Abstract
In previous work, Eli Aljadeff and the first-named author attached an algebra \mathcal 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 \mathcal 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 \mathcal 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 \mathcal V_H maps injectively into \mathcal B_H . In the second half of this paper, we show that \mathcal B_H is a localization of \mathcal 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.
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