Emergent version of Drinfeld's associator equations

Abstract

The works of Alekseev and Torossian [AT] and Alekseev, Enriquez, and Torossian [AET] show that any solution of Drinfeld's associator equations gives rise to a solution of the Kashiwara-Vergne equations in an explicit way. We introduce a weak version of Drinfeld's associator equations that we call the emergent version of the original equations. It is shown that solutions to the resulting linearized emergent Drinfeld's equations still lead to solutions to the linearized Kashiwara-Vergne equations. The emergent Drinfeld equations arise within a natural topological context of emergent braids, which we discuss. Our results are adjacent to the results of Bar-Natan, Dancso, Hogan, Liu and Scherich [BDHLS] on the relationship between emergent tangles and the Goldman-Turaev Lie bialgebra. We hope that in time our results will play a role in relating several bodies of work, on Drinfeld associators, Kashiwara-Vergne equations, and on expansions for classical tangles, for w-tangles, and for the Goldman-Turaev Lie bialgebra.