Pentagon and hexagon equations following Furusho

TL;DR

This paper provides a simpler proof demonstrating that the pentagon equation implies the hexagon equations for associators, utilizing algebraic geometry and Grothendieck-Teichmuller groups.

Contribution

It offers a more straightforward proof of Furusho's theorem, reducing complexity by avoiding spherical braids and clarifying the underlying principles.

Findings

Pentagon equation implies the two hexagon equations for associators.

Simplified proof reduces reliance on spherical braids.

Utilizes algebraic geometry and Grothendieck-Teichmuller groups effectively.

Abstract

In [F] H. Furusho proves the beautiful result that of the three defining equations for associators, the pentagon implies the two hexagons (see also [W]). In this note we present a simpler proof for this theorem (although our paper is less dense, and hence only slightly shorter). In particular, we package the use of algebraic geometry and Groethendieck-Teichmuller groups into a useful and previously known principle, and, less significantly, we eliminate the use of spherical braids.