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Article overview
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Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection | Anton Alekseev
; Florian Naef
; | Date: |
10 Aug 2017 | Abstract: | For an oriented 2-dimensional manifold $Sigma$ of genus $g$ with $n$
boundary components the space
$mathbb{C}pi_1(Sigma)/[mathbb{C}pi_1(Sigma), mathbb{C}pi_1(Sigma)]$
carries the Goldman-Turaev Lie bialgebra structure defined in terms of
intersections and self-intersections of curves. Its associated graded (under
the natural filtration) is described by cyclic words in $H_1(Sigma)$ and
carries the structure of a necklace Schedler Lie bialgebra. The isomorphism
between these two structures in genus zero has been established in [G.
Massuyeau, Formal descriptions of Turaev’s loop operations] using Kontsevich
integrals and in [A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef, The
Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem]
using solutions of the Kashiwara-Vergne problem.
In this note we give an elementary proof of this isomorphism over
$mathbb{C}$. It uses the Knizhnik-Zamolodchikov connection on
$mathbb{C}ackslash{ z_1, dots z_n}$. The proof of the isomorphism for Lie
brackets is a version of the classical result by Hitchin. Surprisingly, it
turns out that a similar proof applies to cobrackets. | Source: | arXiv, 1708.3119 | Services: | Forum | Review | PDF | Favorites |
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