Koszulity of a certain dioperad

Abstract

We establish that the dioperad $Y^{(n)}$, encoding bialgebras with a product of degree zero, a coproduct of degree $(1-n)$ and a rank three cyclic tensor, which satisfy a deformed version of the balanced infinitesimal bialgebra condition, is Koszul. This result is established by studying specific subcomplexes of the assocoipahedra of Poirier and Tradler. These subcomplexes are related to a certain type of meromorphic quadratic differential on $\mathbb{CP}^1$, which we call cloven Strebel differentials. Using that geometric interpretation, we can control the topology of the relevant subcomplexes and deduce the vanishing of higher cohomology of the corresponding dioperadic bar complexes.