Liouville Extensions of Artinian Simple Module Algebras

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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 define the notion of Liouville extensions of AS D-module algebras and to characterize the corresponding Picard–Vessiot group schemes.