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On the Kashiwara–Vergne conjecture

Published inInventiones mathematicae, vol. 164, no. 3, p. 615-634
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  • Open Access - Licence nationale Springer
Publication date2006
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 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.

Citation (ISO format)
ALEKSEEV, Anton, MEINRENKEN, E. On the Kashiwara–Vergne conjecture. In: Inventiones mathematicae, 2006, vol. 164, n° 3, p. 615–634. doi: 10.1007/s00222-005-0486-4
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ISSN of the journal0020-9910
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