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

TL;DR

This paper establishes an explicit equivalence between two star products on the symmetric algebra of a Lie algebra, linking the differential operator realizing this equivalence to the Grothendieck-Teichmüller group.

Contribution

It provides a concrete connection between two star products and the Grothendieck-Teichmüller group in the linear case.

Findings

Explicit equivalence between star products $ullet$ and $ullet_{log}$.

Differential operator related to the Grothendieck-Teichmüller group.

Clarification of the algebraic structure in the linear case.

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.