U.N. Iyer, C. Kassel Generic Base Algebras and Universal Comodule Algebras for some finite-dimensional Hopf algebras

algebra H an algebra BH of rational fractions, which is a finitely generated smooth domain of Krull dimension equal to the dimension of H. The algebra BH is called the generic base algebra associated to H. The generic base algebra is known for very few Hopf algebras: it has been described for finite group algebras in a work by Aljadeff, Haile, Natapov and for the four-dimensional Sweedler algebra by C. Kassel. In this talk, I will describe BH for other finite-dimensional Hopf algebras such as the Taft algebras, the Hopf algebras E(n) and certain monomial Hopf algebras, all natural generalizations of the Sweedler algebra. This is joint work with C. Kassel. A theory of polynomial identities for comodule algebras was also worked out by Aljadeff and Kassel. It leads naturally to a universal comodule algebra UH, the analogue of the “relatively free algebra ” in the classical theory of polynomial identities. The subalgebra of coinvariants VH of UH maps injectively into BH. In the few known cases, the injection turns BH into a localization of VH. We show that this also holds for the Hopf algebras considered here. Finally for the same Hopf algebras we also describe a suitable central localization of UH as a BH-module