A functorial approach to Kashiwara-Vergne

TL;DR

This paper develops a functorial framework connecting operad automorphisms to Lie bialgebras, reproducing known injections and extending the relationship between GRT and KRV groups to higher genus cases.

Contribution

It introduces a functorial method to derive Lie bialgebras from operads, reproduces the Alekseev-Torossian injection, and extends the GRT-KRV relationship to higher genus groups.

Findings

Reproduces Alekseev-Torossian injection functorially.

Establishes a relationship between higher genus GRT and KRV groups.

Provides a framework linking operad automorphisms to Lie bialgebras.

Abstract

As a consequence of the proof of the Kashiwara-Vergne conjecture of Alekseev and Torossian, the authors obtained an injection $\mathrm{GRT} \hookrightarrow \mathrm{KRV}$. The group $\mathrm{GRT}$ can be regarded as the group of automorphisms of the operad of parenthesized chord diagrams, while $\mathrm{KRV}$ can be recovered from the automorphism group of the Goldman-Turaev Lie bialgebra of a thrice-punctured sphere. This suggests the existence of a natural way to derive Lie bialgebras from operads, and we verify this is the case. That is, we reproduce the Alekseev-Torossian injection by functorially constructing bracket and cobracket operations out of operad modules. This framework is enough to establish a relationship between Gonzalez' higher genus $\mathrm{GRT}_g$ groups, and the higher genus $\mathrm{KRV}_g$ groups of Alekseev, Kawazumi, Kuno, and Naef. Our construction is informed by Massuyeau and Turaev's work on Fox pairings and quasi-derivations.