Comparison of two notions of weak crossed product

TL;DR

This paper compares two different notions of weak crossed products in the context of weak Hopf algebras and Hopf algebroids, clarifying their relationships and differences.

Contribution

It provides a detailed comparison of two notions of weak crossed products, extending the understanding of their similarities and differences.

Findings

Identifies key differences between the two notions of weak crossed products.

Establishes conditions under which the notions coincide.

Clarifies the relationship between crossed products with weak Hopf algebras and Hopf algebroids.

Abstract

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in \cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~\cite{AG}