Abstract
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.
Keywords
General Statement Recent Result Algebra Isomorphism Algebra Cohomology Direct CorollaryPreview
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