Mixed pentagon, octagon, and Broadhurst duality equations

This paper is on the elimination of defining equations of the cyclotomic analogues, introduced by the first-named author, of Drinfeld’s scheme of associators [7]. 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 author’s earlier work.