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

TL;DR

This paper explores the deformation theory of the wheeled properad of strongly homotopy Lie bialgebras, revealing a connection to the Grothendieck-Teichmuller Lie algebra and graph complexes, with implications for understanding derivations and symmetries.

Contribution

It establishes a quasi-isomorphism between the derivation complex of the wheeled closure of homotopy Lie bialgebras and a version of the Kontsevich graph complex, revealing a doubled Grothendieck-Teichmuller algebra structure.

Findings

Derivation complex of wheeled properad is quasi-isomorphic to a graph complex.

The Lie algebra of derivations is isomorphic to two copies of rak{grt}.

Explicit action of the tetrahedron class as a derivation.

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 $\mathcal{H}olieb^\circlearrowleft$ (and of its degree shifted version $\mathcal{H}olieb_{p,q}^\circlearrowleft,\ \forall p,q\in\mathbb{Z}$) 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 $\mathcal{H}olieb^{\circlearrowleft}$ 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 $\mathcal{H}olieb^{\circlearrowleft}$ in two homotopy inequivalent ways.