Let H be a quasitriangular weak Hopf algebra. It is proved that the centralizer subalgebra of its source subalgebra in H is a braided group (or Hopf algebra in the category of left H-modules), which is cocommutative and also a left braided Lie algebra in the sense of Majid.
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.
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.