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Weak Hopf algebras and quantum groupoids

Peter Schauenburg
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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.

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Peter Schauenburg. 2002-04-13. Weak Hopf algebras and quantum groupoids. https://arxiv.org/abs/math/0204180

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