  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
$\mathbb{C}\backslash\{ z_1, \dots z_n\}$. 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.
