The Noether problem for Hopf algebras

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.