The Frobenius properad is Koszul (Q345178)

\textit{B. Vallette} [Trans. Am. Math. Soc. 359, No. 10, 4865--4943 (2007; Zbl 1140.18006)] and \textit{M. Markl} and \textit{A. A. Voronov} [Prog. Math. 270, 249--281 (2009; Zbl 1208.18008)] independently proved that the properad of Lie bialgebra is Koszul. The Koszulness of the properad of involutive Lie bialgebra, conjectured by the authors of this paper in 2009, is here proved. By Koszul duality, this implies that the properad of (nonunital) Frobenius algebras is also Koszul; extra work gives the Koszulness of the properad of unital-counital Frobenius algebras.NEWLINENEWLINEIt is proved that the Grothendieck-Teichmüller group \(\mathrm{GRT}_1\) acts, in a nontrivial way, over minimal models of the properads of involutive Lie bialgebras/Frobenius algebras, and hence and homotopy involutive Lie bialgebras/Frobenius structures of a given vector space; this gives a large class of deformations of involutive Lie bialgebras.NEWLINENEWLINEFinally, for a given homotopy involutive Lie bialgebra, a homotopy Batalin-Vilkovisky algebra structure is built on the associated Chevalley-Eilenberg complex.