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 algebra 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 implies a positive answer
for the classical Noether problem for the group. We show that the generalized
Noether problem has a positive answer for all pointed finite-dimensional Hopf
algebras over a field of characteristic zero. We actually give a precise
description of B_H for such a Hopf algebra, including a bound on the degrees of
the generators.
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 again H is a pointed finite-dimensional 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.Comment: 19 pages. Section 4.3 and three references have been added to Version