On linearised and elliptic versions of the Kashiwara–Vergne Lie algebra

The goal of this article is to define a linearized or depth-graded version 𝔩𝔨𝔯𝔳 , and a closely related elliptic version 𝔨𝔯𝔳 ell , of the Kashiwara–Vergne Lie algebra 𝔨𝔯𝔳 originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara–Vergne problem. We show how the elliptic Lie algebra 𝔨𝔯𝔳 ell is related to earlier constructions of elliptic versions 𝔤𝔯𝔱 ell and 𝔡𝔰 ell of the Grothendieck–Teichmüller Lie algebra 𝔤𝔯𝔱 and the double shuffle Lie algebra 𝔡𝔰 respectively. Based on the known relationships between the three Lie algebra 𝔤𝔯𝔱 , 𝔡𝔰 and 𝔨𝔯𝔳 , we discuss the corresponding relationships between the linearized versions, and also between the elliptic versions.