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.
