Double categories and quantum groupoids

TL;DR

This paper constructs a new class of weak Hopf algebras, or quantum groupoids, using double groupoids and cocycle data, extending classical Hopf algebra constructions to a more general setting.

Contribution

It introduces a novel construction of quantum groupoids from matched pairs of groupoids and cocycle data, generalizing Kac's classical Hopf algebra framework.

Findings

Established a new method for constructing quantum groupoids

Extended classical Hopf algebra theory to double groupoids

Provided examples illustrating the construction

Abstract

We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now well-known construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.