The logarithmic star product in the linear case and the Grothendieck--Teichm\"uller group

The purpose of this short note is to establish an explicit equivalence between two star products $\star$ and ${\star }_{\log }$ on the symmetric algebra $\mathrm{S}(\mathfrak{g})$ of a finite-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb{K}\supset \mathbb{C}$ of characteristic 0: the differential operator of infinite order with constant coefficients realizing the equivalence is related to the Grothendieck-Teichm"uller group.