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 ${\mathbf{Lieb}}_{c,d}$ to its polydifferential version $\mathcal{D}({\mathbf{Lieb}}_{c,d})$ and show that it is quasi-isomorphic to the oriented graph complex ${\mathbf{GC}}_{c+d+1}^{\text{or}}$, up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex ${\mathbf{GC}}_{c+d}$, we conclude that 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 ${\mathbf{Lieb}}_{c,d}$ to $\mathcal{D}({\mathbf{Lieb}}_{c,d})$. The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product $H({\mathbf{GC}}_c)\otimes H({\mathbf{GC}}_d)$ of cohomologies of Kontsevich graph complexes ${\mathbf{GC}}_c\otimes {\mathbf{GC}}_d$.

11 pages