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 KRV0n, and a group KRV2 which con-tains KRV02 as a normal subgroup. We show that KRV2 also contains the Grothendieck-Teichmüller group GRT1 as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Furthermore, we prove that SolKV is isomorphic to a direct product of affine line A1 and the set of solutions of the pentagon equation with values in the group KRV03. 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.