Mixed Pentagon, octagon and Broadhurst duality equation

TL;DR

This paper explores the relationships between various algebraic equations and dualities in the context of cyclotomic analogues of associators, providing new proofs and formalism to understand their structure.

Contribution

It demonstrates that the mixed pentagon equation implies the octagon equation for N=2, and develops a formalism of infinitesimal module categories for proof derivation.

Findings

Mixed pentagon implies octagon for N=2

Broadhurst duality is compatible with torsor structure

Developed formalism of infinitesimal module categories

Abstract

This paper is on elimination of defining equations of the cyclotomic analogues, introduced by the first author, of Drinfeld's scheme of associators. We show that the mixed pentagon equation implies the octagon equation for N=2 and the particular distribution relation. We also explain that Broadhurst duality is compatible with the torsor structure. We develop a formalism of infinitesimal module categories and use it for deriving a proof left implicit in the first named author's earlier work.