On the Kashiwara-Vergne conjecture

Details On the Kashiwara-Vergne conjecture On the Kashiwara-Vergne conjecture

2005-06-24

arxiv.org/abs/math/0506499v3

Invent. Math. 164, 615-634 (2006)

Mathematics

Quantum Algebra

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

Scientific paper

10.1007/s00222-005-0486-4

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.

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