MASCHKE-TYPE THEOREM AND DUALITY THEOREM FOR WEAK TWISTED SMASH PRODUCTS

Abstract

Let $H$ be a weak Hopf algebra in the sense of Böhm and Szlachányi [3] and $A$ a weak $H$-bimodule algebra. Then in this paper we first introduce the notion of a weak twisted smash product $A \star H$ and then find some sufficient and necessary conditions making it into a weak bialgebra. Furthermore, we give a Maschke-type theorem for the weak twisted smash product over semisimple weak Hopf algebra $H$, which generalizes the well-known Maschke-type theorem in [5, 15, 17]. Finally, we obtain an analogue of the duality theorem for the weak twisted smash products.