On the irrationality of certain coefficients of the Alekseev-Torossian associator

TL;DR

This paper provides explicit formulas for certain coefficients of the Alekseev-Torossian and a second Drinfeld associator, revealing their irrationality under a conjecture on multiple zeta values.

Contribution

It offers explicit formulas for initial coefficients of specific associators and demonstrates their irrationality assuming a conjecture on multiple zeta values.

Findings

Explicit formulas for initial coefficients of associators

Both associators are irrational under a conjecture

Analysis of the Grothendieck-Teichmüller group's action

Abstract

We give explicit formulas for the first few coefficients of the Alekseev-Torossian associator and a second Drinfeld associator. This is done by analyzing the free and transitive action of the Grothendieck-Teichm\"uller group and its Lie algebra $\mathfrak{grt}_1$ on the set of Drinfeld associators. As a result we obtain that, up to a conjecture on multiple zeta values, both associators are not rational.