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On the Kashiwara-Vergne conjecture

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arxiv math/0506499 v3 pith:KSRHW4BN submitted 2005-06-24 math.QA math.RT

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keywords algebraconjecturegroupkashiwara-vergnedistributionsdufloisomorphismbi-invariant
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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 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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  1. The formality of the Goldman-Turaev Lie bialgebra on a closed surface

    math.QA 2025-02 accept novelty 8.0 of 10

    The pro-unipotent automorphism group of the associated graded Goldman-Turaev Lie bialgebra on a closed surface is explicitly described in terms of a divergence map and the kernel of a reduced coproduct.

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