Double shuffle relation for associators

TL;DR

This paper proves that Drinfel'd's pentagon equation implies the generalized double shuffle relation, establishing an embedding of the Grothendieck-Teichmüller group into the double shuffle group and deriving the gamma factorization formula.

Contribution

It demonstrates that the pentagon equation implies the double shuffle relation and constructs an embedding of GRT_1 into DMR_0, settling a significant conjecture.

Findings

Drinfel'd's pentagon equation implies the generalized double shuffle relation

An embedding from GRT_1 to DMR_0 is established

The gamma factorization formula is derived from the double shuffle relation

Abstract

It is proved that Drinfel'd's pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichm\"uller group $GRT_1$ into Racinet's double shuffle group $DMR_0$ is obtained, which settles the project of Deligne-Terasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation.