Deformation theory of the wheeled properad of strongly homotopy Lie bialgebras and graph complexes

Abstract

It is well-known that the Lie algebra of homotopy non-trivial degree zero derivations of the properad of strongly homotopy Lie bialgebras $\mathcal{H}olieb$ can be identified with the Grothendieck-Teichmuller Lie algebra $\mathfrak{grt}$. We study in this paper the derivation complex of the wheeled closure $\mathcal{H}olie{b}^{\circlearrowleft }$ (and of its degree shifted version $\mathcal{H}olie{b}_{p,q}^{\circlearrowleft },\forall p,q\in \mathbb{Z}$) and establishing a quasi-isomorphism to a version of the Kontsevich graph complex. This result leads us to a surprising conclusion that the Lie algebra of homotopy non-trivial derivations of the wheeled properad $\mathcal{H}olie{b}^{\circlearrowleft }$ can be identified with the direct sum of \textit{two} copies of $\mathfrak{grt}$. As an illustrative example, we describe explicitly how the famous tetrahedron class in $\mathfrak{grt}$ acts as a derivation of $\mathcal{H}olie{b}^{\circlearrowleft }$ in two homotopy inequivalent ways.