On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra (ProQuest: Formulae and/or non-USASCII text omitted; see image) . In 1977, Duflo constructed a homomorphism of (ProQuest: Formulae and/or non-USASCII text omitted; see image) -modules (ProQuest: Formulae and/or non-USASCII text omitted; see image) , w...

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Bibliographic Details
Published in:Inventiones mathematicae 2006-06, Vol.164 (3), p.615-634
Main Authors: Alekseev, A., Meinrenken, E.
Format: Article
Language:English
Subjects:
Algebra
Lie groups
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Summary:Let G be a connected Lie group, with Lie algebra (ProQuest: Formulae and/or non-USASCII text omitted; see image) . In 1977, Duflo constructed a homomorphism of (ProQuest: Formulae and/or non-USASCII text omitted; see image) -modules (ProQuest: Formulae and/or non-USASCII text omitted; see image) , 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 (ProQuest: Formulae and/or non-USASCII text omitted; see image) , as well as a more general statement for distributions. [PUBLICATION ABSTRACT]
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-005-0486-4