17,21,25]. It has been observed that the decorated planar rooted forests H RT (Ω) whose vertices are decorated by a set Ω, together with a set of grafting operations {B + ω | ω ∈ Ω}), is a free object on the empty set in the category of Ω-operated algebras [31,40]. Particularly, the noncommutative Connes-Kreimer Hopf algebra H RT of planar rooted forests equipped with the grafting operation B + is a free operated algebra [22].…”

Hopf algebras of planar binary trees: an operated algebra approach

“…The Connes-Kreimer Hopf algebra of rooted trees can be viewed as an operated algebra, where the operator is the grafting operation B + . More generally, the decorated (planar) rooted trees with vertices decorated by a set Ω, together with a set of grafting operations {B + α | α ∈ Ω}), is an Ω-operated algebra [25,38]. Indeed it is the free Ω-operated algebra on the empty set or equivalently the initial object in the category of Ω-operated algebras.…”

“…It is well-known that the noncommutative Connes-Kreimer Hopf algebra of planar rooted trees is isomorphic to the Loday-Ronco Hopf algebra of planar binary trees [14,22]. Now the former can be treated in the framework of operated algebras [38]. So there should be an analogy of operated alebras on top of planar binary trees, which is introduced and explored in the present paper by the name of ∨-algebras or more generally ∨ Ω -algebras.…”