We obtain the structure theorem for -Hopf bimodules over Hopf algebroids, where H is the total algebra of the Hopf algebroid . Based on this theorem, we investigate the structure theorem for comodule algebras over Hopf algebroids.
We prove that the conjugate convolution operators can be used to calculate jumps for functions. Our results generalize the theorems established by He and Shi. Furthermore, by using Lukács and Móricz's idea, we solve an open question posed by Shi and Hu.
The paper is concerned with endomorphism algebras for weak Doi-Hopf modules. Under the condition “weak Hopf-Galois extensions”,
we present the structure theorem of endomorphism algebras for weak Doi-Hopf modules, which extends Theorem 3.2 given by Schneider
in . As applications of the structure theorem, we obtain the Kreimer-Takeuchi theorem (see Theorem 1.7 in ) and the
Nikshych duality theorem (see Theorem 3.3 in ) in the case of weak Hopf algebras, respectively.