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

TL;DR

This paper explores the deformation theory of quasi-Lie bialgebras using graph complexes, extending classical Lie bialgebra concepts to more general structures with applications in algebra and geometry.

Contribution

It introduces a framework for analyzing the derivation complex of homotopy quasi-Lie bialgebras via Kontsevich graph complexes, including wheeled and unwheeled cases.

Findings

Computed the cohomology of the derivation complex in terms of Kontsevich graph complexes.

Extended deformation theory to quasi-Lie bialgebras with wheeled and unwheeled structures.

Provided new insights into the algebraic and geometric applications of these structures.

Abstract

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 $\phi$. 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.