The Frobenius properad is Koszul

Abstract

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, with the help of an earlier result of one of the authors, that there is a highly nontrivial action of the Grothendieck–Teichmüller group ${\mathrm{GRT}}_1$ on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms. As a corollary, one obtains a large class of universal deformations of (involutive) Lie bialgebras and Frobenius algebras, parameterized by elements of the Grothendieck–Teichmüller Lie algebra. We also prove that for any given homotopy involutive Lie bialgebra structure on a vector space, there is an associated homotopy Batalin–Vilkovisky algebra structure on the associated Chevalley–Eilenberg complex.