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A functorial approach to Kashiwara-Vergne

Rodrigo Navarro-Betancourt

TL;DR

The paper develops a functorial framework to derive Lie brackets and cobrackets from operad modules, connecting operadic automorphisms (GRT) with the Kashiwara–Vergne groups (KRV) in both genus zero and higher-genus settings. It constructs a functorial bridge from framed operads of parenthesized chord diagrams to the Goldman–Turaev Lie bialgebra via Fox pairings and quasi-derivations, organized through relative Lie cohomology and decorated operadic modules. This yields a natural, functorial realization of the Alekseev–Torossian injection GRT_g^f → KRV_g,n+1^f, and identifies how higher-genus automorphism groups act on the associated graded Goldman–Turaev bialgebras. The work unifies operadic, cohomological, and topological perspectives to expose the algebraic underpinnings of the GV/KV landscape across genus, framing, and puncture configurations, with explicit constructions for framed genus diagrams PaCD^f_g and their GK–KRV echoes.

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.

A functorial approach to Kashiwara-Vergne

TL;DR

The paper develops a functorial framework to derive Lie brackets and cobrackets from operad modules, connecting operadic automorphisms (GRT) with the Kashiwara–Vergne groups (KRV) in both genus zero and higher-genus settings. It constructs a functorial bridge from framed operads of parenthesized chord diagrams to the Goldman–Turaev Lie bialgebra via Fox pairings and quasi-derivations, organized through relative Lie cohomology and decorated operadic modules. This yields a natural, functorial realization of the Alekseev–Torossian injection GRT_g^f → KRV_g,n+1^f, and identifies how higher-genus automorphism groups act on the associated graded Goldman–Turaev bialgebras. The work unifies operadic, cohomological, and topological perspectives to expose the algebraic underpinnings of the GV/KV landscape across genus, framing, and puncture configurations, with explicit constructions for framed genus diagrams PaCD^f_g and their GK–KRV echoes.

Abstract

As a consequence of the proof of the Kashiwara-Vergne conjecture of Alekseev and Torossian, the authors obtained an injection . The group can be regarded as the group of automorphisms of the operad of parenthesized chord diagrams, while 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 groups, and the higher genus groups of Alekseev, Kawazumi, Kuno, and Naef. Our construction is informed by Massuyeau and Turaev's work on Fox pairings and quasi-derivations.

Paper Structure

This paper contains 26 sections, 84 theorems, 450 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Fox pairings and quasi-derivations
  4. Fox pairings
  5. Quasi-derivations
  6. Brackets and cobrackets
  7. Lie cohomology
  8. Relative cohomology
  9. Relative extensions
  10. Revisiting $\mathbf{Fox}$
  11. Operads
  12. Operad modules
  13. Infinitesimal braid algebras
  14. Parenthesized chord diagrams
  15. Higher genus Kashiwara-Vergne groups
  16. ...and 11 more sections

Key Result

Theorem 1.1

For $n\geq 1$, there exist functors $\Lambda_{n+1}: \mathrm{OpR}^\Delta_f(n+1) \rightarrow \mathbf{GoTu}$ such that $\Lambda_{n+1}(\mathbf{PaCD}^f, \mathbf{PaCD}^f)$ is isomorphic to the associated graded of the Goldman-Turaev bialgebra of the compact surface of genus $0$ with $n+1$ boundary compone

Theorems & Definitions (195)

  • Theorem 1.1: Theorem \ref{['big_adp']}
  • Theorem 1.2: Theorem \ref{['PaCD_is_GT']}
  • Theorem 1.3: Theorem \ref{['GRT_in_KRV']}
  • Theorem 1.4: Theorem \ref{['quasi-iso']}
  • Theorem 1.5: Theorem \ref{['diff_char']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • ...and 185 more