The Noether problem for Hopf algebras

In previous work, Eli Aljadeff and the first-named author attached an algebra B-H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebras H is B-H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group G implies a positive answer to the classical Noether problem for G. We show that the generalized Noether problem has a positive answer for all finite-dimensional pointed Hopf algebras over a field of characteristic zero (we actually give a precise description of B-H for such a Hopf algebra). A theory of polynomial identities for comodule algebras over a Hopf algebra H gives rise to a universal comodule algebra whose subalgebra of coinvariants V-H maps injectively into B-H. In the second half of this paper, we show that B-H is a localization of V-H when H is a finite-dimensional pointed Hopf algebra in characteristic zero. We also report on a result by Uma Iyer showing that the same localization result holds when H is the algebra of functions on a finite group.