In a previous article (Amano and Masuoka, 2005), the author and Masuoka developed a Picard-Vessiot theory for module algebras over a cocommutative pointed smooth Hopf algebra D. By using the notion of Artinian simple (AS) D-module algebras, it generalizes and unifies the standard Picard-Vessiot theories for linear differential and difference equations. The purpose of this article is to de. ne the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picard-Vessiot group schemes.