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Genus Zero Kashiwara-Vergne Solutions from Braids

Zsuzsanna Dancso, Iva Halacheva, Guillaume Laplante-Anfossi, Marcy Robertson, Chandan Singh

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--Teichmüller 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.

Genus Zero Kashiwara-Vergne Solutions from Braids

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--Teichmüller 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.

Paper Structure

This paper contains 33 sections, 70 theorems, 466 equations, 21 figures.

Table of Contents

  1. Operad expansions: the operad of parenthesized braids and Drinfeld associators
  2. Operads
  3. The operad of parenthesized braids
  4. Completion and homomorphic expansions
  5. Associated graded of a group
  6. Prounipotent completion of groupoids and operads
  7. Associated graded structures for groupoids and operads
  8. Expansions in action: chord diagrams and Drinfeld associators
  9. The Grothendieck--Teichmüller groups
  10. Moperad expansions: The moperad of parenthesized braids with a frozen strand
  11. Moperads
  12. Completion and homomorphic expansions for moperads
  13. Parenthesized braids with a frozen strand
  14. Associators and moperad expansions
  15. A variation on the Grothendieck--Teichmüller groups
  16. ...and 18 more sections

Key Result

Theorem 1

Any moperad homomorphic expansion $(\varphi^1,\varphi):\mathsf{PaB}_{\mathbb{K}}^1 \rightarrow \mathsf{CD}^+$ with $\mu=1$ restricts to a tangential automorphism $F_{\varphi^1}$ on the free group $F_2 \subseteq \mathsf{PaB}^1((0(12)),(0(12)))$, and $F_{\varphi^1}$ is a symmetric KV solution. $\black

Figures (21)

  • Figure 1: A braid in $\mathsf{B}_3$, and the generators of $\mathsf{B}_3$. The braid on the left is drawn from bottom to top and can be written in terms of generators as $\beta_1\beta_2\beta_1^{-1}$.
  • Figure 2: A parenthesized permutation represented by a planar binary rooted tree with labeled leaves.
  • Figure 3: An example of a parenthesized braid.
  • Figure 4: The operadic composition of parenthesized braids.
  • Figure 5: The generating morphisms of the operad $\mathsf{PaB}$.
  • ...and 16 more figures

Theorems & Definitions (200)

  • Theorem : \ref{['thm: the construction is good']}
  • Theorem : Theorems \ref{['thm:KV-action-F-012']} and \ref{['thm:KRV']}
  • Theorem : \ref{['prop: symmetric KV solutions give KV solutions', 'cor: moperad map factors if KV Ass is Drinf Ass']}
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Example 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 190 more