Total Integrals for Weak Doi–Koppinen Data

Abstract

In this paper, we introduce the concept of total integrals in the case of weak Hopf algebras, which is a generalization of the total integral introduced by Menini and Militaru. We give two necessary and sufficient conditions for the existence of a total integral, and study when the category \(\mathcal {M}_{A^{\rm coH}}\) of right modules and the category \(^{H}\mathcal {M}_{A}\) of weak Doi–Hopf modules are equivalent under the assumption that there exists a total integral, where AcoH denotes the coinvariant subalgebra of the weak left H-comodule algebra A. In the end, we give some applications of the affineness criterion for weak Doi–Hopf modules.