From the Lie operad to the Grothendieck-Teichm\"uller group

TL;DR

This paper explores the deformation theory of Lie operads and their polydifferential versions, establishing connections with the Grothendieck-Teichmüller group and universal enveloping algebras, revealing new symmetries and rigidity properties.

Contribution

It demonstrates that the deformation complex of the morphism from the Lie operad to its polydifferential version is quasi-isomorphic to the Kontsevich graph complex, linking operad deformations to graph complexes.

Findings

The Grothendieck-Teichmüller group acts as a symmetry group in the case d=2.

Unique non-trivial homotopy deformation exists for d=1, described explicitly.

The universal enveloping functor exhibits rigidity in the homotopy setting.

Abstract

We study the deformation complex of the standard morphism from the degree $d$ shifted Lie operad to its polydifferential version, and prove that it is quasi-isomorphic to the Kontsevich graph complex $\mathbf{GC}_d$. In particular, we show that in the case $d=2$ the Grothendieck-Teichm\"uller group $\mathbf{GRT_1}$ is a symmetry group (up to homotopy) of the aforementioned morphism. We also prove that in the case $d=1$ corresponding to the usual Lie algebras the standard morphism admits a unique homotopy non-trivial deformation which is described explicitly with the help of the universal enveloping construction. Finally we prove the rigidity of the strongly homotopy version of the universal enveloping functor in the Lie theory.