Deformation theory of Lie bialgebra properads

TL;DR

This paper computes the homotopy derivations of properads related to Lie bialgebras, revealing the action of the Grothendieck-Teichmüller group and classifying automorphisms and deformations, with implications for deformation theory and graph complexes.

Contribution

It explicitly determines the homotopy derivations of Lie bialgebra properads and describes their automorphisms and deformations, connecting these to Kontsevich graph complexes.

Findings

Grothendieck-Teichmüller group acts faithfully on properads

Odd Lie bialgebra properad has a unique non-trivial automorphism

Explicit description of the unique non-trivial deformation

Abstract

We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck-Teichm\"uller group acts faithfully (and essentially transitively) on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that by contrast to the even case the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism - the standard rescaling automorphism, and that it has precisely one non-trivial deformation which we describe explicitly.