The pentagon equation and the confluence relations

TL;DR

This paper establishes an equivalence between the pentagon equation and confluence relations, leading to a new presentation of the Grothendieck-Teichmüller group and associators without the pentagon equation.

Contribution

It demonstrates the equivalence between two fundamental relations in algebraic structures and provides a pentagon-free description of key mathematical groups.

Findings

Equivalence of Drinfeld's pentagon equation and confluence relations.

Derived a pentagon-free presentation of the Grothendieck-Teichmüller group.

Simplified understanding of associators and related algebraic structures.

Abstract

We show an equivalence of Drinfeld's pentagon equation and Hirose-Sato's confluence relations. As a corollary, we obtain a pentagon-free presentation of the Grothendieck-Teichm\"{u}ller group $GRT_1$ and associators.