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

Abstract

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.