A Hopf algebra on nonplanar binary forests

TL;DR

This paper introduces a new Hopf algebra structure on nonplanar binary forests, refining existing models and establishing connections with stochastic process renormalization and Lie algebra structures.

Contribution

It defines a novel coproduct on nonplanar binary forests that refines Connes-Kreimer's cuts and links to a Lie algebra via a pre-Lie operator.

Findings

The coproduct disallows cutting both children of a vertex.

The Hopf algebra connects with a Hopf algebra used in stochastic renormalization.

It is dual to the universal enveloping algebra of a Lie algebra from a pre-Lie operator.

Abstract

We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of admissible cuts. We show that the terms in this coproduct have an additional combinatorial interpretation in terms of subsets of leaves, which facilitates the construction of Hopf algebra morphisms involving this Hopf algebra, and creates a connection with a Hopf algebra of Bruned used in the renormalization of stochastic processes. Finally, we show that this Hopf algebra is dual to the universal enveloping algebra of a Lie algebra arising from a pre-Lie operator on binary trees based on edge-insertion.