On the Kashiwara-Vergne conjecture

Abstract

Let $G$ be a connected Lie group, with Lie algebra $g$. In 1977, Duflo constructed a homomorphism of $g$-modules $Duf:S(g)->U(g)$, which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group $G$. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara-Vergne conjecture for any Lie group $G$. (2) We give a reformulation of the Kashiwara-Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism $H(g,S(g))->H(g,U(g))$, as well as a more general statement for distributions.

18 pages, final version, to be published in Inventiones Math

References in corpus (4)

• Deformation quantization of Poisson manifolds, I
• Sur la conjecture combinatoire de Kashiwara-Vergne
• Isomorphisme de Duflo et la cohomologie tangentielle
• Uniqueness in the Kashiwara-Vergne conjecture

Cited by in corpus (14)

• Kontsevich deformation quantization and flat connections
• Biquantization of symmetric pairs and the quantum shift
• The Kashiwara-Vergne conjecture and Drinfeld's associators
• Solution non universelle pour le problème KV-78
• Formality theorems for Hochschild complexes and their applications
• Uniqueness in the Kashiwara-Vergne conjecture
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• A Lie-algebraic approach to the local index theorem on compact homogeneous spaces
• Ribbon 2-Knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras
• Eulerian idempotent and Kashiwara-Vergne conjecture
• The Kashiwara-Vergne conjecture
• On the Jacobson element and generators of the Lie algebra $\mathfrak{grt}$ in nonzero characteristic
• Kashiwara-Vergne solutions degree by degree
• On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras