Maschke-Type Theorem, Duality Theorem, and the Global Dimension for Weak Crossed Products

Abstract

The notion of a crossed product with a Hopf algebroid was introduced by Böhm and Brzeziński [12]. It is well-known that weak Hopf algebras (see Böhm et al. [10]) is an example of Hopf algebroids. Then we get the definition and some property of a weak crossed product A#σ H over an algebra A and a weak Hopf algebra H. Next we give a Maschke-type theorem for the weak crossed product over a semisimple weak Hopf algebra H. Furthermore, we obtain an analogue of the Nikshych's duality theorem for weak crossed products. Finally, using this duality theorem, we prove that the global dimension of A equals to the global dimension of A#σ H if H and H* are both semisimple.