Journal of Noncommutative Geometry
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Published online: 2016-06-28
The Noether problem for Hopf algebras
Christian Kassel[1] and Akira Masuoka[2] (1) Université de Strasbourg, France(2) University of Tsukuba, Ibaraki, Japan
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.
Keywords: Hopf algebra, Noether problem, invariant theory, rationality, polynomial identities, localization
Kassel Christian, Masuoka Akira: The Noether problem for Hopf algebras. J. Noncommut. Geom. 10 (2016), 405-428. doi: 10.4171/JNCG/237