On Associators and the Grothendieck-Teichmuller Group I

TL;DR

This paper introduces a formalism that clarifies the relationship between associators and the Grothendieck-Teichmuller group, simplifying Drinfel'd's work and providing a new perspective on their connection.

Contribution

It presents a new formalism that makes the relationship between associators and the Grothendieck-Teichmuller group more natural and straightforward, and re-proves the existence of rational associators.

Findings

Established a natural formalism linking associators and the Grothendieck-Teichmuller group

Re-proved the existence and constructibility of rational associators

Simplified aspects of Drinfel'd's original work on associators

Abstract

We present a formalism within which the relationship (discovered by Drinfel'd) between associators (for quasi-triangular quasi-Hopf algebras) and (a variant of) the Grothendieck-Teichmuller group becomes simple and natural, leading to a simplification of Drinfel'd's original work. In particular, we re-prove that rational associators exist and can be constructed iteratively, though the proof itself still depends on the apriori knowledge that a not-necessarily-rational associator exists.