The formality of the Goldman-Turaev Lie bialgebra on a closed surface

TL;DR

This paper investigates the Goldman-Turaev Lie bialgebra on closed surfaces, reformulating related algebraic structures and determining its automorphism group using non-commutative connections.

Contribution

It extends the Kashiwara-Vergne framework to higher genera and explicitly characterizes the automorphism group for closed surfaces.

Findings

Determined the pro-unipotent automorphism group of the Goldman-Turaev Lie bialgebra.

Reformulated Kashiwara-Vergne groups in higher genus contexts.

Connected algebraic structures with non-commutative connection tools.

Abstract

We reformulate the Kashiwara-Vergne groups and associators in higher genera, introduced in Alekseev-Kawazumi-Kuno-Naef, in terms of non-commutative connections using the tools developed in a previous paper. As the main result, the case of closed surfaces is dealt with to determine the pro-unipotent automorphism group of the associated graded of the Goldman-Turaev Lie bialgebra.