Quantizations of Lie bialgebras, duality involution and oriented graph complexes

TL;DR

This paper demonstrates that the Grothendieck-Teichmüller group action on Lie bialgebra properads commutes with a duality involution, and shows universal quantizations are homotopy equivalent to those respecting this involution, using new results in graph complex cohomology.

Contribution

It establishes the compatibility of the Grothendieck-Teichmüller group action with duality involution and introduces a novel cohomology invariance result for oriented graph complexes.

Findings

Grothendieck-Teichmüller group action commutes with duality involution

Universal quantizations are homotopy equivalent to involution-compatible ones

Involution on graph complexes induces identity on cohomology

Abstract

We prove that the action of the Grothendieck-Teichm\"uller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the identity map on its cohomology.