The Kashiwara-Vergne conjecture and Drinfeld's associators
type: Item
instance of: scholarly article
title: The Kashiwara-Vergne conjecture and Drinfeld's associators
zbMATH Open document ID: 1243.22009
author: Charles Torossian
author: Anton Alekseev
published in: Annals of Mathematics. Second Series
publication date: 2012-04-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0802.4300
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 6024996
zbMATH Keywords: Campbell-Hausdorff series
zbMATH Keywords: Drinfeld associators
zbMATH Keywords: Kashiwara-Vergne conjecture
Wikidata QID: Q122912438
reviewed by: Juan Núñez Valdés
MaRDI profile type: Publication
OpenAlex ID: W2125748670
arXiv ID: 0802.4300
cites work: Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations
cites work: Solution non universelle pour le probl\`eme KV-78
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cites work: Pentagon and hexagon equations
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DOI: 10.4007/ANNALS.2012.175.2.1
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review text: 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.
description: scientific article; zbMATH DE number 6024996