Polydifferential Lie bialgebras and graph complexes

Abstract

We study the deformation complex of a canonical morphism i from the properad of (degree shifted) Lie bialgebras \(\textbf{Lieb}_{c,d}\) to its polydifferential version \(\mathcal {D}(\textbf{Lieb}_{c,d})\) and show that it is quasi-isomorphic to the oriented graph complex \(\textbf{GC}^{{\text {or}}}_{c+d+1}\), up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex \(\textbf{GC}_{c+d}\), we conclude that for \(c+d=2 \) the space of homotopy non-trivial infinitesimal deformations of the canonical map i can be identified with the Grothendieck–Teichmüller Lie algebra \(\mathfrak {grt}\); moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism i from \(\textbf{Lieb}_{c,d}\) to \(\mathcal {D}(\textbf{Lieb}_{c,d})\). The full deformation complex is described with the help of a new graph complex of so called entangled graphs.