Examples of weak Hopf algebras arising from vacant double groupoids

TL;DR

This paper constructs explicit examples of weak Hopf algebras using vacant double groupoids, linking their structure to group cohomology and providing concrete instances.

Contribution

It introduces a method to construct weak Hopf algebras from vacant double groupoids and relates their classification to group cohomology, offering explicit examples.

Findings

Explicit examples of weak Hopf algebras from vacant double groupoids

Connection between Kac exact sequence and group cohomology

Finite vacant double groupoids explicitly described

Abstract

We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi) via vacant double groupoids as explained in \http://arxiv.org/abs/math.QA/0308228. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.