Wheeled PROPs, graph complexes and the master equation

Motivated by the universal obstruction to the deformation quantization of Poisson structures in infinite dimensions we introduce the notion of quantizable odd Lie bialgebra. The main result of the paper is a construction of a highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so called ...

Added: September 30, 2026

Formality Theorem for Quantizations of Lie Bialgebras

Merkulov S., Letters in Mathematical Physics 2016 Vol. 106 No. 2 P. 169–195

Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras. ...

Added: September 30, 2026

The Frobenius properad is Koszul

Merkulov S., Willwacher T., Campos R., Duke Mathematical Journal 2016 Vol. 165 No. 1 P. 2921–2989

We show the Koszulness of the properad governing involutive Lie bialgebras and also of the properads governing nonunital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, ...

Added: September 30, 2026

Pre-Calabi-Yau algebras and oriented gravity properad

Merkulov S., Journal of Pure and Applied Algebra 2026 Vol. 230 P. 1–19

We study the dual cyclic Hochschild complex $Cyc(A,\K)$ of a (possibly, infinite-dimensional) $A_\infty$-algebra $(A,\mu)$ and prove that any pre-Calabi-Yau extension $\pi$ of the given $A_\infty$ structure $\mu$ in $A$ induces on the cyclic cohomology of $(A,\mu)$ a representation of a new dg properad of {\em oriented}\, ribbon graphs. We compute the cohomology of that properad in terms of ...

Added: September 29, 2026

On Quantizable Odd Lie Bialgebras

Khoroshkin A., Mekulov S., Willwacher T., Letters in Mathematical Physics 2016 Vol. 106 No. 9 P. 1199–1215

Motivated by the obstruction to the deformation quantization of Poisson structures in infinitedimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures. ...

Added: September 11, 2016