On the Kashiwara-Vergne conjecture

• 成果类型: Article
• 署名作者: Alekseev, A; Meinrenken, E
• 署名单位: University of Geneva; University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-005-0486-4
• 发表日期: 2006
• 页码: 615-634
• 关键词: deformation quantization; invariant distributions; PROOF

摘要

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 biinvariant 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.