This paper generalizes the operadic construction of the Connes-Kreimer Hopf
algebra of rooted trees by Moerdijk. Examples of Hopf algebras obtained in this
way include the Loday-Ronco Hopf algebra of planar binary trees and the
Brouder-Frabetti pruning Hopf algebra. In some examples we obtain a natural
pre-Lie or dendriform algebra structure on the dual Hopf algebra. v2: Proof in
section 8 corrected. v3: more corrections.Comment: 24 pages, 3 figure

van der Laan, Pepijn

English

arXiv

In the literature several Hopf algebras that can be described in terms of trees have been studied. This paper tries to answer the question whether one can understand some of these Hopf algebras in terms of a single mathematical construction.
The starting point is the Hopf algebra of rooted trees as dened by Connes and Kreimer in [3] (section 2). Apart from its physical relevance, it has a universal property in Hochschild cohomology. We generalize the operadic construction by Moerdijk [12] of this Hopf algebra to more general trees (with colored edges), and prove a universal property in coalgebra Hochschild cohomology (sections 3, 4, and 5). For a Hopf operad P , the construction is based on the operad P [n] obtained from P by adjoining a free n-ary operation. The dierence with respect to [12] is that we also prove that the operad P [n] is under suitable restrictions a Hopf operad. This presentation simplies some proofs and assures that relevant  unctorial properties come for free. Section 6 shows that we can do with less restrictions if we are only interested in the the initial algebra P [n](0)

Laan, P. van der

Utrecht University Repository

Some Hopf algebras of trees

Pepijn Van Der Laan

CiteSeerX

Some Hopf Algebras of Trees

In the literature several Hopf algebras that can be described in terms of trees have been studied. This paper tries to answer the question whether one can understand some of these Hopf algebras in terms of a single mathematical construction.

The starting point is the Hopf algebra of rooted trees as dened by Connes and Kreimer in [3] (section 2). Apart from its physical relevance, it has a universal property in Hochschild cohomology. We generalize the operadic construction by Moerdijk [12] of this Hopf algebra to more general trees (with colored edges), and prove a universal property in coalgebra Hochschild cohomology (sections 3, 4, and 5). For a Hopf operad P , the construction is based on the operad P [n] obtained from P by adjoining a free n-ary operation. The dierence with respect to [12] is that we also prove that the operad P [n] is under suitable restrictions a Hopf operad. This presentation simplies some proofs and assures that relevant  unctorial properties come for free. Section 6 shows that we can do with less restrictions if we are only interested in the the initial algebra P [n](0)

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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.3043

Some Hopf Algebras of Trees

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