J. Noncommut. Geom. 10:2 (2016), 405-428; arXiv:1404.4941
Mathematics Subject Classification (2000): 16T05, 16W22, 16R50, 14E08 (Primary); 13A50, 13B30, 12F20 (Secondary)
Abstract.
In previous work, Eli Aljadeff and the first-named author
attached an algebra BH of rational fractions to each Hopf algebra H.
The generalized Noether problem is the following: for which finite-dimensional Hopf algebras H
is BH 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 for 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 BH 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 VH maps injectively into BH.
In the second half of this paper, we show that BH is a localization of VH
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.
Downloadable pdf file from (téléchargeable de) from arXiv:1404.4941
(5 octobre 2018)