The Kashiwara-Vergne conjecture and Drinfeld’s associators

Abstract

The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite-dimensional groups ${\rm KRV}^0_n$, and a group ${\rm KRV}_2$ which contains ${\rm KRV}^0_2$ as a normal subgroup. We show that ${\rm KRV}_2$ also contains the Grothendieck-Teichmüller group ${\rm GRT}_1$ as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem ${\rm SolKV}$. Furthermore, we prove that ${\rm SolKV}$ is isomorphic to a direct product of affine line $\mathbb{A}^1$ and the set of solutions of the pentagon equation with values in the group ${\rm KRV}^0_3$. The latter contains the set of Drinfeld’s associators as a subset. As a by-product of our construction, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld’s theorem on existence of associators.

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Authors

Anton Alekseev

Section de mathématiques
Université de Genève
2-4 rue du Lièvre
Case postale 64
CH-1211 Genève 4
Switzerland

Charles Torossian

Université Denis-Diderot-Paris 7
Institut de mathématiques de Jussieu
175 rue du Chevaleret
75013 Paris
France