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 KV_n, and an extension \hat{KV}_2 of the group KV_2. We show that the group \hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem Sol(KV). Furthermore, we prove that Sol(KV) is isomorphic to a direct product of a line \k (\k being a field of characteristic zero) and the set of solutions of the pentagon equation with values in the group KV_3. The latter contains the set of Drinfeld's associators as a subset. As a by-product, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators.