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On triviality of the Kashiwara-Vergne problem for quadratic Lie algebras | Anton Alekseev
; Charles Torossian
; | Date: |
21 Sep 2009 | Abstract: | We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras
(that is, Lie algebras admitting an invariant scalar product) reduces to the
problem of representing the Campbell-Hausdorff series in the form
ln(e^xe^y)=x+y+[x,a(x,y)]+[y,b(x,y)], where a(x,y) and b(x,y) are Lie series in
x and y. This observation explains the existence of explicit rational solutions
of the quadratic KV problem (see M. Vergne, C.R.A.S. 329 (1999), no. 9,
767--772 and A. Alekseev, E. Meinrenken, C.R.A.S. 335 (2002), no. 9, 723--728
arXiv:math/0209346), whereas constructing an explicit rational solution of the
full KV problem would probably require the knowledge of a rational Drinfeld
associator. It also gives, in the case of quadratic Lie algebras, a direct
proof of the Duflo theorem (implied by the KV problem). | Source: | arXiv, 0909.3743 | Services: | Forum | Review | PDF | Favorites |
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