The Noether problem for Hopf algebras

  • Christian Kassel

    Université de Strasbourg, France
  • Akira Masuoka

    University of Tsukuba, Ibaraki, Japan


In previous work, Eli Aljadeff and the first-named author attached an algebra of rational fractions to each Hopf algebra . The generalized Noether problem is the following: for which finite-dimensional Hopf algebras is the localization of a polynomial algebra? A positive answer to this question when is the algebra of functions on a finite group implies a positive answer to the classical Noether problem for . 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 for such a Hopf algebra).

A theory of polynomial identities for comodule algebras over a Hopf algebra gives rise to a universal comodule algebra whose subalgebra of coinvariants maps injectively into . In the second half of this paper, we show that is a localization of when 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