The Kashiwara-Vergne conjecture and Drinfeld’s associators

By A. Alekseev, C. Torossian. 2008

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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 KRV 0 n , and a group KRV 2 which contains KRV 0 2 as a normal subgroup. We show that KRV 2 also contains the Grothendieck-Teichmuller group GRT 1 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 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. 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a5c8e9a163c4fea4865757cf3ae8cad45fadf86

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations (2009, methodology): “is a pronilpotent Lie algebra, the group exp(a) is isomorphic to a, equipped with the Campbell–Baker–Hausdorff product. We now describe the approach of [ AT2 ].”
Equivalence of formalities of the little discs operad (2009, background): “The Lie algebra morphism tn → sdern, H(CG(n))→ H(CGtree(n)), is injective, as proved in [1].”
M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra (2010, methodology): “Graphsn = S(ICGn[1]) where ICGn is spanned by internally connected graphs, shifted in degree by 1.”
Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, methodology): “Thus, the first named author, DBN, was absolutely delighted when in January 2008 Anton Alekseev described to him his joint work [2] with Charles Torossian: Anton told DBN that they found a relationship between the Kashiwara–Vergne conjecture [52], a cousin of the Duflo isomorphism (which DBN already”