Koszulity of a certain dioperad

TL;DR

This paper proves that a specific dioperad related to bialgebras with certain degree conditions is Koszul by analyzing associated geometric structures called cloven Strebel differentials.

Contribution

It introduces a novel geometric approach using cloven Strebel differentials to establish Koszulity of the dioperad encoding these bialgebras.

Findings

The dioperad $Y^{(n)}$ is Koszul.

Subcomplexes of assocoipahedra are related to cloven Strebel differentials.

Vanishing of higher cohomology of dioperadic bar complexes is demonstrated.

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.