# Realizations of Hopf algebras of graphs by alphabets

## TL;DR

This paper constructs polynomial realizations of Hopf algebras related to graphs and posets using totally quasi-ordered alphabets, providing explicit algebraic embeddings and coproduct structures.

## Contribution

It introduces new polynomial realizations of Hopf algebras on graphs and posets via totally quasi-ordered alphabets, with explicit coproduct definitions.

## Key findings

- Polynomial embeddings of graph-related Hopf algebras into polynomial algebras.
- Explicit coproduct structures via alphabet doubling and squaring.
- Identification of cointeracting bialgebras in the commutative case.

## Abstract

We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here considered are totally quasi-ordered. The coproducts are given by doubling the alphabets; a second coproduct is defined by squaring the alphabets, and we obtain cointeracting bialgebras in the commutative case.