• DOI:10.1007/s00222-005-0486-4
• Corpus ID: 8052211

On the Kashiwara–Vergne conjecture

@article{Alekseev2005OnTK,
  title={On the Kashiwara–Vergne conjecture},
  author={A. Alekseev and Eckhard Meinrenken},
  journal={Inventiones mathematicae},
  year={2005},
  volume={164},
  pages={615-634}
}

• A. Alekseev, E. Meinrenken
• Published 24 June 2005
• Mathematics
• Inventiones mathematicae

Let G be a connected Lie group, with Lie algebra $\mathfrak{g}$. In 1977, Duflo constructed a homomorphism of $\mathfrak{g}$-modules $\text{Duf}\colon S(\mathfrak{g})\to U(\mathfrak{g})$, 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 $H(\mathfrak{g},S(\mathfrak{g}))\to H(\mathfrak{g},U(\mathfrak{g}))$, as well as a more general statement for distributions.