The Kashiwara-Vergne conjecture and Drinfeld's associators
The authors study the uniqueness issue for the Kashiwara-Vergne (KV) conjecture. This problem is a property of the Campbell-Hausdorff series put forward in 1978, which has many implications in Lie theory and harmonic analysis. The problem was settled in the positive by \textit{E. Meinrenken} and the first author [J. Lie Theory 16, No. 3, 531--538 (2006; Zbl 1111.22017)]. To do this study, the authors introduce a family of infinite-dimensional groups KRV\(^0_n\), and a group KRV\(_2\) which contains KRV\(^0_2\) as a normal subgroup, and show that KRV\(_2\) also contains the Grothendieck-Teichmüller group GRT\(_1\) as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem SolKV. Besides, they prove that SolKV is isomorphic to a direct product of the affine line \(\mathbb{A}^1\) and the set of solutions of the pentagon equation with values in the group KRV\(^0_3\). The latter contains the set of Drinfeld's associators as a subset.NEWLINENEWLINEIn the last part of the paper, which is very extended and complete, and as a by-product of their construction, they obtain a new proof of the Kashiwara-Vergne conjecture based on Drinfeld's theorem on the existence of associators. In the paper under review they use several tools, as free Lie algebras and their derivations and a family of particular subalgebras, called Kashiwara-Vergne Lie algebras and deal with associators. Finally, they include two Appendices. In the first of them they collect the cohomology computations needed in the paper.
scientific article; zbMATH DE number 6024996
scientific article