# The Noether problem for Hopf algebras

### Christian Kassel

Université de Strasbourg, France### Akira Masuoka

University of Tsukuba, Ibaraki, Japan

## Abstract

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.

## Cite this article

Christian Kassel, Akira Masuoka, The Noether problem for Hopf algebras. J. Noncommut. Geom. 10 (2016), no. 2, pp. 405–428

DOI 10.4171/JNCG/237