Weak Hopf algebras and quantum groupoids

TL;DR

This paper compares weak Hopf algebras and quantum groupoids, clarifying their relationships and extending the theory to modules, comodules, duality, and Hopf algebroids.

Contribution

It provides a detailed comparison between weak Hopf algebras and $ imes_R$-bialgebras, clarifying their equivalences and extending the theory to related structures.

Findings

Weak bialgebras are equivalent to $ imes_R$-bialgebras with Frobenius-separable $R$.

Extended the comparison to module and comodule theories.

Discussed conditions for bialgebroids to be called Hopf algebroids.

Abstract

We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a $\times_R$-bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a $\times_R$-bialgebra in which $R$ is Frobenius-separable. We extend the comparison to cover module and comodule theory, duality, and the question when a bialgebroid should be called a Hopf algebroid.