On rational Drinfeld associators

TL;DR

This paper establishes bounds on denominators of rational Drinfeld associators, leveraging p-adic associators and Galois actions, with applications to Duflo's question and Kontsevich knot invariants.

Contribution

It provides the first denominator estimates for rational Drinfeld associators and applies these results to solve Duflo's question and improve bounds on Kontsevich invariants.

Findings

Proved denominator bounds for rational Drinfeld associators

Resolved Duflo's question on Jacobson element factorizations in characteristic p

Derived new estimates for Kontsevich knot invariants

Abstract

We prove an estimate on denominators of rational Drinfeld associators. To obtain this result, we prove the corresponding estimate for the p-adic associators stable under the action of suitable elements of Gal(\bar{Q}/Q). As an application, we settle in the positive Duflo's question on the Kashiwara--Vergne factorizations of the Jacobson element J_p(x,y)=(x+y)^p-x^p-y^p in the free Lie algebra over a field of characteristic p. Another application is a new estimate on denominators of the Kontsevich knot invariant.