Polynomial realizations of Hopf algebras built from nonsymmetric operads

TL;DR

This paper develops polynomial realizations of natural Hopf algebras derived from nonsymmetric operads, linking them to various well-known noncommutative Hopf algebras through specific alphabet constructions.

Contribution

It introduces a method to realize these Hopf algebras polynomially using noncommutative variables with relations, connecting them to existing algebraic structures.

Findings

Established polynomial realizations for Hopf algebras from operads.

Linked these realizations to known noncommutative Hopf algebras.

Provided a framework for understanding relations between operad-based and classical Hopf algebras.

Abstract

The natural Hopf algebra $\mathbf{N} \cdot \mathcal{O}$ of an operad $\mathcal{O}$ is a Hopf algebra whose bases are indexed by some words on $\mathcal{O}$. We construct polynomial realizations of $\mathbf{N} \cdot \mathcal{O}$ by using alphabets of noncommutative variables endowed with unary and binary relations. By using particular alphabets, we establish links between $\mathbf{N} \cdot \mathcal{O}$ and some other Hopf algebras including the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the noncommutative Connes-Kreimer Hopf algebra of Foissy, the noncommutative Fa\`a di Bruno Hopf algebra and its deformations, the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon, and the double tensor Hopf algebra of Ebrahimi-Fard and Patras.