Some Hopf Algebras of Trees
Pepijn van der Laan
TL;DR
This paper generalizes the construction of Hopf algebras of trees, unifying various examples like the Connes-Kreimer, Loday-Ronco, and Brouder-Frabetti algebras, and explores their algebraic structures.
Contribution
It introduces a generalized operadic framework for constructing Hopf algebras of trees, encompassing several known examples and revealing new algebraic structures.
Findings
Unified construction of multiple Hopf algebras of trees
Identification of pre-Lie and dendriform structures on duals
Corrections to previous proofs in the framework
Abstract
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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology