Genus Zero Kashiwara-Vergne Solutions from Braids

TL;DR

This paper reinterprets the construction of Kashiwara--Vergne solutions using moperads and braids, establishing connections with associators, Grothendieck--Teichmüller groups, and automorphisms of free Lie algebras.

Contribution

It introduces a new operadic framework linking braids, chord diagrams, and KV solutions, revealing how group actions influence these structures.

Findings

Equivalence between moperads yields genus zero KV solutions generated by a single classical solution.

Grothendieck--Teichmüller groups act on KV solutions, intertwining with symmetry groups.

Symmetric KV solutions correspond to module maps factoring through chord diagrams if associated with Drinfeld associators.

Abstract

Using the language of moperads -- monoids in the category of right modules over an operad -- we reinterpret the Alekseev--Enriquez--Torossian construction of Kashiwara--Vergne (KV) solutions from associators. We show that any equivalence between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck--Teichm\"uller module groups act on the latter, intertwining the actions of the KV symmetry groups. In the other direction, we show that any symmetric KV solution gives rise to a module map from parenthesized braids with a frozen strand to tangential automorphisms of free Lie algebras. This map factors through the moperad of chord diagrams if and only if the associated KV associator is a Drinfeld associator.