Oskar Frost | Searcharxiv

arXiv2024

Graph complexes and Deformation theories of the (wheeled) properads of quasi- and pseudo-Lie bialgebras

Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor $ϕ$. This structure was introduced by Drinfeld while studying deformation theory of universal enveloping algebras and has since seen many other applications in algebra and geometry. In this paper we study the derivation complex of strongly homotopy quasi-Lie bialgebra, both in the unwheeled (i.e standard) and wheeled case, and compute its cohomology in terms of Kontsevich graph complexes.

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arXiv2024

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

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.

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