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A Hopf algebra on nonplanar binary forests

Elizabeth Xiao

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.

A Hopf algebra on nonplanar binary forests

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.
Paper Structure (10 sections, 2 theorems, 23 equations, 1 figure)

This paper contains 10 sections, 2 theorems, 23 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Rooted trees
  3. Hopf algebras
  4. The Connes-Kreimer Hopf algebra
  5. Variations of Connes-Kreimer for binary trees
  6. Issues with $\Delta^c$
  7. Issues with $\Delta^d$
  8. Pre-Lie algebras
  9. The vertex-insertion pre-Lie algebra
  10. The Hopf algebra of subsets of leaves

Key Result

Proposition 2.1

The terms of $\Delta(T)$ are in bijection with subsets of leaves of $T$.

Figures (1)

  • Figure :

Theorems & Definitions (11)

  • Example 1.1
  • Remark 1.1
  • Example 1.2
  • Example 1.3
  • Remark 2.1
  • Definition 2.1
  • Example 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • ...and 1 more