Some Quasi-Hopf Algebras With Gauge Transformation

Let H be a quasi-Hopf algebra, A be an H - module algebra and F∈.H(?) H be a gauge transformation. We prove that A#H ≌ A_F-1., #H_F ,(H_F )_F-1= H , Maschke type theorem, the category of A#H -module and the category of A_F-1#H_F^- module are isomorphic, and give a definition of A be a rightA#H -module in the quasi-Hopf sense. Let H be a Hopf algebra and A an H - module algebra. In this thesis we prove that A is a Gorenstein algebra if and only if A^H is a Gorenstein algebra in the Hopf sense. We also give a condition underwhich A/A^H is a Frobenius extension.