485 research outputs found

### Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

We define polynomial H-identities for comodule algebras over a Hopf algebra H
and establish general properties for the corresponding T-ideals. In the case H
is a Taft algebra or the Hopf algebra E(n), we exhibit a finite set of
polynomial H-identities which distinguish the Galois objects over H up to
isomorphism.Comment: 12 pages. V2 is an extended version of v1: Sections 2.3 and 3 are
new; title has been changed and references added. V3: a few typos correcte

### The Noether problem for Hopf algebras

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 algebra 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 implies a positive answer
for the classical Noether problem for the group. We show that the generalized
Noether problem has a positive answer for all pointed finite-dimensional Hopf
algebras over a field of characteristic zero. We actually give a precise
description of B_H for such a Hopf algebra, including a bound on the degrees of
the generators.
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 again H is a pointed finite-dimensional 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.Comment: 19 pages. Section 4.3 and three references have been added to Version

### Norm formulas for finite groups and induction from elementary abelian subgroups

It is known that the norm map N_G for a finite group G acting on a ring R is
surjective if and only if for every elementary abelian subgroup E of G the norm
map N_E for E is surjective. Equivalently, there exists an element x_G in R
with N_G(x_G) = 1 if and only for every elementary abelian subgroup E there
exists an element x_E in R such that N_E(x_E) = 1. When the ring R is
noncommutative, it is an open problem to find an explicit formula for x_G in
terms of the elements x_E. In this paper we present a method to solve this
problem for an arbitrary group G and an arbitrary group action on a ring.Using
this method, we obtain a complete solution of the problem for the quaternion
and the dihedral 2-groups,and for a group of order 27. We also show how to
reduce the problem to the class of (almost) extraspecial p-groups.Comment: 31 pages. In Section 1 a universal ring and the proof of the
existence of formulas for any finite group were adde

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