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

Toyo Taniguchi

TL;DR

This work extends the Kashiwara–Vergne formality program to higher genus surfaces by reformulating KV groups and associators in terms of non-commutative connections on Hopf groupoids and their boundaries, exploiting a factorisation of the Turaev cobracket into a Hamiltonian flow and a divergence. It provides a concrete description of the pro-unipotent automorphism group of the associated graded Goldman–Turaev Lie bialgebra for closed surfaces as $\mathrm{KRV}_{(g,0)} = \exp(\mathfrak{krv}_{(g,0)})$ with $\mathfrak{krv}_{(g,0)} = \{ g\in\mathrm{Der}^+(\hat{L}(H)_{\omega}) : \mathrm{div}^{\nabla'_{\bullet,H}}(g) \in \ker(|\bar{\Delta}_{\omega}|)\}$, and develops a basis for $|T(H)_{\omega}|$ via rewriting rules, enabling explicit low-degree kernel computations of the reduced coproduct. The paper also establishes a framework connecting KV theory with non-commutative geometry through connections and divergences, providing tools for analyzing higher-genus associators and their automorphism groups. These advances offer a rigorous bridge between formality in the genus-zero theory and its higher-genus counterpart, with potential implications for elliptic and higher-genus associator constructions and related quantum algebra structures.

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.

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

TL;DR

This work extends the Kashiwara–Vergne formality program to higher genus surfaces by reformulating KV groups and associators in terms of non-commutative connections on Hopf groupoids and their boundaries, exploiting a factorisation of the Turaev cobracket into a Hamiltonian flow and a divergence. It provides a concrete description of the pro-unipotent automorphism group of the associated graded Goldman–Turaev Lie bialgebra for closed surfaces as with , and develops a basis for via rewriting rules, enabling explicit low-degree kernel computations of the reduced coproduct. The paper also establishes a framework connecting KV theory with non-commutative geometry through connections and divergences, providing tools for analyzing higher-genus associators and their automorphism groups. These advances offer a rigorous bridge between formality in the genus-zero theory and its higher-genus counterpart, with potential implications for elliptic and higher-genus associator constructions and related quantum algebra structures.

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.

Paper Structure

This paper contains 12 sections, 36 theorems, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Goldman--Turaev Lie Bialgebra
  3. The Formality and the Kashiwara--Vergne Problem
  4. Reformulation of KV Groups
  5. Tangential Derivations
  6. Connections on Modules
  7. Integration and Connections
  8. Rewriting Rules
  9. The Closed Surface Case
  10. The Kernel of the Reduced Coproduct
  11. The Case of a Free Associative Algebra
  12. Proof of Proposition \ref{['prop:ckom']}.

Key Result

Theorem 1

The KV groups and the set of KV associators are expressed as follows:

Figures (3)

  • Figure 1: The configuration of beads corresponding to the vertex $|(y_g^{[1]}x_g^{[i_1]}\cdots x_g^{[i_2-1]})\cdots (y_g^{[t]}x_g^{[i_t]}\cdots x_g^{[i_1-1]})|$.
  • Figure 2: The move $m_j$.
  • Figure 3: The collection move and the re-distribution move.

Theorems & Definitions (70)

  • Theorem : Theorem \ref{['thm:kvboundary']}
  • Theorem : Theorem \ref{['thm:kvclosed']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: goldman,turaev,akkn
  • Theorem 2.5: sigma
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 60 more