The Betti side of the double shuffle theory. II. Double shuffle relations for associators

TL;DR

This paper establishes a connection between associators and double shuffle relations by exploring stabilizer torsors and their algebraic properties, completing a key goal in the series on the Betti side of double shuffle theory.

Contribution

It introduces stabilizer torsors for associators and demonstrates their relationships, advancing the understanding of double shuffle relations in the context of associators.

Findings

Inclusion of the torsor of associators into double shuffle relations.

Description of double shuffle torsor via module stabilizer torsor.

Containment of the module stabilizer torsor within the algebra stabilizer torsor.

Abstract

We derive from the compatibility of associators with the module harmonic coproduct, obtained in Part I of the series, the inclusion of the torsor of associators into that of double shuffle relations, which completes one of the aims of this series. We define two stabilizer torsors using the module and algebra harmonic coproducts from Part I. We show that the double shuffle torsor can be described using the module stabilizer torsor, and that the latter torsor is contained in the algebra stabilizer torsor.